aa r X i v : . [ m a t h - ph ] S e p GAUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES
GANG TIAN AND GUANGBO XU
Abstract
We construct a cohomological field theory for a gauged linear sigma model space in a geometricphase, using the method of gauge theory and differential geometry. The cohomological field theoryis expected to match the Gromov–Witten theory of the classical vacuum up to a change of variable,and is expected to match various other algebraic geometric constructions.
Keywords : cohomological field theory, CohFT, gauged linear sigma model, GLSM,gauged Witten equation, vortices, moduli spaces, virtual cycle
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1. The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Comparison with algebraic geometric approaches . . . . . . . . . . . . . . . . . . . . . . . . 61.3. Virtual cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5. Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82. Moduli Spaces of Stable r -Spin Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1. Marked nodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2. Resolution data and gluing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3. r -spin curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4. Unfoldings of r -spin curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5. Universal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193. Vortex Equation and Gauged Witten Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1. The GLSM space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2. Chen–Ruan cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3. Gauged maps and vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4. Gauged Witten equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5. Energy and bounded solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6. The relation with vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294. Vortices over Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.1. Critical loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2. Isoperimetric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3. Vortex equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4. Annulus lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5. Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345. Analytical Properties of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.1. Holomorphicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Date : September 5, 2018.The second named author is partially supported by the Simons Foundation through the HomologicalMirror Symmetry Collaboration grant and AMS-Simons Travel Grant. The majority of this paper isfinished when the second named author was in Department of Mathematics, Princeton University. C bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4. Energy identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436. Moduli Space of Stable Solutions and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 446.1. Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2. Decorated dual graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.3. Stable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.4. Topology of the moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517. Topological Virtual Orbifolds and Virtual Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.1. Topological manifolds and transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2. Topological orbifolds and orbibundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.3. Virtual orbifold atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.4. Good coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.5. Shrinking good coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.6. Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.7. Branched manifolds and cobordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.8. Strongly continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 748. Virtual Cycles and Cohomological Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.1. Cohomological field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 758.2. Relation with orbifold Gromov–Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 778.3. Relation with the mirror symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799. Constructing the Virtual Cycle. I. Fredholm Theory . . . . . . . . . . . . . . . . . . . . . . . . . 799.1. Moduli spaces with prescribed asymptotic constrains . . . . . . . . . . . . . . . . . . . . 799.2. The Banach manifold and the Banach bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 809.3. Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.4. The index formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8510. Constructing the Virtual Cycle. II. Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8910.1. Stabilizers of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9010.2. Thickening data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9210.3. Nearby solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9910.4. Thickened moduli space and gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10310.5. Proof of Proposition 10.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.6. Last modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211. Constructing the Virtual Cycle. III. The Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11211.1. The inductive construction of charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11311.2. Coordinate changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11611.3. Constructing a good coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11911.4. The virtual cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12012. Properties of the Virtual Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.1. The dimension property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.2. Disconnected graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.3. Cutting edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12812.4. Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 3 Introduction
The
GAUGED LINEAR SIGMA MODEL (GLSM) is a two-dimensional supersymmetric quan-tum field theory introduced by Witten [Wit93] in 1993. It has stimulated many importantdevelopments in both the mathematical and the physical studies of string theory and mir-ror symmetry. For example, it plays a fundamental role in physicists argument of mirrorsymmetry [HV00][GS18]. Its idea is also of crucial importance in the verification of genuszero mirror symmetry for quintic threefold [Giv96][LLY97]. Since 2012 the authors have initiated a project aiming at constructing a mathemati-cally rigorous theory for GLSM, using mainly the method from symplectic and differentialgeometry. In our previous works [TX15, TX16a, TX16b], we have constructed certaincorrelation function (i.e. Gromov–Witten type invariants) under certain special condi-tions: the gauge group is U p q and the superpotential is a Lagrange multiplier. Thiscorrelation function is though rather restricted, as for example, we do not know if theysatisfy splitting axioms or not.A major difficulty in the study of the gauged Witten equation lies in the so-called BROAD case, in which the Fredholm property of the equation is problematic. This issuealso causes difficulties in various algebraic approaches. Now we have realized that if werestrict to the so-called
GEOMETRIC PHASE , then the issue about broad case disappearsin our symplectic setting. Our construction in the geometric phase has been outlined in[TX17]. It ends up at constructing a
COHOMOLOGICAL FIELD THEORY (CohFT) , namelya collection of correlation functions which satisfy the splitting axioms. The detailedconstruction in this scenario is the main objective of the current paper.A natural question is the relation between the GLSM correlation functions and theGromov–Witten invariants. They are both multilinear functions on the cohomologygroup of the classical vacuum of the GLSM space, which is a compact K¨ahler manifoldor orbifold. The relation between them is a generalization of the quantum Kirwan map ,proposed by D. Salamon, studied by Ziltener [Zil14], and proved by Woodward [Woo15]in the algebraic case. In short words, the two types of invariants are related by a “coor-dinate change,” where the coordinate change is defined by counting point-like instantons.The point-like instantons are solutions to the gauged Witten equation over the complexplane C , a generalization of affine vortices in the study of the quantum Kirwan map.These instantons appear as bubbles in the adiabatic limit process of the gauged Wittenequation. There has also been a physics explanation by Morrison–Plesser [MP95]. In theforthcoming work [TX] we will rigorously construct such coordinate change and provethe expected relation between GLSM correlation functions and the Gromov–Witten in-variants.A large proportion of this paper is devoted to the detailed construction of a virtualfundamental cycle of the compactified moduli space of solutions to the gauged Wittenequation. Guided by the original work of Jun Li and the first named author [LT98b], wefilled in many details in the previous work [TX16b] about the abstract theory. In thispaper more details are provided which suits the much more general geometric situationof the current problem.1.1. The main theorem.
The concept of cohomological field theory was introduced byManin [Man99] to axiomize Gromov–Witten invariants. Let us first recall its definition. By the name of “gauged linear sigma model,” we always mean the case that the superpotential W isnonzero. On the technical level, a nonzero W makes a significant difference from the case when W “ W “ AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 4
Definition 1.1. (Cohomological field theory) Let Λ be a field of characteristic zero thatcontains Q . A cohomological field theory on a Z -graded vector space V consists of anondegenerate bilinear form and a collection of multilinear maps (correlations) x¨y g,n : V b n b H ˚ p M g,n ; Λ q Ñ Λ , @ g, n ě , g ` n ě splitting axioms .(a) (NON-SEPARATING NODE) Suppose 2 g ` n ą
3. Let γ P H p M g,n ; Q q be the classdual to the divisor of configurations obtained by shrinking a non-separating loop,which is the image of a map ι γ : M g ´ ,n ` Ñ M g,n . Then x α b ¨ ¨ ¨ b α n ; β Y γ y g,n “ x α b ¨ ¨ ¨ b α n b ∆; ι ˚ γ β y g ´ ,n ` . Here ∆ “ ř j δ j b δ j is the “diagonal” representing the bilinear form.(b) (SEPARATING NODE) Let γ P H p M g,n ; Q q be class dual to the divisor of con-figurations obtained by shrinking a separating loop, which is the image of amap ι γ : M g ,n ` ˆ M g ,n ` Ñ M g,n , also characterized by a decomposition t , . . . , n u “ I \ I with | I | “ n , | I | “ n . Suppose 2 g i ´ ` n i ě
0. Thenfor β P H ˚ p M g,n ; Q q , if we write ι ˚ γ β “ ř l β ,l b β ,l by K¨unneth decomposition,then x α b ¨ ¨ ¨ b α n ; β Y γ y g,n “ ǫ p I , I q ÿ j,l x α I b δ j ; β ,l y g ,n ` x α I b δ j ; β ,l y g ,n ` Here ǫ p I , I q is the sign of permutations of odd-dimensional α i ’s.In practice, the coefficient field is usually the Novikov field Λ “ ! ÿ i “ a i T λ i | a i P Q , lim i Ñ8 λ i “ `8 ) in which T is a formal variable. Then for any compact symplectic manifold p X, ω q ,the Gromov–Witten invariants gives a CohFT on the cohomology H ˚ p X ; Λ q which isequipped with the Poincar´e dual pairing. There are also other instances of CohFT.(a) Given a symplectic orbifold p X, ω q , the orbifold Gromov–Witten invariants, con-structed by Chen–Ruan [CR02] and [AGV02], is a CohFT on the Chen–Ruancohomology H ˚ CR p X ; Λ q .(b) Given a nondegenerate quasihomogeneous polynomial W : C N Ñ C and an ad-missible symmetry group G of W , the Fan–Jarvis–Ruan–Witten invariants, con-structed by Fan–Jarvis–Ruan [FJR13, FJR] (alternatively by [PV16]), is a CohFTon the “state space” H W,G associated to p W, G q .Both the Gromov–Witten invariants and the FJRW invariants are defined by con-structing certain virtual fundamental cycles on the moduli spaces of solutions to certainelliptic partial differential equation (the pseudoholomorphic curve equation and the Wit-ten equation). The splitting axioms rely on relations of the virtual fundamental cyclesof moduli spaces of different genera.We would like to construct a CohFT for a GLSM space in geometric phase. A GLSMspace is a quadruple p X, G, W, µ q where X is a noncompact K¨ahler manifold, G is a Usually one imposes the S n -symmetry axiom which says the correlation functions are (su-per)symmetric with respect to permuting the marked points. We do not include this condition here,nor prove it for the GLSM correlation function. Nonetheless, this symmetry will follow from the ex-pected relation between the GLSM correlation functions and the Gromov–Witten invariants. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 5 reductive Lie group with maximal compact subgroup K , W (the superpotential ) is a G -invariant holomorphic function on X , and µ is a moment map of the K -action, whichspecifies a stability condition. We also impose other conditions (see details in Section 3),such as the existence of an R-symmetry: an R-symmetry is a C ˚ -action on X commutingwith G with respect to which W is homogeneous of degree r ě
1. The elliptic equationwe use is called the gauged Witten equation , defined over r -spin curves. An r -spin curve is a quadruple C “ p Σ C , z C , L C , ϕ C q where Σ C is an orbifold smooth or nodal Riemannsurface, z C is the set of orbifold marked points, L C is an orbifold line bundle over Σ C ,and ϕ C is an isomorphism from L b r C to the log-canonical bundle of p Σ C , z C q (see detailsin Section 2).The variables of the gauged Witten equation are called gauged maps. A gauged map from an r -spin curve C to the GLSM space p X, G, W, µ q is a triple p P, A, u q , where P is aprincipal K -bundle over Σ C , A is a connection on P , and u is a section of a fibre bundle Y Ñ Σ C associated to P and L C (with structure group K ˆ U p q and fibre X ). Such atriple p P, A, u q leads to the following quantities:(a) The covariant derivative d A u P Ω p Σ C , u ˚ T vert Y q .(b) The curvature F A P Ω p Σ C , ad P q .(c) The moment map µ p u q P Ω p Σ C , ad P _ q .(d) The gradient ∇ W p u q P Ω , p Σ C , u ˚ T vert Y q .Upon choosing a metric on Σ C and a metric on the Lie algebra k , a natural energyfunctional of p P, A, u q can be defined as E p P, A, u q “ ż Σ C ´ } d A u } ` } µ p u q} ` } F A } ` } ∇ W p u q} ¯ vol Σ C . The gauged Witten equation is roughly, though not exactly, the equation of motion, i.e.,the equation of minimizers of the above energy functional. It reads p d A u q , ` ∇ W p u q “ , ˚ F A ` µ p u q “ . Here we use the metric on Σ C and the associated Hodge star operator to identify F A witha zero-form, and use the metric on k to identify µ p u q with a section of ad P . The gaugedWitten equation is a generalization of the SYMPLECTIC VORTEX EQUATION introduced byMundet [Mun99, Mun03] and Cieliebak–Gaio–Salamon [CGS00]. It has a gauge symme-try, and is elliptic modulo gauge transformations. The ellipticity roughly implies thatthe moduli space of gauge equivalence classes of solutions should be finite dimensional,and that there should be a well-behaved Fredholm theory (obstruction theory).A crucial point in our framework is that the metrics on r -spin curves are of cylindricaltype near marked points and nodes, and the family of metrics vary smoothly over themoduli space M rg,n of stable r -spin curves. Such a choice has the following importantimplications. Any finite energy solution should converge at marked points to a point inCrit W X µ ´ p q , and the limit is well-defined as a point in the classical vacuum ¯ X W : “ Crit W {{ G “ p Crit W X µ ´ p qq{ K. When p X, G, W, µ q is in geometric phase , namely when the semistable part of the criticallocus p Crit W q ss is smooth. It implies that the gauged Witten equation has a Morse–Botttype asymptotic behavior near marked points. This fact resolves the previous difficulty of If the metric has finite area at marked points, then the limit is only in Crit W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 6 the gauged Witten equation studied in [TX15, TX16a, TX16b], namely in the broad caseone does not have Fredholm property without using a perturbation of the superpotential.The remaining construction can be carried out via the standard, if not at all sim-ple, procedure. For each given energy bound, one obtains a compactified moduli space M rg,n p X, G, W, µ q of gauge equivalence classes of solutions. The only bubbles are solu-tions over the infinite cylinder, which we call solitons . The C bound is guaranteed bycertain assumption on the geometry of X at infinity. There is a continuous evaluationmap M rg,n p X, G, W, µ q Ñ p I ¯ X W q n ˆ M g,n . Here I ¯ X W is the inertia orbifold of the classical vacuum ¯ X W (which is a compact K¨ahlermanifold or orbifold. The Fredholm property of the equation allows us to carry outthe sophisticated virtual cycle construction (which we give more detailed remarks inSubsection 1.3). The whole construction can be concluded in the following main theorem. Theorem 1.2.
For a GLSM space p X, G, W, µ q in geometric phase, there is a cohomo-logical field theory on the Chen–Ruan cohomology H ˚ CR p ¯ X W ; Λ q of the classical vacuum ¯ X W defined by virtual counting of gauge equivalence classes ofsolutions to the gauged Witten equation. Of course, the above theorem is meaningless if we cannot show that the GLSM CohFTis nontrivial. Its nontriviality can be seen via its relation with the Gromov–Witten theory,which is the topic of a forthcoming paper [TX]. Indeed, one studies the adiabatic limit ofthe gauged Witten equation by blowing up the cylindrical metrics on the surfaces. Similarto the case of [GS05], solutions to the gauged Witten equation converge to holomorphiccurves in ¯ X W , with the bubbling of point-like instanton . These instantons are solutions tothe gauged Witten equation over the complex plane. Hence we conjectured (see [TX17])that the GLSM CohFT agrees with the Gromov–Witten theory of ¯ X W up to a correction,where the correction should be defined by counting point-like instantons. Such a pictureis a generalization of the quantum Kirwan map conjectured by Salamon and studied byZiltener [Zil05, Zil14] and Woodward [Woo15], and should be a mathematical formulationof the work of Morrison–Plesser [MP95] (see Subsection 8.2 for more discussions).1.2. Comparison with algebraic geometric approaches.
Besides our symplecticapproaches, there are other approaches toward mathematical theories of the gauged linearsigma model using algebraic geometry. A general advantage of our symplectic approachis that the method can be generalized the open-string situation. An example is the workof the second-named author and Woodward [WX] on the open quantum Kirwan map inthe situation when the superpotential W is zero. When having a nonzero superpotential,one can still study the gauged Witten equation over bordered surfaces with Lagrangianboundary condition while assuming certain compatibility between W and the Lagrangian L . For example, when W has real coefficients and L is the real locus of X .1.2.1. Quasimap theory.
The first algebraic approach we would like to mention is thetheory of
QUASIMAPS developed in recent years in [CK10, CKM14, CK14, CCK15, CK17,CK16]. In this theory one considers an affine variety W acted by a reductive group G .A quasimap to the (projective) GIT quotient W {{ G is an algebraic analogue of a gaugedmap. The moduli space of stable quasimaps admits a perfect obstruction theory, and thevirtual cycle gives rise to a cohomological field theory on H ˚ p W {{ G q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 7
The quasimap theory is similar to our theory in many aspects. However, one differ-ence lies in the notion of ǫ -stability in the quasimap theory. For ǫ “ 8 the quasimaptheory coincides with the Gromov–Witten theory of W {{ G , while a finite ǫ constrains thedegrees of “rational tails” and the degrees of base points. The wall-crossing between dif-ferent ǫ -stability chambers is used to relate the quasimap invariants and Gromov–Witteninvariants of W {{ G (see [CK]). Our expectation is that rational tails in the quasimaptheory correspond to point-like instantons in our GLSM theory, and the wall-crossingcontribution in the quasimap theory should correspond to the counting of point-likeinstantons.1.2.2. Mixed-spin- p -fields. In the case of the Fermat quintic hypersurface, Chang–Li–Li–Liu [CLLL15, CLLL16] developed the framework called the mixed-spin- p -fields towardscomputing Gromov–Witten invariants by relating to FJRW invariants, via certain kind of“master space” construction. It is worth mentioning that their virtual cycle is constructedvia the so-called COSECTION LOCALIZATION invented by Kiem–Li [KL13]. More precisely,one first considers a nonproper moduli space of maps into an ambient space, which hasa perfect obstruction theory. Then the superpotential, or rather its gradient, defines acosection of the obstruction sheaf, which can localize the virtual cycle to a proper modulispace. Such an idea is similar to our virtual cycle construction. To describe the ourmoduli spaces one does not need the ambient geometry of the critical locus of W northe gauged Witten equation, however the obstruction theory depends on the ambientinformation.1.2.3. Fan–Jarvis–Ruan.
In [FJR18] Fan–Jarvis–Ruan constructed correlation functions x τ l p α k q , . . . , τ l k p α k qy in the context of gauged linear sigma mode. Here α i P H W,G are certain cohomological“states” associated to a GLSM space and l i are powers of the ψ -classes. In their approach,they have to impose the restriction that α i are of compact type , namely, all cohomologyclasses from narrow sectors and certain (but not all) cohomology classes from broad sec-tors (see [FJR18, Example 4.4.2]). Their construction extended the notion of quasimapsto Landau–Ginzburg phases and uses the cosection localization technique of Kiem–Li[KL13]. For similar reasons as in the case of Fan–Jarvis–Ruan–Witten theory, withoutincluding all broad states, their algebraic methods cannot construct a CohFT.1.2.4. Matrix factorization and categorical construction.
The B-model open string ana-logue (in genus zero) of Fan–Jarvis–Ruan–Witten theory is the theory of matrix factor-izations. We remind the reader that in A-model, the Fukaya category (viewed as certaingenus zero open string invariant of symplectic manifold) is related to the quantum co-homology (closed string counterpart) via the Hochschild homology. Quite analogously,in the Landau–Ginzburg side, Polishchuk–Vaintrob [PV16] constructed a cohomologicalfield theory on the Hochschild homology of the category of (equivariant) matrix factor-izations as an algebraic version of Fan–Jarvis–Ruan–Witten theory. This leads to theconstruction of Ciocan-Fontanine–Favero–Gu´er´e–Kim–Shoemaker [CFG ` ] of a cohomo-logical field theory associated to a GLSM space.1.3. Virtual cycle.
The construction of the correlation functions in the CohFT is basedon the theory of virtual cycles. In symplectic geometry, the virtual cycle theory arose inpeople’s efforts in defining Gromov–Witten invariants for general symplectic manifolds.Jun Li and the first named author have their approach in both the algebraic case [LT98a]and the symplectic case [LT98b]. Meanwhile there were also other people’s approach such
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 8 as the method of Kuranishi structure of Fukaya–Ono [FO99]. Until recently, there havebeen various new developments in the virtual cycle theory, such as the polyfold methodof Hofer–Wysocki–Zehnder [HWZ07, HWZ09a, HWZ09b] and the algebraic topologicalapproach of Pardon [Par16].Our approach follows the topological nature of Li–Tian’s method, which is based onthe fact that transversality can also be achieved in the topological category. The concretetreatment, though, looks similar to the Kuranishi approach. The main difference fromthe Kuranishi approach, besides that we are in the topological rather than the smoothcategory, is that we directly construct a finite good coordinate system while bypassingthe Kuranishi structure.1.4.
Outline.
This paper is divided into two parts. In Part I we focus on the setup ofthe gauged Witten equation and the property of solutions, ending at the definition andthe topology of the moduli spaces. In Part II we concentrate on constructing the virtualcycles on the moduli spaces and prove various properties of the virtual cycles.1.5.
Acknowledgement.
We thank Mauricio Romo for helpful discussions on the physicsof GLSM. We thank Alexander Kupers for kindly answering questions about topologicaltransversality.
Part I
In Part I we provide the geometric setup of the problem and prove properties ofsolutions to the gauged Witten equation and their moduli spaces. It is organized asfollows. In Section 2 we review the notion of r -spin curves and results about theirmoduli spaces. We also construct a family of cylindrical metrics in order to define thegauged Witten equation. In Section 3 we introduce the notion of gauged linear sigmamodel spaces and define the gauged Witten equation. In certain parts we compare thisequation with the symplectic vortex equation. In Section 4 we include results aboutvortices over cylinders, which play important roles in later construction such as theasymptotic behavior of solutions. In Section 5 we prove properties of solutions to thegauged Witten equation such as their asymptotic behavior and the energy inequality. InSection 6 we study the topology of the set of gauge equivalence classes of solutions andits compactification.2. Moduli Spaces of Stable r -Spin Curves In this section we recall the notions of r -spin curves and their moduli spaces. We alsoset up notions and notations which are useful for the later constructions.2.1. Marked nodal curves.
Suppose g, n ě g ` n ě
3. The moduli spaceof smooth genus g Riemann surfaces with n marked points has a well-known Deligne–Mumford compactification, denoted by M g,n . It is a compact complex orbifold, which isan effective orbifold except for p g, n q “ p , q and p , q (see [RS06] for related facts froman analytical point of view).2.1.1. Marked nodal curves.
We denote a representative of a point of M g,n by p Σ , ~ z q where Σ is a compact smooth or nodal curve of genus g , and ~ z “ p z , . . . , z n q is anordered set of marked points which are distinct and disjoint from the nodes. Given suchΣ, let V Σ be the set of irreducible components, whose elements are denoted by ν . Let E Σ be the set of its nodal points, whose elements are denoted by w , . . . , w m . Let π : ˜Σ Ñ Σbe the normalization, which is a possibly disconnected smooth Riemann surface with
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 9 components t Σ ν | ν P V Σ u . π is generically one-to-one and the preimages of the markedpoints z a are still denoted by z a . On the other hand, each node of Σ has two preimagesand we denote by ˜ w a to be a preimage of some node.2.1.2. Universal unfoldings.
The Deligne–Mumford spaces (as well as the moduli spacesof stable r -spin curves) have complete algebraic geometric description as certain Deligne–Mumford stacks. However, to fit in the differential geometric construction, we would liketo give a differential geometric construction. Here we follow the approach of Robbin–Salamon [RS06] which gives a differential geometric construction of the moduli space M g,n of stable marked Riemann surfaces.A holomorphic family of complex curves consists of open complex manifolds U , V anda proper holomorphic map π : U Ñ V with relative complex dimension one. It is calleda nodal family if for every critical point p P U of π , there exist holomorphic coordinates p w , w , . . . , w s q of U around p and holomorphic coordinates of V around π p p q such thatlocally π p w , w , . . . , w s q “ p w w , w , . . . , w s q . Given n ě
0, an n - marked nodal family consists of a nodal family π : U Ñ V togetherwith holomorphic sections Z , . . . , Z n : V Ñ U whose images are mutually disjoint andare disjoint from critical points of π . Then for each b P V , p π ´ p b q , Z p b q , . . . , Z n p b qq is amarked nodal curve.Given two n -marked nodal families p π i : U i Ñ V i , Z i, , . . . , Z i,n q , p i “ , q , we abbre-viate the data as U and U . A morphism from U to U is a commutative diagram ofholomorphic maps U ϕ / / π (cid:15) (cid:15) U π (cid:15) (cid:15) V ϕ / / Z ,i I I V Z ,i U U . Definition 2.1.
Let p Σ , ~ z q be a marked nodal curve. An unfolding of p Σ , ~ z q consists of anodal family π : U Ñ V with markings Z , . . . , Z n , a base point b P V and an isomorphism p Σ , z , . . . , z n q » p U b , Z p b q , . . . , Z n p b qq . We can define morphisms of unfoldings as morphisms of marked nodal families thatrespect the central fibre identifications. We can also define germs of unfoldings andgerms of morphisms of unfoldings. An unfolding is universal if for every other unfoldingconsisting a nodal family π : U Ñ V , markings Z , . . . , Z n , a base point b P V , andfor every isomorphism ϕ : U b » U b , there is a unique germ of morphism from p U , b q to p U , b q whose restriction to U b coincides with ϕ .By one of the main theorems of [RS06], a marked nodal curve p Σ , ~ z q has a unique germof universal unfolding if and only if it is stable. Notice that for any universal unfolding π : U Ñ V , for any automorphism γ of p Σ , ~ z q , γ induces an action on U and V thatpreserves the markings. Denote the actions by γ U : U Ñ U , γ V : V Ñ V . Hence π : U Ñ V becomes an Aut p Σ , ~ z q -equivariant family. Hence from now on, forall universal unfoldings U Ñ V of a stable marked curve p Σ , ~ z q , we always assume it isAut p Σ , ~ z q -equivariant.On the other hand, there is a natural continuous injection V { Aut p Σ , ~ z q ã Ñ M g,n AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 10 hence a local universal unfolding provides a local orbifold chart of the Deligne–Mumfordspace. It also implies the following facts. Suppose there is an isomorphism ϕ : p U b , Z p b q , . . . , Z n p b qq » p U b , Z p b q , . . . , Z n p b qq where b , b P V . Then there must be a unique automorphism γ P Aut p Σ , ~ z q such that γ V p b q “ b and ϕ “ γ U | U b .2.1.3. Dualizing sheaf.
Given a holomorphic family π : U Ñ V of complex curves, the dualizing sheaf is the line bundle ω U { V : “ K U b π ˚ K _ V . Away from critical points of π , ω U { V is canonically isomorphic to the fibrewise canonicalbundle. We consider the particular case that V is a point and U “ Σ is a nodal curve.Then if we pull back the dualizing sheaf ω Σ Ñ Σ to the normalization ˜Σ, then it becomesisomorphic to K ˜Σ b O p ˜ w q b ¨ ¨ ¨ b O p ˜ w m q where ˜ w , . . . , ˜ w m are all preimages of nodes under the normalization. This isomorphismis not canonical but up to a C ˚ -action. What is canonical is the isomorphism ω Σ | Σ ˚ » K Σ ˚ where Σ ˚ Ă Σ is the complement of nodes and K Σ ˚ is the canonical bundle. To beconsistent, from now on, for a smooth or nodal Riemann surface K Σ , let K Σ be thedualizing sheaf, and call it the canonical bundle of Σ.2.1.4. Generalized marked curves.
In the study of pseudoholomorphic curves, one usuallyneeds to add marked points to unstable components to obtain a stable domain curve. Thenewly added marked points are unordered. It is then convenient to generalize the notionof marked curves.
Definition 2.2. A generalized marked curve is a triple p Σ , ~ z , y q where p Σ , ~ z q is markedcurve and y is a set of points which does not intersect with ~ z nor nodes. Two generalizedmarked curves p Σ , ~ z , y q and p Σ , ~ z , y q are isomorphic if there is an isomorphismsbetween p Σ , ~ z q and p Σ , ~ z q that maps y bijectively onto y . A generalized markedcurve is stable if it becomes a stable marked curve after arbitrarily ordering ~ z Y y .If the genus of Σ is g , ~ z “ n and y “ l , then we call it a generalized marked curveof type p g, n, l q .Let the moduli space of generalized marked curves of type p g, n, l q be M g,n,l . It canbe viewed as the quotient M g,n,l : “ M g,n ` l { S l . More generally, we can consider the moduli M g,n,m ,...,m r : “ M g,n ` m `¨¨¨` m r { S m ˆ ¨ ¨ ¨ ˆ S m r . From universal unfoldings of stable marked curves one can easily obtain universalunfoldings of stable generalized marked curves. Indeed, let p Σ , ~ z , y q be a stable general-ized marked curve. Give an arbitrary order of y , this produces a stable marked curve p Σ , ~ z Y ~ y q , hence admits a universal unfolding U Ñ V with sections Z , . . . , Z n , Y , . . . , Y l .We also have the relations between the automorphism groupsAut p Σ , ~ z Y ~ y q Ă Aut p Σ , ~ z , y q where the former contains biholomorphic maps that preserves the ordering on y . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 11
Now for each h P S l , consider the marked curves p Σ , ~ z Y h~ y q obtained by permuting theordering on y . Then for each h we can choose a universal unfolding U h Ñ V h . Moreover,we can choose U h ’s in such a way that the following conditions are satisfied.(a) If p Σ , ~ z Y h ~ y q and p Σ , ~ z Y h ~ y q are not isomorphic as stable marked curves, thenthe underlying sets r V h s and r V h s are disjoint in M g,n ` l .(b) If γ : p Σ , ~ z Y h ~ y q » p Σ , ~ z , Y h ~ y q is an isomorphism, then there is an isomorphismof nodal families U h / / (cid:15) (cid:15) U h (cid:15) (cid:15) V h / / V h . (The existence of the above commutative diagram on the germ level is guaranteedby the universality of the unfoldings).Then the original U Ñ V can be viewed as a universal unfolding of the generalized markedcurve, which is Aut p Σ , ~ z , y q -equivariant.2.2. Resolution data and gluing parameters. (See [TF, Remark 5.2.3]) Consider a(possibly) nodal stable generalized marked p Σ , ~ z , y q of type p g, n, l q . Its normalization isanother generalized marked p ˜Σ , ~ z , w , y q of type p g, n, m, l q where m { C . Notice that the preimages of the nodes are unordered. It is not necessarilyconnected. For every vertex v P V Σ , denote the smooth generalized r -spin curve by p ˜Σ v , ~ z v , w v , y v q . It is a stable generalized r -spin curve of type p n v , m v , l v q (see Definition 2.2 and discussionafterwards). Then there exist universal unfoldings π v : U v Ñ V v where we suppressedthe data of an identification of C v with the central fibre. The product of these unfoldingsparametrizes deformations of C that do not resolve the nodes. Moreover, the automor-phism group Γ C of p Σ C , ~ z , y q acts on the normalization ğ v P V C p ˜Σ v , ~ z v , w v , y v q . Hence can properly shrink V v so that Γ C also acts on the disjoint union of the universalunfoldings ğ v P V C U v Ñ ğ v P V C V v . (2.1)For the purpose of gluing, we need to have more explicit description of how to resolvethe nodes. Lemma 2.3. [TF, Lemma 5.24]
There exist a collection of universal unfoldings π v : U v Ñ V v , a collection of open subsets N v Ă U v , which we call nodal neighborhoods ,and a collection of holomorphic functions ζ v : N v Ñ C for all v P V Σ satisfying thefollowing conditions.(a) N v is the disjoint union of open neighborhoods of points in w v , denoted by N v “ ď ˜ w P w v N v, ˜ w . (b) For any automorphism γ of C there is an isomorphism ϕ γ : U v » U γv Remember γ acts on the set of irreducible components. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 12 such that ϕ γ ˝ ϕ γ “ ϕ γγ .(c) For any automorphism γ of C and ˜ w P w v we have ϕ γ p N v, ˜ w q “ N γv,γ ˜ w . Moreover, ζ γv,γ ˜ w ˝ ϕ γ “ ζ v, ˜ w . (2.2) (d) For any node w of C with preimages ˜ w ´ , ˜ w ` belonging to components v ´ , v ` (whichcould be equal if w is non-separating), for every automorphism γ of C of order m γ ,there is an m γ -th root of unity µ γ,w such that ` ζ γv ´ ,γ ˜ w ´ ˝ ϕ γ ˘ ` ζ γv ` ,γ ˜ w ` ˝ ϕ γ ˘ “ µ γ,w ζ v ´ , ˜ w ´ ζ v ` , ˜ w ` . Proof.
Only (2.2) is not stated in [TF, Lemma 5.24], but it is a consequence of theconstruction in their proof. (cid:3)
The functions ζ v,w are viewed as fibrewise coordinates around the nodes. Choosingthe data π v : U v Ñ V v , ζ v : N v Ñ C and µ γ,w that satisfy conditions in Lemma 2.3, wecan produce a special type of universal unfolding of C . The following notations resemblethose in [FO99]. Define V def “ ź v P V Σ V v , V res “ ź w P E p Σ q C . Then the automorphism group Γ C of C acts on V def . It also acts on V res in the followingway. The coordinates of any ζ P V res are denoted by ζ w . For each γ P Γ C and ζ P V res ,define p γζ q w “ µ γ,w ζ γw . Denote variables in V def by η and variables in V res by ζ , call them deformation parameters and gluing parameters . We construct a universal unfolding of p Σ , ~ z , y q in the followingway.For each deformation parameter η , one has the corresponding fibre p Σ η , ~ z η , y η q ob-tained attaching the preimages of the nodes together. Now for each (small) gluing pa-rameter ζ , define an object p Σ η,ζ , ~ z η,ζ , y η,ζ q as follows. For each node w and correspondinggluing parameter ζ w , if ζ w ‰
0, then replace the union N w ´ Y N w ` Ă Σ η by the annulus N w,ζ : “ ! p ξ ´ , ξ ` q P N w ´ ˆ N w ` | ξ ´ ξ ` “ ζ w ) . This provides a nodal (or smooth) curveΣ η,ζ : “ ´ Σ η r ď ζ w ‰ N w ´ Y N w ` ¯ Y ď ζ w ‰ N w,ζ . The positions of ~ z η,ζ and y η,ζ are the same as ~ z η and y η .It is a well-known fact that this gives a universal unfolding of p Σ , ~ z , y q . We then definethe union of all N v, ˜ w and the neck region N w,ζ as the thin part, denoted by U thin Ă U ;the closure of the complement of U thin is called the thick part, denoted by U thick . Then U “ U thin Y U thick . Remember that γ acts on the set of preimages of nodes. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 13
From the construction we know that V def is contractible. Hence there exists a smoothtrivialization U thick » V ˆ Σ thick . Definition 2.4. (Resolution data for stable curves) Let p Σ , ~ z , y q be a stable generalizedmarked curve. A resolution data of p Σ , ~ z , y q consists of the following objects.(a) A universal unfolding ˜ U Ñ ˜ V of its normalization of the form (2.1).(b) A collection of holomorphic functions ζ v : N v Ñ C and µ γ,w as in Lemma 2.3(c) A smooth trivialization˜ U r ˜ N » ˜ V ˆ ` Σ r p ˜ U b X ˜ N q ˘ . which is holomorphic near the boundary of ˜ N .In simple words, a resolution datum is a much more enhanced object than a universalunfolding. We actually have shown the existence of resolution datum of any stablegeneralized marked curve. As we have seen, a resolution datum provides the followingobjects.(a) A universal unfolding U Ñ V of ¯ C .(b) Fibrewise holomorphic coordinates near nodes and markings that are invariantunder automorphisms.(c) Trivializations of the thick part of U .2.3. r -spin curves. Orbifold curves and orbifold bundles.
A one-dimensional complex orbifold is calledan orbifold Riemann surface or an orbifold curve. In this paper, we impose the followingconditions and conventions for orbifold curves.(a) We always assume that orbifold curves are effective orbifolds, and has finitelymany orbifold points.(b) We always assume that an orbifold curve is marked, and denoted by p Σ , ~ z q . More-over, the set of orbifold points are contained in the set of markings ~ z , however,each z a P ~ z may not be a strict orbifold point.Near each z a P ~ z there exists an orbifold chart of the form p U a , Γ a q » p D ǫ , Z r a q , r a ě . The Z r a action on D ǫ is the standard action, by viewing Z r a as a subgroup of U p q .The notion of nodal orbifold curves is more complicated. If we use the notion ofgroupoid, then a nodal orbifold curve is not a Lie groupoid, as the set of objects mayhave nodal singularities. In an equivalent description, at a node w the nodal orbifoldcurve has a chart of the form p U w , Γ w q » ` tp ξ ´ , ξ ` q P D ǫ | ξ ´ ξ ` “ ( , Z r w ˘ , r w ě , where the Z r w -action on U w is given by γ p ξ ´ , ξ ` q “ p γ ´ ξ ´ , γξ ` q . For a smooth or nodal orbifold curve p Σ , ~ z q , there is an underlying coarse curve , whichis a marked smooth or nodal Riemann surface, denoted by p ¯Σ , ~ z q . There is a holomorphicmap Σ Ñ ¯Σ. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 14
Given a smooth orbifold curve p Σ , ~ z q , an orbifold line bundle is L is a complex orbifoldwith a holomorphic map π : L Ñ Σ which, over non-orbifold points has local trivializa-tions as an ordinary holomorphic line bundle, while over an orbifold chart p U a , Γ a q thereis a chart of the form p ˜ U a , Γ a q » p U a ˆ C , Z r a q where the Z r a -action on the C -factor is given by a weight n a , i.e. γ p z, t q “ p γz, γ n a t q . We call the element e πq i P Z r , where q “ n a r a the monodromy of the line bundle at z a .When Σ is nodal, we require that an orbifold line bundle L Ñ Σ has local charts at anode w of the form p ˜ U w , Γ w q » ` tp ξ ´ , ξ ` , t q P D ǫ ˆ D ǫ ˆ C | ξ ´ ξ ` “ u , Z r w ˘ (2.3)where the C -factor still has the linear action by the local group Z r w . Then with respectto the normalization map π : ˜Σ Ñ Σ, the pull-back bundle˜ L : “ π ˚ L Ñ ˜Σis an orbifold line bundle whose monodromies at two opposite preimages of a node areopposite. Definition 2.5. (Log-canonical bundle) Let p Σ , ~ z q be a marked smooth or nodal curvewith n markings z , . . . , z n . Its log-canonical bundle is the holomorphic line bundle K Σ , log “ K Σ b â z a P Σ v O p z a q . For an orbifold curve p Σ , ~ z q , its log-canonical bundle is the pull-back K Σ , log : “ π ˚ K ¯Σ , log , where π : Σ Ñ ¯Σ is the desingularization map. Remark . When we discuss a single curve p Σ , ~ z q , over the complement of markings andnodes we can trivialize O p z a q hence K Σ , log can be regarded as the canonical bundle of thepunctured surface. However, when we discuss a family of curves, there is no canonicalfamily of trivializations of the bundles O p z a q over the punctured surface. We make suchchoices in Subsection 2.5.2.3.2. r -spin curves. Now we introduce a central concept of this paper, called r -spincurves. Definition 2.7. ( r -spin curve) Let r be a positive integer.(a) A smooth or nodal r -spin curve of type p g, n q is denoted by C “ p Σ C , ~ z C , L C , ϕ C q where p Σ C , ~ z C q is an n -marked smooth or nodal orbifold curve, L C Ñ Σ C is aholomorphic orbifold line bundle, and ϕ C is an isomorphism ϕ C : L b r C » K C , log : “ K Σ C , log . Indeed at a node K Σ , log has a local chart of the form (2.3). AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 15 (b) Let C i “ p Σ C i , ~ z C i , L C i , ϕ C i q , i “ , r -spin curves. An isomorphism from C to C consists of an isomorphism of orbifold bundles (represented by the followingcommutative diagram) L C ˜ ρ / / π (cid:15) (cid:15) L C π (cid:15) (cid:15) Σ C ρ / / Σ C such that the following induced diagram commutes L b r C p ˜ ρ q b r / / ϕ C (cid:15) (cid:15) L b r C ϕ C (cid:15) (cid:15) K C , log ρ / / K C , log . (2.4)(c) Let C “ p Σ C , ~ z C , L C , ϕ C q be an r -spin curve. For any open subset U Ă Σ C , the restriction C | U is also an r -spin curve. An open embedding from an r -spin curve C to an r -spin curve C is an isomorphism from C to the restriction of C onto anopen subset of C . Remark . (Minimality of the local group) We require that the orbifold structures arethe markings or nodes of Σ C are minimal in the following sense. Take a local chart of L C at a marking z a p D ǫ ˆ C , Z r a q where the action of Z r a on the C -factor has weight n a . Then the isomorphism ϕ C : L b r C » K C , log implies that rm a { r a is an integer, hence there exists an integer m a P t , , . . . , r ´ u such that n a r a “ m a r . We require that n a and r a are coprime. Similar requirement is imposed for nodes. Inother words, the group generated by the monodromies of L C at a marking or a node isthe same as the local group of the orbifold curve at that marking or node. In particular,when the monodromy of L C is trivial, the marking or node is not a strict orbifold point.2.3.3. Infinite cylinders.
Denote the infinite cylinder by Θ : “ R ˆ S . Let the standard cylindrical coordinates be s ` i t where we also regard t P r , π s . Hencethere is a complex coordinate z “ e s ` i t which identifies Θ with C ˚ . For any cyclic group Z k , it acts on Θ by rotating the t -coordinate. The orbifold Θ { Z k is still isomorphic to Θ , hence we may either regard the infinite cylinder as a smooth object, or regard it asan orbifold with local group Z k at ˘8 for certain k ě
1. When we are in the latterperspective, we call it an orbifold cylinder.The isomorphism classes of r -spin structures on an infinite are classified by their mon-odromies. Indeed, for any m P t , , . . . , r ´ u , there is an orbifold line bundle L m Ñ Θ whose monodromy at ´8 is e πq i where q “ mr , together with a holomorphic isomor-phism ϕ : L b rm » K Θ . If we regard z as the local coordinate near ´8 , then over eachcontractible open subset of Θ , there is a holomorphic section e of L m such that ϕ p e b r q “ z m dzz . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 16
The monodromy at `8 of L m is e ´ πq i .2.3.4. Automorphisms.
Given an r -spin curve C , its automorphisms form a group, de-noted by Aut C . C is called stable if Aut C is finite. Every automorphism of C inducesan automorphism of the underlying orbifold nodal curve Σ C . Let Aut p L C q Ă Aut C bethe subgroup of automorphisms which induce the identity of p Σ C , ~ z q , namely the set ofbundle automorphisms. Then there is a tautological exact sequence of groups1 / / Aut L C / / Aut C / / Aut p Σ C , ~ z q . It is easy to see that when C is smooth, Aut L C » Z r (see Definition 2.7 and (2.4)). Moregenerally, suppose C is not smooth. Then for every irreducible component v P V C andevery preimage ˜ w P ˜Σ v of a node, there is a restriction map r ˜ w : Aut L C v Ñ Z r . as the automorphism of the fibre of L C v at ˜ w . For each node w P E p Σ C q , choose an orderof its preimages as p ˜ w ´ , ˜ w ` q , belonging to components v ´ and v ` respectively. Thendefine r : ź v P V C Aut L ˜ C v Ñ ź w P E p Σ C q Z r { Z r w , ` γ v ˘ v ÞÑ ` r ˜ w ` p γ v ´ q r ˜ w ´ p γ v ´ q ´ ˘ w where Z r w is the local group at the node w . Then there is an exact sequence1 / / Aut L C / / ź v P V C Aut L C v / / ź w P E C Z r { Z r w . (2.5) Example . Consider a nodal r -spin curve p Σ C , L C , ϕ C q which has no marking but onlyone node which separates the nodal curve into two irreducible components. Suppose alocal chart of the L C at the node w is of the form p ˜ U w , Γ w q » ` p ξ ´ , ξ ` , t q P D ǫ ˆ D ǫ ˆ C | ξ ´ ξ ` “ ( , Z r ˘ where the action on ξ ´ , ξ ` , t has weights ´ , ,
1. Then consider the Z r -equivariant mapfrom ˜ U w to itself defined by p ξ ´ , ξ ` , t q ÞÑ p ξ ´ , e ´ π i r ξ ` , t q . This defines an automorphism of L C locally near the node. Let the irreducible componentcorresponding to the branch ξ ` “ ξ ´ “
0) be C ´ (resp. C ` ). Then this localautomorphism extends to the identity of L C | C ´ and extends to the automorphism on L C | C ` by fibrewise multiplication of e π i r . This is an example of automorphisms of L C that has different scaler factors on different irreducible components.2.3.5. Generalized stable r -spin curves. As in Definition 2.2, a generalized r -spin curve is an object C “ p Σ C , L C , ϕ C , ~ z , y q where the object without y is an r -spin curve and y is an unordered set of points on Σ C that are disjoint from special points. If the genus of Σ C is g , ~ z “ n and y “ l , thenwe call it a generalized r -spin curve of type p g, n, l q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 17
Unfoldings of r -spin curves. We do not define the notion of unfoldings of (gen-eralized) r -spin curves in the algebraic-geometric (stacky) fashion. Instead, we constructuniversal unfoldings of stable r -spin curves from universal unfoldings of their underlyingcoarse curves.Let C “ p Σ C , L C , ϕ C , ~ z , y q be a stable generalized r -spin curve. Let the underlyingcoarse curve be p ¯Σ C , ~ z , y q which is a stable generalized marked curve. Given a resolutiondatum r . Then we obtained an unfolding ¯ U Ñ V » V def ˆ V res . We define an orbifold U out of ¯ U as follows. Indeed, we can decompose the thin part ¯ U thin as¯ U thin “ n ğ a “ ¯ N z a \ m ğ b “ ¯ N w b . The resolution data provides holomorphic identifications¯ N z a » D r ˆ V , ¯ N w b » p ξ ´ , ξ ` q P D r | | ξ ´ || ξ ` | ď ǫ ( ˆ V def . We glue in orbifold charts N z a and N w b as follows. Suppose the local group of z a is Z r a .Then N z a is just the r a -fold cover of D r multiplying V . Suppose the local group of w b is Z r b . Then define N w b “ p ˜ ξ ´ , ˜ ξ ` q P D r | | ˜ ξ ´ || ˜ ξ ` | ď ǫ ( . Define the Z r b -action on N w b by γ p ˜ ξ ´ , ˜ ξ ` q “ p γ ´ ˜ ξ ´ , γ ˜ ξ ` q . Hence we obtained a complex orbifold U together with a holomorphic map U Ñ V . Thethick-thin decomposition on ¯ U induce a thick-thin decomposition U “ U thick Y U thin . (2.6)Moreover, it is easy to define the dualizing sheaf ω U { V as a holomorphic line bundle, andthe relative log-canonical bundle, whose restriction to each fibre is isomorphic to thelog-canonical bundle of that fibre.Now we construct a holomorphic orbifold line bundle L U Ñ U and an isomorphism ϕ C : L b r U Ñ K U { V , log . Indeed, the central fibre is already equipped with the line bundle L C which is an r -throot of the log-canonical bundle of the central fibre. Choose a collection of coordinatecharts tp W β , w β qu on Σ C which cover the thick part Σ thick C . Moreover, assume that foreach two charts in this collection, the overlap is contractible. The transition function of ω C are holomorphic functions g ββ : W β X W β Ñ C ˚ while the transition functions of L C are holomorphic functions h ββ : W β X W β Ñ C ˚ such that g ββ “ h rββ . The smooth trivialization (2.6) makes w β smooth coordinates oneach fibre, but not necessarily holomorphic. But still there are unique sections of thecanonical bundle of each fibre that over each W β of the form dw β ` ǫd ¯ w β where ǫ is a smooth function defined on W β ˆ V . Hence g ββ extends to a smooth functionon p W β X W β q ˆ V that is fibrewise holomorphic.The bundle L C also extends to a smooth complex line bundle over U thick via (2.6).Over W β X W β , we can take r -th root of the transition function g ββ which is fibrewise AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 18 holomorphic. Since the overlap is contractible, there is a unique r -th root that continu-ously extends the value on the central fibre. Therefore we obtain a fibrewise holomorphicline bundle L over the thick part. It is also naturally a holomorphic bundle since thecanonical bundle is holomorphic over the family.On the other hand, one can define the bundle over the thin part. Indeed, take a localchart of L C near a node w by p ξ ´ , ξ ` , t q P D r ˆ D r ˆ C | ξ ´ ξ ` “ ( where the local group is Z r w . Then gluing a bundle chart over U by p ξ ´ , ξ ` , t q P D r ˆ D r ˆ C | | ξ ´ || ξ ` | ă ǫ ( where the action of Z r w on the three coordinates has the same weights as before. Thisdefines an orbifold line bundle L U Ñ U . The isomorphism ϕ U : L b r U Ñ K U { V , log isimmediate. Remark . As in the case of universal unfoldings of stable marked curves, after ap-propriate shrinkings of the family U Ñ V the automorphism group Aut C also acts onthis family. More precisely, there is a Γ C -action on the line bundle L U that descendsto actions on U and V . We only remark on the somewhat unusual way of the action ofAut L C Ă Aut C on this family. Given an element γ P Aut L C which over each irreduciblecomponent v P V C is an element γ v P Z r . For each node w P E C where a local chart of L U has coordinates p ξ ´ , ξ ` , t q and the Z r w -action has weights ´ , , n w . Then we define γ p ξ ´ , ξ ` , t q “ p ξ ´ , γ v ` γ ´ v ´ ξ ` , γ v ´ t q “ γ v ` γ ´ v ´ p γ v ` γ ´ v ´ ξ ´ , ξ ` , γ v ` t q . Notice that by the exact sequence (2.5), γ w : “ γ v ` γ ´ v ´ P Z r w . This is a well-defined mapon the total space of L U . Notice that this action will move the fibre over the gluingparameter ζ w “ ξ ´ ξ ` to the fibre over another gluing parameter γ v ` γ ´ v ´ ζ w .Because of the above construction, a resolution datum of a stable generalized r -spincurve is identified with a resolution datum of its underlying coarse curve.The above construction also provides a way to define the topology on the moduli spaceof stable r -spin curves.Now given a stable generalized r -spin curve C and a resolution data r C which containsthe fibration U Ñ V , there is a smooth part U ‚ Ă U which is the complement of the orbifold markings and the nodes.2.4.1. Bundles over r -spin curves. Let K be a connected compact Lie group. In thisdiscussion we also need to include a smooth K -bundle over C . Since our surfaces havenodes and orbifold points, we need to clarify the meaning of K -bundles over r -spincurves. In this paper, a principal K -bundle over an r -spin curve C always means asmooth principal K -bundle over the punctured smooth Riemann surface Σ ˚ C . Howeverwe still denote it as P Ñ C . Definition 2.11. ( Resolution data ) Let C be a generalized r -spin curve and P Ñ C be a smooth K -bundle. A resolution data of p C , P q , denoted by r “ p r C , r P q , consistsof a resolution data r C of C and a resolution data r P , where the latter means smooth AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 19 K -bundle P Ñ U ‚ and an isomorphism of K -bundles P / / (cid:15) (cid:15) P b (cid:15) (cid:15) Σ ‚ C / / U ‚ b , a smooth trivialization over the thin part, t thin P : P thin “ P | U thin » U thin ˆ K, and a trivialization over the thick part t thick P : P thick “ P | U thick » V ˆ P | Σ thick C . Moreover, they satisfy the following conditions. For any automorphism p γ, γ P q P Γ “ Aut p C , P q where γ : C Ñ C is an automorphism of the generalized r -spin curve C inducingan action on the family U , and γ P : P Ñ P is a smooth K -bundle isomorphism covering γ : U Ñ U . Then we require that P thin γ P / / t thin P (cid:15) (cid:15) P thin t thin P (cid:15) (cid:15) U thin ˆ K γ / / U thin ˆ K , P thick γ P / / t thick P (cid:15) (cid:15) P thick t thick P (cid:15) (cid:15) V ˆ P | Σ thick C γ / / V ˆ P | Σ thick C . (2.7)2.5. Universal structures.
In this subsection we construct several structures for all r -spin curves that vary at least smoothly over the moduli space M rg,n . First we construct afamily of conformal Riemannian metrics that are of cylindrical type over all r -spin curves.Let g ě n ě g ` n ě
3. For any smooth genus g curve with n marked points,denote the punctured Riemann surface by Σ ˚ . Then for any local holomorphic coordinate z centered at a marked point z a , the coordinate s ` i t “ ´ log z is called a cylindricalcoordinate on Σ ˚ near the puncture z a . The punctured neighborhood identified with acylinder r T, `8q ˆ S is called a cylindrical end .We would like to specify certain type of Riemannian metrics on punctured Riemannsurfaces that are of cylindrical type. We always take metrics whose conformal classbelongs to the conformal class defined by the complex structure, hence a conformalRiemannian metric is equivalent to its area two-form. Definition 2.12.
An conformal Riemannian metric on Σ ˚ is called a cylindrical metric of perimeter 2 πλ if near each puncture z a , there exists a cylindrical coordinate s ` i t such that the area two-form is ν “ σ p s, t q dsdt where σ : r S, `8q ˆ S Ñ R ` satisfyingsup r S, `8qˆ S ” e s ˇˇ ∇ l p σ ´ λ q ˇˇı ď C l , @ l ě . (2.8)It is easy to verify that the above exponential decay condition is independent of thechoice of the local holomorphic coordinate z “ e s ` i t . Definition 2.13.
Let g, n ě g ` n ě λ ą M g,n is a collection of conformal Riemannian metricson all Σ ˚ where Σ ˚ is the complement of special points of a stable n -marked genus g Riemann surface. These metrics satisfy the following conditions.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 20 (a) The restriction of g Σ to each irreducible component is a cylindrical metric ofperimeter 2 πλ .(b) If ρ : p Σ , ~ z q Ñ p Σ , ~ z q is an isomorphism, then it induces a diffeomorphism ρ : Σ ˚ Ñ Σ ˚ . We require that ρ ˚ g Σ “ g Σ . In particular, g Σ is invariant withrespect to automorphisms.(c) For any nodal family U Ñ V of stable curves with genus g and n markings, thefibrewise metric defines a smooth metric on the vertical tangent bundle away fromthe special points.Our purpose is to construct families of cylindrical metrics over M g,n for all p g, n q .Similar construction is sketched in [Ven15]. It is also essentially used by Fan–Jarvis–Ruan [FJR] in the construction of FJRW invariants. It is possible that more delicateconstruction can produce metrics that are compatible with the operations that relatemoduli spaces for different p g, n q , but we do not require such conditions.For each p P M g,n , choose a representative p Σ p , ~ z p q and a resolution data r p whichcontains a local universal unfolding U p Ñ V p . The resolution data also specifies a thinpart U thin p and a thick part U thick p , with a smooth trivialization U thick p » Σ thick p ˆ V p . Then one can equip an area form on U thick p by pulling back one on Σ thick p . On the otherhand, for each component of the thin part, if it is corresponds to a marking, then one canuse the local fibrewise coordinate w to define the volume form i dww d ¯ w ¯ w . If the componentof the thin part corresponds to a node, then for each gluing parameter ζ , the thin part ofthe fibre over ζ is identified with a long cylinder w ´ w ` “ ζ . Then the long cylinder can beequipped with the volume form i dw ` w ` d ¯ w ` ¯ w ` “ i dw ´ w ´ d ¯ w ´ ¯ w ´ . This way we construct fibrewisearea forms over the thin part. One can use a smooth cut-off function to interpolate thearea forms on the thick part and thin part so that one has a fibrewise area form over U ˚ p .Finally, one can use this action of Γ p over U p to symmetrize so we obtain a Γ p -invariantfibrewise area form that is of cylindrical type.Now for each p , the resolution data r p covers an open subset W p Ă M g,n . Since M g,n is compact, one can find a finite cover W p , . . . , W p s of M g,n . Choose a smooth partitionof unity (in the orbifold sense) subordinate to this open cover. Now we can define g Σ for r Σ s “ p P M g,n , define ν Σ as follows. For each p i with p P W p i , there exists an embeddingΣ ã Ñ U p i , by identification with a fibre. Different embeddings differ by an action by anautomorphism in Γ p i . Then we can pull back two-forms ν Σ ,p i P Ω p Σ ˚ q . It does not depends on the embeddings we choose. Then define ν Σ “ s ÿ i “ ρ i p p q ν Σ ,p i . So ν Σ is invariant under the automorphism group of Σ. Moreover, the exponential de-cay condition (2.8) is conserved under convex combination. Hence ν Σ is indeed a validcylindrical metric. The check of Item (c) of Definition 2.13 is left to the reader.From now on, for each pair g, n with 2 g ` n ě
3, we fix the choice of such collection ofcylindrical metrics (of perimeter 1) on stable genus g , n -marked Riemann surfaces. Whenwe have a stable r -spin curve, we equip the underlying punctured Riemann surface thecylindrical metric we choose. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 21
The family of cylindrical metric can be viewed as a family of Hermitian metrics onthe canonical bundle K ¯ C of ¯ C restricted to the complements of the markings and nodes.However, in the concept of r -spin structures we used the notion of log-canonical bundles.Over each stable r -spin curve C , one has an isomorphism K C , log | Σ ˚ C » K Σ ˚ C (2.9)because the divisor O p z q b ¨ ¨ ¨ b O p z n q is trivial away from the markings. However thereis no canonical isomorphism.Consider the universal curve U g,n Ñ M g,n . The markings give sections S , . . . , S n : M g,n Ñ U g,n whose images are divisors D , . . . , D n Ă U g,n » M g,n ` . Then one canchoose a holomorphic section of the line bundle r D `¨ ¨ ¨` D n s that vanishes exactly alongthese divisors. Then restrict to each fibre, this section provides a family of trivializationsof O p z q away from the markings which are invariant under automorphisms and vary ina holomorphic way over M g,n . From now on we choose such a section, upon which theisomorphism (2.9) is regarded as canonical.We claim that the cylindrical metrics and the canonical isomorphisms (2.9) induce oneach stable r -spin curve C “ p ¯ C , L C , ϕ C q a Hermitian metric on L C away from puncturesand nodes. Indeed, given a local holomorphic coordinate z on Σ C and a local there existsa local holomorphic section e of L C such that ϕ C p e b r q “ dz. Then define } e } “ p} dz }q r . Different choices of e differ by an element of Z r hence this is awell-defined metric on L C . Further, near each marking or node at which the monodromyof L C is e πm i r , for any holomorphic coordinate w , there is a unique (up to Z r ) section e of L C such that ϕ C p e b r q “ dww w m . The metric near this marking is } e } “ | w | m ´ p} dw }q r “ | w | m ˆ } dww } ˙ r . (2.10)Namely, } e } behaves like | w | m near the marking.Let P C Ñ Σ C be the unit circle bundle of L C , which is a principal U p q -bundle overthe punctured surface. The Hermitian metric on L C induces the Chern connection A C P A p P C q . (2.10) implies that at a puncture where m ‰
0, the connection is singular, and,under the trivialization induced by the unitary frame e {} e } , the connection is A C “ d ` i mr dt ` α “ d ` i qdt ` α where α decays exponentially in the cylindrical coordinates.Late we will use the following result, whose proof is left to the reader. Lemma 2.14.
For fixed g, n , given a family of cylindrical metrics over M g,n in the senseof Definition 2.13, there exists C ą such that, for all stable genus g , n -marked r -spincurves C , we have } F A C } L p Σ ˚ C q ď C where the norm is taken with respect to the cylindrical metric on Σ ˚ C . Moreover, we canchoose the family of metrics for all g, n such that for the same C the above inequality istrue for all g, n . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 22
We also need to specify the universal structures over infinite cylinders. As discussedbefore, the r -spin structures over an infinite cylinder Θ is classified by the monodromiesof a line bundle L m Ñ Θ at ´8 . Let the cylindrical coordinate be s ` i t . Then thereexists a global holomorphic sections of L m such that ϕ m p e b r q “ z m dzz . Denote q “ mr . Equip with Θ the standard cylindrical metric of perimeter 2 π . Thendefine the Hermitian metric on L m by } e } ” | z | q . Then ǫ m : “ | z | ´ q e is a global smoothsection of unit length. Trivialize L m by ǫ m , we see that the Chern connection reads A C “ d ` i qdt. Vortex Equation and Gauged Witten Equation
In this section we set up the gauged Witten equation as the classical equation of motionfor gauged linear sigma model. This equation is essentially induced from the GLSMLagrangian which is due to Witten [Wit93]. The current setting has appeared in [TX17],while a slightly different version is used in the series of papers [TX15, TX16a, TX16b] ofthe authors.3.1.
The GLSM space.
Let K be a compact Lie group, G be its complexification, withLie algebras k and g . We choose an Ad-invariant metric on k , so that k is identified withits dual space.We do not require that K is connected, as we can view FJRW theory as a special caseof our theory where K is a finite group.Consider a noncompact K¨ahler manifold p X, ω X , J X q with the following additionalstructures.(a) A holomorphic C ˚ -action, which is usually referred to as the R-symmetry .(b) A holomorphic G -action which restricts to a Hamiltonian K -action, with a mo-ment map µ : X Ñ k ˚ » k .(c) A holomorphic function W : X Ñ C .We require that these structures are compatible in the following sense. Hypothesis . (a) The R-symmetry commutes with the G -action.(b) W is homogeneous of degree r , i.e., there is a positive integer r such that W p ξx q “ ξ r W p x q , @ ξ P C ˚ , x P X. (c) W is invariant with respect to the G -action.(d) 0 P k is a regular value of µ . Definition 3.2.
We call the quadruple p X, G, W, µ q a GLSM space . It is said to be inthe geometric phase if µ ´ p q intersects only with the smooth part of Crit W and theintersection is transverse. Here Crit W is an analytic subvariety of X ; its smooth part iswhere it is smooth and dW is normally hyperbolic.We also make the following technical assumption. Hypothesis . Let X W be the union of all irreducible components of Crit W that havenonempty intersections with µ ´ p q . There is a homomorphism ι W : C ˚ Ñ G such that ξ ¨ x “ ι W p ξ q ¨ x, @ x P X W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 23
All above hypothesis allow us to consider the symplectic reductions¯ X : “ µ ´ p q{ K, ¯ X W : “ r X W X µ ´ p qs{ K. (3.1)Here ¯ X is a compact K¨ahler orbifold and ¯ X W Ă ¯ X is a closed suborbifold.The following shows an important property of W . Lemma 3.4.
Any critical value of W must be zero.Proof. By the homogeneity of W with respect to the R-symmetry, at a critical point x of W , we have 0 “ X η W p x q “ rηW p x q . where X η is the infinitesimal C ˚ -action for η P Lie C ˚ . Hence W p x q “ (cid:3) We also have the following assumption on the geometry at infinity.
Hypothesis . There exists ξ W in the center Z p k q Ă k , and a continuous function τ ÞÑ c W p τ q (for τ P Z p k q ) satisfying the following condition. If we define F W : “ µ ¨ ξ W , then F W is proper and x P X W , ξ P T x X F W p x q ě c W p τ q ùñ x ∇ ξ ∇F W p x q , ξ y ` x ∇ Jξ ∇F W p x q , J ξ y ě , x ∇F W p x q , J Y µ p x q´ τ p x qy ě . (3.2)3.1.1. Example: hypersurfaces in weighted projective spaces.
Consider a quasihomoge-neous polynomial Q : C N Ñ C . This means that there are integers r , . . . , r N , r suchthat for all ξ P C ˚ and p x , . . . , x N q P C N , Q p ξ r x , . . . , ξ r N x N q “ ξ r Q p x , . . . , x N q . It is called nondegenerate if 0 P C N is the only critical point of Q . It induces a C ˚ -action(the R-symmetry) on C N by ξ ¨ p x , . . . , x N q “ p ξ r x , . . . , ξ r N x N q . (3.3)The corresponding weighted projective space is P N ´ p r , . . . , r N q “ p C N r t uq{ C ˚ . The nondegenerate assumption implies that the vanishing of Q defines a smooth hyper-surface (in the orbifold sense) ¯ X Q Ă P N ´ p r , . . . , r N q .To realize the target ¯ X Q from a GLSM space, following Witten [Wit93], consider X “ C N ` with coordinates p p, x , . . . , x N q . The R-symmetry extends to C N ` byacting trivially on p . Introduce W : C N ` Ñ C which is defined as W p p, x , . . . , x N q “ pQ p x , . . . , x N q . Let G be C ˚ and act on X by g ¨ p p, x , . . . , x N q “ p g ´ r p, g r x , . . . , g r N x N q . It is Hamiltonian and a moment map is µ p p, x , . . . , x N q “ ´ i ” r | x | ` ¨ ¨ ¨ ` r N | x N | ´ r | p | ´ τ ı (3.4)where τ is a constant, playing the role as a parameter of this theory.It is easy to check that when τ ‰
0, the quadruple p X, G, W, µ q satisfies Hypothesis3.1, and hence a GLSM space. Moreover, when τ ą
0, we are in the geometric phases.Indeed, the critical locus of W decomposes asCrit W “ ! p , x , . . . , x N q | Q p x , . . . , x N q “ ) Y ! p p, , . . . , q | p P C ) . (3.5) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 24
We see that the first component above contains the smooth locus, and when τ ą µ ´ p q is disjoint from the second component above. Hence X W is the affine hypersurface t u ˆ Q ´ p q Ă t u ˆ C N . Moreover, one sees that¯ X “ P N ´ p r , . . . , r N q , ¯ X W “ ¯ X Q We remark that when τ ă
0, we are in the Landau–Ginzburg phase. Witten’s argumentfor Landau–Ginzburg/Calabi–Yau correspondence is based on this observation.3.1.2.
Example: the PAX model.
In [Wit93] there were also GLSM spaces which give riseto complete intersections in toric varieties. They are all abelian GLSMs, namely, G is anabelian group. Here we mention an example with nonabelian G , called the PAX model ,constructed by physicists (see [JKL `
12, JKL ` φ “ p φ , . . . , φ N q be the coordinate of C N and let A “ A p φ q be an n ˆ n complexmatrix whose coefficients are (quasi)homogeneous polynomials of φ of degree r ą R P C n p n ´ k q , P P C p n ´ k q n as an n ˆ p n ´ k q matrixand an p n ´ k q ˆ n matrix respectively.Let our GLSM space be X “ C p n ´ k q n ˆ C N ˆ C n p n ´ k q with coordinates p P, φ, R q .Let the R-symmetry act on X by acting on the variable φ . Let the gauge group be K “ U p qˆ U p n ´ k q , where the U p q factor acting on p P, φ q by e i θ ¨p P, φ q “ p e ´ i rθ P, e i θ φ q and the U p n ´ k q factor acting on p P, R q by U ¨ p P, R q “ p
U P, RU ´ q . The superpotentialis defined as W p P, φ, R q “ tr “ P A p φ q R ‰ . This is why it is called the PAX model where we replaced the letter X by R to distinguishfrom the GLSM space X . The moment map for the K -action is µ p P, φ, R q “ « ř Ni “ | φ i | ´ tr p P : P q ´ τ P P : ´ R : R ´ τ Id p n ´ k qˆp n ´ k q . ff An interesting feature is that there are three different phases depending on values of τ and τ , and all of them are geometric phases and there is no Landau–Ginzburg phase.In the case n “ k “ N “ r “
1, one phase will give rise to a Calabi–Yauthreefold Y Ă P which is a resolution of the rank two locus of a generic section ofHom P p O ‘ P , O P p q ‘ q .3.2. Chen–Ruan cohomology.
In our situation, the theory in the classical level is thecohomology of the orbifold ¯ X W . This is a special type of orbifolds, namely a quotient¯ M “ M { K of a compact smooth manifold by a compact Lie group where the action hasonly finite stabilizers. We review the notion of Chen–Ruan cohomology for an orbifoldof this type. The original introduction of this cohomology theory is [CR04], motivatedfrom physicists’ consideration of string theory in orbifolds [DHVW85, DHVW86]. Herewe refrain from using the language of groupoids.Let M be a compact smooth manifold acted by a compact Lie group K . Suppose theaction has only finite stabilizers. Then the quotient¯ M “ M { K is naturally a smooth orbifold. We know that for any subgroup H Ă K , the fixed pointset M H Ă M is a smooth submanifold of M , and gM H “ M gHg ´ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 25
Let r K s be the set of conjugacy classes of elements of K . We say that a point ¯ x P M isfixed by a conjugacy class r g s P r K s if there is a lift x P M whose stabilizer contains r g s .We denote r g s ¨ ¯ x “ ¯ x .The inertia orbifold of ¯ M is the disjoint union I ¯ M “ ğ r g sPr K s ¯ M r g s : “ ğ r g sPr K s ! ¯ x P ¯ M | r g s ¨ ¯ x “ ¯ x ) . For the identity element 1 P K and the conjugacy class r s , the component ¯ M r s , whichis identified with ¯ M itself, is called the untwisted sector . For other nontrivial conjugacyclasses r g s , the components ¯ M r g s are called twisted sectors . Sometimes we also regard¯ M r s as a twisted sector.The Chen–Ruan cohomology of ¯ M is roughly H ˚ CR p ¯ M ; Q q : “ à r g sPr K s H ˚ p ¯ M r g s ; Q q . An important feature is the so-called degree shifting . We need to assume that ¯ M is almostcomplex . Indeed for every conjugacy class p g q , there is a rational number ι r g s P Q and we define H d CR p ¯ M ; Q q “ à r g sPr K s H d ´ ι r g s p ¯ M r g s ; Q q . This degree shifting number ι r g s carries information of the normal direction of ¯ M p g q . Thisnumber is defined as follows. One can choose a point x P M which is generic enough sothat the fixed point set is the finite subgroup x g y Ă K . Then there is a local uniformizerat ¯ x P ¯ X which is a triple p V, x g y , ϕ q where V is a smooth manifold acted by x g y and ϕ : V {x g y Ñ ¯ M is a homeomorphismonto an open neighborhood of ¯ x . Then the fixed point set of g is a smooth submanifold V g Ă V . The action makes the normal bundle of V g a complex representation of theabelian group x g y . Hence we can decompose the normal bundle into the direct sum ofone-dimensional representations, so that g acts asdiag p e π i m { m g , . . . , e π i m s { m g q , ď m , . . . , m s ă m g . Then define ι p g q “ s ÿ l “ m l m g P Q . Indeed ι r g s is independent of the choice of local uniformizers. An important property isthat ι r g s ` ι r g s ´ “ codim C ` ¯ M r g s , ¯ M ˘ : “ dim C ¯ M ´ dim C ¯ M r g s . (3.6)3.3. Gauged maps and vortices.
We review the basic notions about gauged mapsfrom surfaces to a manifold with group action. Let K be a compact Lie group. Let Σbe a surface and P Ñ K be a smooth principal K -bundle. Let X be a smooth manifoldacted by K . The infinitesimal K -action is denoted by X η P Γ p T X q , @ η P k . Here the convention is that η ÞÑ X η is an anti-homomorphism of Lie derivatives. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 26
Let Y “ P ˆ K X be the associated fibre bundle over Σ, where the projection map isdenoted by π Y : Y Ñ Σ. Then there is a vertical tangent bundle T vert Y Ñ Y. Any connection A P A p P q induces a splitting of0 / / T vert Y / / T Y / / π ˚ Y T Σ / / T Y Ñ T vert Y. A gauged map from Σ to X is a triple v “ p P, A, u q where P Ñ Σ is a principal K -bundle, A P A p P q is a connection (or called a gauge field ) and u P S p Y q is a section of Y (or called a matter field ). When P is fixed in the context, we also call the pair v “ p A, u q a gauged map from P to X . If u is (weakly) differentiable, one has the ordinary derivative du P Γ p Σ , Λ b u ˚ T Y q . Using the projection
T Y Ñ T vert Y induced by A , one has the covariant derivative d A u P Γ p Σ , Λ b u ˚ T vert Y q “ Γ p Σ , Hom p T Σ , u ˚ T ver Y qq . If X has a K -invariant almost complex structure J X , then it induces a complex struc-ture on T vert Y . Meanwhile if Σ has a complex structure so it becomes a Riemann surface,then one can take the p , q -part of the covariant derivative, which is denoted by B A u P Ω , p u ˚ T vert Y q » Γ p Σ , Hom , p T Σ , u ˚ T vert Y qq . Associated to the adjoint representation there is the adjoint bundlead P “ P ˆ ad k . The space connections A p P q is an affine space modelled on the linear space Ω p Σ , ad P q .The curvature of A is a 2-form F A P Ω p Σ , ad P q . A smooth gauge transformation of P is a smooth map g : P Ñ K satisfying g p ph q “ h ´ g p p q h, @ h P K. Here ph denotes the canonical right K -action on the principal bundle P . Gauge transfor-mations naturally form an infinite-dimensional Lie group G p P q , using the multiplicationof G and they can be viewed as automorphisms of P (i.e., smooth fibre bundle auto-morphisms that respect the right K -action). It acts on the spaces A p P q and S p Y q byreparametrization. Hence we regard gauge transformations as right actions by G p P q ,denoted by p A, u q ¨ g “ p g ˚ A, g ˚ u q . The covariant derivatives and curvatures transform naturally with respect to gauge trans-formations. Namely, for any A P A p P q , u P S p Y q and g P G p P q , d g ˚ A g ˚ u “ g ´ d A u, F g ˚ A “ Ad ´ g F A . We only consider the case that the K -action on X is Hamiltonian. Namely, X is asymplectic manifold with a symplectic form ω X , the K -action preserves ω X and there isa moment map µ : X Ñ k ˚ which means ω p X η , ¨q “ d x µ, η y , @ η P k as 1-forms on X , and we require µ is K -equivariant, which means µ p gx q “ Ad ˚ g µ p x q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 27
The co-adjoint action on k ˚ induces a vector bundle ad P ˚ which is naturally the dual ofad P . Then µ induces an element µ Y P C Σ p Y, ad P ˚ q where the (non-standard) notation means smooth maps between fibre bundles over Σ.Then if u P S p Y q is a section, the composition µ p u q P Γ p Σ , ad P ˚ q is naturally defined.Now we can review the basic set up of the symplectic vortex equation. This is an equa-tion introduced first by Mundet [Mun99, Mun03] and Cieliebak–Gaio–Salamon [CGS00].Assume that the K -action is Hamiltonian with moment map µ : X Ñ k ˚ and there isa K -invariant almost complex structure J X . Choose an adjoint-invariant metric on k sothat we can identify k » k ˚ . Choose an area form on Σ ˚ so we can identify a zero-formwith a two-form on Σ by the Hodge star operator ˚ . A vortex is a triple p P, A, u q where P Ñ Σ is a principal K -bundle, p A, u q is a gauged map satisfying the following equation B A u “ , ˚ F A ` µ p u q “ . Here the second equation is viewed as an equality in the space Γ p Σ , ad P q . This equa-tion is gauge-invariant in the following sense. Given a gauge map p A, u q and a gaugetransformation g P G p P q , one has B g ˚ A g ˚ u “ g ´ pB A u q , ˚ F g ˚ A ` µ p g ˚ u q “ Ad ´ g p˚ F A ` µ p u qq . There is also a notion of energy, called the
Yang–Mills–Higgs functional for gaugedmaps. It is defined as E p A, u q “ ” } d A u } L ` } F A } L ` } µ p u q} L ı . Here the energy density is computed using the metric on X induced from the symplecticform and the almost complex structure, and the metric on Σ induced from the complexstructure and the area form.Indeed the vortex equation can be viewed as the equation of motion with respect to theabove energy functional. To see we need to have a short review of equivariant topology.The equivariant (co)homology of X is defined to be the usual (co)homology of the Borelconstruction X K “ EK ˆ K X, where EK Ñ BK is the universal K -bundle over the classifying space BK . Then forany gauged map p P, A, u q over Σ, P induces a homotopy class of maps of K -bundles P / / (cid:15) (cid:15) EK (cid:15) (cid:15) Σ / / BK .
The section u is equivalent to an equivariant map Φ u : P Ñ X . Hence there is a well-defined homotopy class of equivariant maps P Ñ EK ˆ X. It descends to a map Σ Ñ X K . If Σ is a closed surface. Then it gives a homology class A P H K p X ; Z q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 28
Gauged Witten equation.
We first provide the formulation of the gauged Wittenequation over a smooth r -spin curve C “ p Σ C , L C , ϕ C , ~ z q . Let Σ ˚ C Ă Σ C be the complementof the markings and nodes, which is a smooth open Riemann surface. Recall that inSubsection 2.5 we have chosen a Hermitian metric on L C which only depends on theisomorphism class of C . Then P C Ñ Σ ˚ C is the associated U p q -bundle and A C P A p P C q isthe Chern connection. Definition 3.6.
Let C “ p Σ C , L C , ϕ C , ~ z q be a smooth or nodal r -spin curve. A gaugedmap from C to X is a triple v “ p P, A, u q where P Ñ Σ ˚ C is a smooth principal K -bundle, A P A p P q is a connection, and u P S p Y q is a section of Y C : “ ¯ P C ˆ ¯ K X where ¯ P C Ñ Σ ˚ C is the ¯ K -bundle constructed from P C and P .Fix P Ñ Σ ˚ C . We can lift the superpotential W : X Ñ C to a section W C P Γ p Y C , π ˚ K C , log q » Γ p Y C , π ˚ K Σ ˚ C q as follows. A point of Y C is represented by a triple r p C , p, x s with the equivalence r p C h C , ph, x s “ r p C , p, h C hx s , @ h C P C ˚ , h P K. Then define W C pr p C , p, x sq “ W p x q ϕ C pp p C q r q . By the fact that W is homogeneous of degree r and K -invariant, W C is a well-definedsection.Consider arbitrary connections A P A p P q and let ¯ A “ p A C , A q be the induced con-nection on ¯ P . Since we are in two dimensions, any ¯ A P A p ¯ P C q induces a holomorphicbundle structure on the complexification of ¯ P C . Moreover, since the ¯ G -action on X isholomorphic, it induces a holomorphic structure on the total space Y C . One has thefollowing important property of W C . Lemma 3.7.
For any A P A p P q , W C is a holomorphic section of π ˚ K Σ ˚ C with respect tothe holomorphic structure on Y C induced from ¯ A . Since the K¨ahler metric is ¯ K -invariant, it induces a Hermitian metric on the verti-cal tangent bundle T vert Y C Ñ Y C . Therefore, one can dualize the differential d W C P Γ p Y C , π ˚ K Σ ˚ C b p T vert Y C q _ q , obtaining the gradient ∇W C P Γ p Y C , π ˚ K _ Σ ˚ C b T vert Y C q . On the other hand, for each A P A p P q and u P S p Y C q , one can take the covariantderivative of u with respect to ¯ A , and take its p , q part. We regard this as an operationon the pair p A, u q and hence denote the covariant derivative by d A u and its p , q part as B A u P Γ p Σ ˚ C , K Σ ˚ C b u ˚ T vert Y C q “ : Ω , p Σ ˚ C , u ˚ T vert Y C q . It lies in the same vector space as u ˚ ∇W C “ : ∇W C p u q . We have the Witten equation B A u ` ∇W C p u q “ . (3.7)We need to fixed the complex gauge by imposing a curvature condition. Let ad P bethe adjoint bundle. Then the curvature form F A of A is a section of ad P with two formcoefficients. On the other hand, since the R-symmetry commutes with the K -action, forany u P S p Y q , µ p u q is a well-defined section of ad P . The vortex equation is ˚ F A ` µ p u q “ . (3.8) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 29
The gauged Witten equation over C is the system on p P, A, u qB A u ` ∇W C p u q “ , ˚ F A ` µ p u q “ . (3.9)Basic local properties of solutions to the gauged Witten equation still hold as forvortices. For example, there is a gauge symmetry of (3.9) for gauge transformations g : P Ñ K . Moreover, for any weak solution, there exists a gauge transformation makingit a smooth solution.3.5. Energy and bounded solutions.
Fix the smooth r -spin curve C . Choose p ą p A, u q P A ,ploc p P q ˆ S ,ploc p Y C q . With respect to a local trivialization of P over a chart U Ă Σ ˚ C , u can be viewed as a map from U to X and A is identified with d ` φds ` ψdt , where φ, ψ P W ,ploc p U, ¯ k q and s, t are local coordinates of U . The covariantderivative of u with respect to A , in local coordinates, reads d A u “ ds b pB s u ` X φ p u qq ` dt b pB t u ` X ψ p u qq . For a gauged map v “ p P, A, u q from C to X , define its energy as E p P, A, u q “ ´ } d A u } L ` } µ p u q} L ` } F A } L ¯ ` } ∇W C p u q} L “ «ż Σ ˚ C ´ | d A u | ` | µ p u q| ` | F A | ` | ∇W C p u q| ¯ σ c ff . (3.10)Let C be a smooth r -spin curve. We say that a gauged map p A, u q is bounded if it hasfinite energy and if there is a ¯ K -invariant compact subset Z Ă X such that u p Σ ˚ C q Ă ¯ P C ˆ ¯ K Z. The relation with vortices.
For any smooth r -spin curve C , there is a subset ofsolutions to the gauged Witten equation p P, A, u q for which B A u “ , ∇W C p u q “ . (3.11)Namely, those solutions which are holomorphic and whose images are contained in Y W : “ ¯ P ˆ ¯ K X W .In Theorem 5.1 we prove that (3.11) is actually equivalent to (3.9). A convenientconsequence is that any bounded solution to the gauged Witten equation produces certaintype of vortices over the puncture Riemann surface Σ C with gauge group K . Indeed, thehomomorphism ι W : C ˚ Ñ G given by Hypothesis 3.3 combines the principal bundles P C and P to a K -bundle P Ñ Σ C , and the connections A C and A to a connection A P A p P q . Then for all sections u : Σ C Ñ Y C with images contained in Y ss W , it is thesame as a section u : P Ñ X W . Furthermore, fixing C the correspondence p P, A, u q ÞÑ p P , A , u q is a one-to-one correspondence. Finally, Hypothesis 3.3 and the above holomorphicity re-sult imply that if p P, A, u q is a solution to the gauged Witten equation, then p P , A , u q is a solution to the equation B A u “ , ˚ F A ` µ p u q “ ˚ ι W p F A C q , u p Σ C q Ă Y ss W . Since F A C is nearly flat over the cylindrical ends, the above equation is a compact per-turbation of the ordinary symplectic vortex equation with target X , together with theconstrain that the image of u has to be contained in the singular subvariety X W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 30
Remark . It seems that one does not need the discussion of r -spin curves to studythe gauged Witten equation. This is true in the level of solutions and moduli spaces.However we do not have a theory for vortex equation with singular target, and one needsthe gauged Witten equation to obtain a well-behaved obstruction theory.4. Vortices over Cylinders
In this section we study the analytical property of vortex equation over a cylinder.Similar treatment and results have been obtained in various literaturs, such as [Xu16][Ven15]. The case that the quotient is an orbifold was first considered in [CW] and later[CWW17]. Here we summarize the necessary results in our own languages.4.1.
Critical loops.
We work under the same assumption as in Subsection 3.3. Overa cylinder r a, b s ˆ S equipped with the trivial r -spin structure and the flat metric, thevortex equation can be viewed as the equation of negative gradient flows over the loopspace L p X q ˆ L k » C p S , X ˆ k q . Indeed, for a loop x “ p x, η q : S Ñ X ˆ k , one can define the action functional A p x q “ ´ ż D u ˚ ω X ` ż S x µ p x p t qq , η p t qy dt. Here u : D Ñ X is a map whose boundary restriction coincides with x . We assume that X is aspherical, hence the action functional is independent of the choice of the extension u . It is easy to verify that, with respect to the L -metric on the loop space induced fromthe metric ω X p¨ , J X ¨q on X and the chosen invariant metric on k , the gradient vector fieldof A is formally ∇A p x q “ « J X ` x p t q ` X η p t q p x p t qq ˘ µ p x p t qq ff . Hence critical points of A are loops that satisfy µ p x p t qq ” , x p t q ` X η p t q p x p t qq “ . The action functional A is invariant under the action of the loop group of K . Let g : S Ñ K be a smooth (or of certain regularity) loop. Define g ˚ x p t q “ p g p t q ´ x p t q , g ´ g p t q ` Ad ´ g p t q η p t qq . Hence the critical point set, denoted by Ă CA (consider only smooth loops) is invariantunder the smooth loop group C p S , K q . Denote the quotient by CA “ Ă CA { C p S , K q . Lemma 4.1.
There is a one-to-one correspondence CA » I ¯ X. Here I ¯ X is the inertia stack of the orbifold ¯ X .Proof. For every critical loop x p t q “ p x p t q , η p t qq , define a map g : r , π s Ñ K by g p q “ Id , g p t q g p t q ´ “ η p t q . Then we have ddt ” g p t q ´ x p t q ı “ g p t q ´ ” x p t q ` X η p t q p x p t qq ı “ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 31
Hence x p q “ g p π q ´ x p π q “ g p π q ´ x p q . Denote g p π q “ g x . Then define the map Ă CA Ñ I ¯ X, x ÞÑ x p q P ¯ X r g x s . It is easy to see that if x and x are gauge equivalent, then g x and g x are in the sameconjugacy class, while x p q and x p q are in the same K -orbit. Hence this descends toa map CA Ñ I ¯ X . We left to the reader the proof of injectivity and surjectivity of thismap. (cid:3) For each conjugacy class r g s P r K s , let CA r g s Ă CA be the subset of critical loops that corresponds to the twisted sector ¯ X r g s .Now we define a distance function on L . For two loops x i “ p x i , η i q , i “ , d L p x , x q “ inf g P C p S ,K q sup t P S ´ d X p x p t q , x p t qq ` | η p t q ´ η p t q| ¯ .d L is clearly gauge invariant. It then induces a (pseudo)distance on subsets of L . Thefollowing two lemmata are left to the reader. Lemma 4.2.
For r g s ‰ r h s , the distance between CA r g s and CA r h s is positive. Lemma 4.3.
For any sufficiently small ǫ ą , there exists δ ą such that for any C loop x “ p x, η q satisfying sup S ´ | d η p t q x p t q| ` | µ p x p t qq| ¯ ă δ there exists a unique conjugacy class r g s P r K s with ¯ X r g s ‰ H such that d L p x , CA r g s q ă ǫ. One can normalize critical loops in the following sense, i.e., fix the gauge. For anyconjugacy class r τ s P r K s , choose a constant λ P k such that e π i λ P r τ s . Then any criticalloop x P CA r τ s is gauge equivalent to a critical loop of the form x p t q “ p x p t q , λ q , µ p x p t qq ” , x p t q ` X λ p x p t qq “ . A critical loop x p t q “ p x p t q , η p t qq with η p t q being a constant is called a normalized criticalloop .4.2. Isoperimetric inequality.
We first consider the untwisted sector. Notice thatevery point of the untwisted sector, which is identified with ¯ X , can be lifted to a constantloop. Lemma 4.4. (cf. [Zil09, Theorem 1.2] ) There is a constant C ą and for every constant c ą , there exists δ c satisfying the following condition. Let r τ s P r K s be a conjugacyclass and let x be a loop such that d L p x , CA r τ s q ď δ c . Suppose the order of r τ s is m r τ s . Then A p x q ď cm r τ s ż π | x p t q ` X η p t q p x p t qq| dt ` C ż π | µ p x p t qq| dt. (4.1) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 32
Proof.
We regard x as defined over R with period 2 π . Abbreviate m r τ s “ m . We firsttreat the case that m “
1, namely r τ s “ r s . Indeed this is covered in [Zil09], althoughZiltener assumed that the quotient ¯ X “ µ ´ p q{ K is smooth. Let us explain. Up to gaugetransformation, we may assume that x is close to the constant loop p x p q , q . Choose aslice S of the K -action through x p q such that S is orthogonal to the infinitesimal K -action at x p q . Then by using an additional gauge transformation we can make x p t q P S for all t . It follows that x x p t q , X η p t q p x p t qqy ď C dist p x p t q , x p qq| x p t q|| X η p t q p x p t qq| (4.2)where the coefficient C dist p x p t q , x p qq can be arbitrarily small.The usual isoperimetric inequality (see [MS04, Theorem 4.4.1]) implies that for any c between c and , for δ c sufficiently small, one has ´ ż D u ˚ ω X ď c π ˆż π | x p t q| dt ˙ . Take ǫ ą c ` ǫ “ c . We have A p x q“ ´ ż D u ˚ ω X ` ż π x µ p x p t qq , η p t qy dt ď c π ˆż π | x p t q| dt ˙ ` } η } L } µ p x q} L ď c ˆż π | x p t q| dt ` ż π | X η p t q p x p t qq| dt ˙ ` C ż π | µ p x p t qq| dt ď p c ` ǫ q ż π | x p t q ` X η p t q p x p t qq| dt ` c ǫ ˇˇˇˇż π x x p t q , X η p t q p x p t qqy dt ˇˇˇˇ ` C ż π | µ p x p t qq| dt ď p c ` ǫ q ż π | x p t q ` X η p t q p x p t qq| dt ` δc ǫ ż π | x p t q|| η p t q| dt ` C ż π | µ p x p t qq| dt. Here the last line follows from (4.2). By making δ sufficiently small, we see A p x q ď p c ` ǫ q ż π | x p t q ` X η p t q p x p t qq| dt ` C ż π | µ p x p t qq| dt. So (4.1) is proved for the case m p g q “ δ c sufficiently small, the m -th iteration of x , defined by x m p t q “ p x m p t q , η m p t qq “ p x p mt q , mη p mt qq , is a loop that is close to CA r s . Then from the untwisted case, one has A p x m q ď cm ż π | x p mt q ` X η p mt q p x p mt qq| dt ` C ż π | µ p x p mt qq| dt “ cm ż π | x p t q ` X η p t q p x p t qq| dt ` C ż π | µ p x p t qq| dt. Moreover, A p x m q “ m A p x q . Hence the general case follows from the untwisted case. (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 33
Vortex equation.
Now we consider the symplectic vortex equation over a cylinder Θ ba “ p a, b q ˆ S where a or b can be infinity. To define the vortex equation one needs tochoose an area form σdsdt. We say that σ is of cylindrical type if there is a constant c ą Θ ba e p ´ τ q| s | ˇˇ ∇ l p σ ´ c q ˇˇ ă `8 , @ τ ą , l ě . (4.3)Notice that the structure group K is not necessarily connected. Hence if we considervortex equation over the cylinder, we need to look at nontrivial K -bundles. However,since K is compact, a multiple cover of any K -bundle over the cylinder must be triv-ial. Since the concern of this section is analytical, we only consider trivial K -bundles.Therefore, we can regard a gauged map p P, A, u q from Θ ba to X as a triple of maps v “ p u, φ, ψ q : Θ ba Ñ X ˆ k ˆ k . We introduce covariant derivatives v s “ B s u ` X φ p u q , v t “ B t u ` X ψ p u q and denote the curvature by κ “ B s ψ ´ B t φ ` r φ, ψ s . Then the symplectic vortex equation reads v s ` J X v t , κ ` σµ p u q “ . (4.4)The energy of a solution is defined as E p v q “ } v s } L p Θ ba q ` }? σµ p u q} L p Θ ba q where the L norm is taken with respect to the standard cylindrical metric. A solutionis called bounded if it has finite energy and if its image has compact closure in X .We introduce more notations under a local trivialization. Let A “ d ` φds ` ψdt be theassociated connection. Define the covariant derivatives ∇ As , ∇ At , ∇ Az , ∇ A ¯ z : Γ p Θ , u ˚ T X q Ñ Γ p Θ , u ˚ T X q by ∇ As ξ “ ∇ s ξ ` ∇ ξ X φ , ∇ At ξ “ ∇ t ξ ` ∇ ξ X φ . ∇ Az ξ “ ´ ∇ As ξ ´ J ∇ At ξ ¯ , ∇ A ¯ z ξ “ ´ ∇ As ξ ` J ∇ At ξ ¯ . On the other hand, define the covariant differentials of a map η : Θ Ñ k to be ∇ As η “ B s η ` r φ K , η s , ∇ At η “ B t η ` r ψ K , η s Convergence.
We define a notion of convergence of vortices. Consider a sequenceof cylinders Θ k : “ Θ b k a k , a k ă b k , lim k Ñ8 a k “ a , lim k Ñ8 b k “ b . Let σ k : Θ k Ñ R ` be a sequence of functions of cylindrical type with a constant c ą c , and satisfyingsup k sup Θ e p ´ τ q s | ∇ l p σ k ´ c q| ă `8 , @ τ ą , l ě . (4.5)Let v k “ p u k , φ k , ψ k q (including k “ 8 ) be a sequence of smooth gauged maps from Θ k to X . Then we say that v k converges to v in c.c.t. (compact convergence topology)if for any compact subset Z Ă Θ , v k | Z converges to v | Z as smooth maps. We say that AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 34 v k converges to v in c.c.t. modulo gauge if there exists a sequence of smooth gaugetransformations g k : Θ k Ñ K such that g ˚ k v k converges in c.c.t. to v .It is easy to establish the following local compactness theorem. The proof is left tothe reader. Proposition 4.5.
Let Θ k , σ k be as above (including k “ 8 ). Suppose v k is a sequenceof smooth vortices, Z Ă Θ is precompact open subset such that lim sup k Ñ8 sup Z e p v k q ă `8 (4.6) then there exist a smooth vortex v over Z and a subsequence (still indexed by k ) suchthat v k | Z converges in c.c.t. to v modulo gauge over Z .Furthermore, if (4.6) holds for all precompact open subset Z Ă Θ , then there exists asmooth vortex v over Θ and a subsequence (still indexed by k ) such that v k convergesto v in c.c.t. modulo gauge. Furthermore, sup Z lim k Ñ8 E p v k , Z q “ E p v q . Annulus lemma.
Now consider the case Θ ba “ Θ T ´ T “ : Θ T . For every such agauged map v and s P r´
T, T s , define the loop x s “ p x p s, t q , ψ p s, t qq P ˜ L . Then we have the following energy identity (cf. [CWW17, Proposition 5.7]) E p v q “ A p x ´ T q ´ A p x T q . Lemma 4.6. (The annulus lemma) For every c ą there exist δ and C ą satisfyingthe following conditions. For any T ą and any solution v “ p u, φ, ψ q to (4.4) over r´ T, T s ˆ S for a function σ ě , if sup s Pr´
T,T s d L p x s , y q ď δ for some y P CA r τ s where r τ s has order m , then sup r´ T ` R,T ´ R s e p s, t q ď Ce ´ Rmc . Proof.
It is the same as the usual argument of deriving the annulus lemma from theisoperimetric inequality. The details are left to the reader. (cid:3)
Asymptotic behavior.
Now consider the vortex equation over the semi-infinitecylinder Θ “ p , `8q ˆ S . Theorem 4.7. (cf. [CWW17, Theorem 1.1])
Let v “ p u, φ, ψ q be a bounded solutionto (4.4) over Θ . Then up to a smooth gauge transformation on Θ , the followingconditions are satisfied.(a) There exists λ P k such that lim s Ñ`8 φ “ , lim s Ñ`8 ψ “ λ. (b) There exists a point x P µ ´ p q such that e πλ x “ x, lim s Ñ`8 e ´ λt u p s, t q “ x. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 35 (c) The above convergences are exponentially fast. Namely, denote γ “ e πλ P K andsuppose the order of γ is m P N , then for all (small) τ ą , we have sup Θ e ´ τm s ´ | φ | ` | ∇ φ | ` | ψ ´ λ | ` | ∇ ψ | ¯ ă `8 . Moreover, if we write u p s, t q “ exp e λt x ζ p s, t q then sup Θ e ´ τm s ´ | ζ | ` | ∇ s ζ | ` | ∇ t ζ | ¯ ă `8 . Moreover, we would like to extend Theorem 4.7 to an uniform estimate for a convergingsequence of solutions. Consider a sequence of area forms σ k on Θ satisfying (4.5). Thefollowing generalization of Theorem 4.7 is left to the reader. Theorem 4.8.
Suppose v k “ p u k , φ k , ψ k q is a sequence of smooth solutions to v k,s ` J X v k,t “ , κ k ` σ k µ p u k q “ . Suppose the sequence v k is uniformly bounded , namely the energies E p v k q are uniformlybounded and the images of u k are all contained in a fixed compact subset of X . Moreover,suppose v k converges to v “ p u , φ , ψ q in c.c.t with no energy loss, i.e., lim k Ñ8 E p v k q “ E p v q . Then for all sufficiently large k (including k “ 8 ), there exist smooth gauge transforma-tions g k : Θ Ñ K such that the following conditions are satisfied.(a) There exist λ P k (independent of k ) and x k P µ ´ p q such that Item (a) andItem (b) of Theorem 4.7 are satisfied with p u, φ, ψ q replaced by p u k , φ k , ψ k q and x replaced by x k for all k .(b) The convergences in Item (c) are uniformly exponentially fast in k . Namely, forall τ ą , one has sup k sup Θ e ´ τm s ´ | φ k | ` | ∇ φ k | ` | ψ k ´ λ | ` | ∇ ψ k | ¯ ă `8 Moreover, if we write u k p s, t q “ exp e λt x k ζ k p s, t q , then sup k sup Θ e ´ τm s ´ | ζ k | ` | ∇ s ζ k | ` | ∇ t ζ k | ¯ ă `8 . (c) x k converges to x . Moreover, if we write x k “ exp x ξ k p8q where ξ k p8q P T x X ,then ξ k p8q P H x X. (d) For k sufficiently large, we can write u k “ exp u ξ k , φ k “ φ ` ξ k , ψ k “ ψ ` ξ k and for δ ă m , we have lim k Ñ8 p} ξ k ´ ξ k p8q} L p,δ ` } D A ξ k } L p,δ q “ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 36 Analytical Properties of Solutions
In this section we prove several properties of solutions to the gauged Witten equationover a smooth r -spin curve, including the holomorphicity of solutions, and the asymptoticbehavior of solutions. A consequence is that solutions represents certain equivarianthomology classes in X in rational coefficients.The properties of solutions over smooth r -spin curves are supposed to imply propertiesof solutions over nodal r -spin curves. However, we need to explain a subtle point. Givena nodal r -spin curve C and consider the gauged Witten equation over C . Each irreduciblecomponent of C , after normalization, is technically not an r -spin curve, since the roles ofmarkings and nodes in the definition of r -spin structures are different. However, to writedown the gauged Witten equation, what we actually need is an isomorphism ϕ C : p L C | Σ ˚ C q b r » K Σ ˚ C of holomorphic line bundles over the punctured surface Σ ˚ C . This data is given by the r -spin structure. Therefore, we still call a normalized irreducible component of a nodal r -spin curve a smooth r -spin curve, although it does not satisfy Definition 2.7 completely.5.1. Holomorphicity.
Our first theorem about analytic properties of solutions says thatall bounded solutions are holomorphic and are contained in the closure of the semistablepart of the critical locus.
Theorem 5.1.
Let v “ p P, A, u q be a bounded solution over a smooth r -spin curve C “ p Σ C , ~ z C , L C , ϕ C q . Then B A u “ and u p Σ ˚ C q Ď Y ss W : “ ¯ P ˆ ¯ K X ss W . Now we start to prove Theorem 5.1. For each puncture z a P Σ C , there exists a localcoordinate w such that with respect to the cylindrical metric, near z a , | d log w | “ | dww | “ z “ s ` i t “ ´ log w , the cylindrical coordinate on a cylindrical end U ˚ a . By thedefinition of r -spin structure, there exists a holomorphic section e of L C | U ˚ a such that ϕ C p e b r q “ w m a dz. Temporarily omit the index a and denote q “ m a { r . If the metric of L C is } e } “ | w | q e h , then by our choice of the metric, h is bounded from below near the puncture and wehave estimates sup U ˚ a | ∇ l h | ď C l , l “ , , , . . . . (5.1)On the other hand, choose an arbitrary trivialization of P over the cylindrical end.Then we obtain trivializations of ¯ P and Y . Under these trivializations, we write A “ d ` φds ` ψdt . Then¯ A “ d ` ¯ φds ` p i q ` ¯ φ q dt “ d ` p i B t h ` φ q ` p´ i q ´ i B s h ` ψ q dt. Moreover, we have W p z, x q “ e ´ i rqt ` rh W p x q dz, ∇W p z, x q “ e i rqt ` rh ∇ W p x q d ¯ z. Define the covariant derivatives ∇ As , ∇ At , ∇ Az , ∇ A ¯ z : Γ p U ˚ a , u ˚ T X q Ñ Γ p U ˚ a , u ˚ T X q by ∇ As ξ “ ∇ s ξ ` ∇ ξ X ¯ φ , ∇ At ξ “ ∇ t ξ ` ∇ ξ X ¯ φ . ∇ Az ξ “ ´ ∇ As ξ ´ J ∇ At ξ ¯ , ∇ A ¯ z ξ “ ´ ∇ As ξ ` J ∇ At ξ ¯ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 37
Also introduce v s “ B s u ` X ¯ φ p u q , v t “ B t u ` X i q ` ¯ ψ p u q , v z “ ` v s ` J X v t ˘ , v ¯ z “ ` v s ´ J X v t ˘ . On the other hand, using the trivializations, we can view sections of ad P as maps to k .We define the covariant differentials of a map η : U ˚ a Ñ k to be ∇ As η “ B s η ` r φ, η s , ∇ At η “ B t η ` r ψ, η s . (5.2)The curvature of ¯ A is written as F ¯ A “ ¯ κdsdt, where ¯ κ “ B s ¯ ψ ´ B t ¯ φ ` r ¯ φ, ¯ ψ s . Lemma 5.2.
In the above notations, ∇ Az ∇W p u q “ ∇ v ¯ z ∇W p u q , ∇ A ¯ z ∇W p u q “ ∇ v z ∇W p u q ` r B h B ¯ z ∇W . Proof.
We have the following straightforward calculations. ∇ As ∇W “ ∇ s ´ e i rqt ` rh ∇ W ¯ ` e i rqt ` rh ∇ ∇ W X ¯ φ “ e i rqt ` rh ´ r B s h ∇ W ` ∇ v s ∇ W ` r ∇ W, X ¯ φ s ¯ “ e i rqt ` rh ´ r B s h ∇ W ` ∇ v s ∇ W ` r ∇ W, X i B t h s ¯ “ e i rqt ` rh ´ r B s h ∇ W ` ∇ v s ∇ W ` p i r B t h q ∇ W ¯ “ ∇ v s e i rqt ` rh ∇ W ` r B h B ¯ z e i rq ` trh ∇ W. For the third equality above we used the fact that W is K -invariant. Similarly, ∇ At ∇W “ ∇ t ´ e i rqt ` rh ∇ W ¯ ` e i rqt ` rh ∇ ∇ W X ¯ ψ “ e i rqt ` rh ´ p i rq ` r B t h q ∇ W ` ∇ v t ∇ W ` r ∇ W, X ¯ ψ s ¯ “ e i rqt ` rh ´ p i rq ` r B t h q ∇ W ` ∇ v t ∇ W ` r ∇ W, X ´ i q ´ i B s h s ¯ “ e i rqt ` rh ´ p i rq ` r B t h q ∇ W ` ∇ v s ∇ W ` p´ i rq ´ i r B s h q ∇ W ¯ “ ∇ v s e i rqt ` rh ∇ W ` r B h B ¯ z e i rqt ` rh ∇ W. Since W is holomorphic, we have ∇ Az ∇W “ ´ ∇ v s e i rqt ` rh ∇ W ´ J X ∇ v t e i rqt ` rh ∇ W ¯ “ ∇ v ¯ z ∇W , and ∇ A ¯ z ∇W “ ∇ v z ∇W ` r B h B ¯ z ∇W . This finishes the proof of Lemma 5.2. (cid:3)
The last lemma is used in proving the following result.
Lemma 5.3. ( cf. [TX15, Proposition 4.5]) Using the above notations, one has lim s Ñ`8 | v s p s, t q| “ lim s Ñ`8 | v t p s, t q| “ lim s Ñ`8 | ∇W p u p s, t qq| “ . (5.3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 38
Proof.
We have the following calculation. p ∇ As q v s ` p ∇ At q v s “ ∇ As ´ ∇ As v s ` ∇ At v t ¯ ´ “ ∇ As , ∇ At ‰ v t ´ ∇ At ´ ∇ As v t ´ ∇ At v s ¯ “ ∇ As ´ ∇ As p´ J X v t ´ ∇W p u qq ¯ ` ∇ As ´ ∇ At p J X v s ` J X ∇W p u qq ¯ ´ R p v s , v t q v t ´ ∇ v t X ¯ κ ´ ∇ At X ¯ κ “ ´ J X ∇ As X ¯ κ ´ ∇ At X ¯ κ ´ ∇ v t X ¯ κ ´ R p v s , v t q v t ´ ∇ As ∇ A ¯ z ∇W p u q“ J X ∇ Az ´ ´ X κ C ` X µ p u q ¯ ` ∇ v t ´ ´ X κ C ` X µ p u q ¯ ´ R p v s , v t q v t ´ ∇ As ´ ∇ v z ∇W p u q ` r B h B ¯ z ∇W ¯ “ J X ∇ Az ´ J X X ∆ h ` X µ p u q ¯ ` ∇ v t ´ J X X ∆ h ` X µ p u q ¯ ´ R p v s , v t q v t ´ ∇ As ´ ∇ v z ∇W p u q ` r B h B ¯ z ∇W ¯ . Using the fact that the solution is bounded and (5.1), one has the following estimatesof each terms of the last express. First, since the infinitesimal actions X and X areholomorphic vector fields and behave equivariantly, we have ∇ Az X ∆ h “ ∇ v z X ∆ h ` X B ∆ h B z “ O p| v s |q ` O p| v t |q ` O p q ; ∇ Az X µ p u q “ ∇ v z X µ p u q ` X ∇ Az µ p u q “ O p| v s |q ` O p| v t |q ` O p q . We also have ∇ v t ´ J X X ∆ h ` X µ p u q ¯ “ O p| v t |q ` O p q ;and R p v s , v t q v t “ O p| v s || v t | q ;and ∇ As ∇ v z ∇W p u q “ O p| ∇ As v s |q ` O p| ∇ As v t |q ` O p| v s | q ` O p| v s || v t |q ` O p q ;and ∇ As B h B ¯ z ∇W p u q “ O p| v s |q ` O p q . Therefore, since ∇ A preserves the metric, one has12 ∆ | v s | “ A p ∇ As q v s ` p ∇ At q v s , v s E ` ˇˇ ∇ As v s ˇˇ ` ˇˇ ∇ At v s ˇˇ ě ´ C ´ ` p| v s | ` | v t | q ¯ ` | ∇ As v s | ` | ∇ At v s | ´ C ´ | ∇ As v s | ` | ∇ As v t | ¯ . Changing s to t , one has a similar estimate12 ∆ | v t | ě ´ C ´ ` p| v s | ` | v t | q ¯ ` | ∇ As v t | ` | ∇ At v t | ´ C ´ | ∇ At v t | ` | ∇ At v s | ¯ . Then adding the above two inequalities together and changing the value of C , one obtains∆ ´ | v s | ` | v t | ¯ ě ´ C ´ ` p| v s | ` | v t | q ¯ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 39
By the mean value estimate, there exists ǫ ą C ) such thatfor r ă p s , t q P U ˚ j , denoting the radius r disk centered at p s , t q by B r , ż B r ´ | v s | ` | v t | ¯ dsdt ď ǫ ùñ v s p s , t q ď πr ż B r ´ | v s | ` | v t | ¯ dsdt ` Cr . One can then derive that | v s | and | v t | converge to 0 as s Ñ `8 . The last equality of(5.3) follows from the Witten equation (3.7). (cid:3)
Since v is bounded, as a consequence, we know thatlim s Ñ`8 dist p u p s, t q , Crit W q “ . (5.4)Then by the holomorphicity of W , we have }B A u } L ` } ∇W p u q} L “ }B A u ` ∇W p u q} L ´ ż Σ ˚ C xB A u, ∇W p u qy σ c “ ´ i ż Σ ˚ C d W p u q ¨ B A u “ i ż Σ ˚ C B “ W p u q ‰ “ i ż Σ ˚ C d ` W p u q ˘ “ i n ÿ j “ lim s Ñ`8 ¿ C s p z j q W p u q . By Lemma 3.4 and (5.4), the integrals of W p u q along the loops appeared above allconverge to zero. It follows that }B A u } L “ } ∇W p u q} L “ Corollary 5.4.
Given an r -spin curve C and a bounded solution p A, u q to (3.9) . Then u is holomorphic with respect to A and u p Σ ˚ C q Ă ¯ P ˆ ¯ K X W . Another part of the C asymptotic behavior for a solution v “ p A, u q is prove that µ p u q converges to zero at punctures. Lemma 5.5.
One has lim s Ñ`8 µ p u p s, t qq “ .Proof. Recall the definition of the covariant differential of maps into k given by (5.2). Bythe holomorphicity of u , one has the following calculation. p ∇ As q µ p u q ` p ∇ At q µ p u q “ ∇ As p dµ ¨ v s q ` ∇ At p dµ ¨ v t q“ ∇ At p dµ ¨ J v s q ´ ∇ As p dµ ¨ J v t q“ ´ ρ ¯ K p v s , v t q ´ dµ ¨ J X ¯ κ “ dµ ¨ ” J X µ p u q ´ J X κ C ı ´ ρ ¯ K p v s , v t q . Therefore, for C ą v ,12 ∆ ˇˇ µ p u q ˇˇ “ @ p ∇ As q µ p u q ` p ∇ At q µ p u q , µ p u q D ` ˇˇ ∇ As µ p u q ˇˇ ` ˇˇ ∇ At µ p u q ˇˇ ě @ dµ ¨ ` J X µ p u q ´ J X κ C ˘ ´ ρ ¯ K p v s , v t q , µ p u q D ě ´ C ` ` | µ p u q| ˘ . Again, this lemma follows from a mean-value estimate and the finite energy condition. (cid:3)
Corollary 5.6. If v “ p A, u q is a bounded solution, then u p Σ ˚ C q Ă Y ss W : “ ¯ P ˆ ¯ K X ss W .Proof. By Corollary 5.4, the image of u is contained in Crit W . Since u is a holomorphicsection and Σ ˚ C is nonsingular, its image can only be contained in a single irreduciblecomponents of X W . Since the limit of u at a puncture must be in µ ´ p q , the image of u must be in the closure of the semi-stable part X ss W . (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 40
Remark . In this paper we emphasize the case that Σ ˚ C has at least one punctures.But there is a way to deal with the case with no punctures, by choosing a large enoughvolume form for the vortex equation. Then somewhere in the domain | µ p u q| must besmall enough so it is in the semi-stable part.5.2. Asymptotic behavior.
Another corollary of above results is the following resulton removal of singularity.
Corollary 5.8.
Let v “ p P, A, u q be a bounded solution of the gauged Witten equationover a smooth r -spin curve C . Let z a be a puncture of C at which the monodromy of L C is e πq a i . Then there exist x a P X W X µ ´ p q , η a P k , and a smooth trivialization of ¯ P over the cylindrical end at z a such that, with respect to this trivialization, we can write A “ d ` φds ` ψdt and regard u as a map u a : U ˚ a Ñ X , satisfying lim s Ñ`8 φ “ , lim s Ñ`8 ψ “ η a , lim s Ñ`8 e ´p i q a ` η a q t u p s, t q “ x a . Proof.
For any ǫ ą
0, define U X p ǫ q : “ x P X | | µ p x q| ă ǫ ( . Assume that u p Σ ˚ C q is contained in ¯ P ˆ ¯ K N where N Ă X is a ¯ K -invariant precompactopen subset. Since 0 is a regular value of µ , there exists ǫ K ą µ p ǫ K q X N contains no critical point of µ (i.e., no point at which dµ is not surjective). Then thereis a holomorphic projection π K : U K p ǫ K q X X W Ñ ¯ X W that annihilates infinitesimal G -actions. More precisely, for x P U K p ǫ K q X X W , a, b P k , dπ K p x q ¨ p X a ` J X b q “
0. By (3) of Hypothesis 3.1, π K also annihilates infinitesimal C ˚ -actions.Corollary 5.4 and Lemma 5.5 imply that, near the puncture, the image of u is containedin ¯ P ˆ ¯ K p X W ˆ U K p ǫ K qq . Take an arbitrary trivialization of ¯ P over the cylindrical end U ˚ a , and denote the map from U ˚ a to X corresponding to u by u a : U ˚ a Ñ X . Denote¯ u a “ π K ˝ u . Then ¯ u a is holomorphic with respect to the complex structure ¯ J , and ¯ u a isindependent of the choice of local trivializations. It is also easy to verify that the energyof ¯ u a as a smooth map is finite. Hence by removal of singularity (for orbifold targets),there exists a point ¯ x a P ¯ X W such thatlim s Ñ`8 ¯ u a p s, t q “ ¯ x a . Then by choosing suitable gauge on the cylindrical end, it is easy to find a smoothtrivialization of ¯ P | U ˚ a which satisfies the prescribed properties. (cid:3) Then over each cylindrical end of Σ ˚ C we can regard a solution to the gauged Wittenequation over C as a special solution to the symplectic vortex equation with target X ss W .We can view the gauge group as either K or ¯ K , because Hypothesis 3.3 says that over X W the K -action and the R-symmetry merge together. Then we can use the results ofSection 4 to refine the asymptotic convergence result of Corollary 5.8, namely solutionsapproach to critical loops exponentially fast. Then we have the following straightforwardcorollary of Theorem 4.7 and Theorem 4.8. Theorem 5.9.
Let v “ p P, A, u q be a bounded solution to the gauged Witten equationover a smooth r -spin curve C . Let z a be a puncture and U ˚ a be a cylindrical end around z a with cylindrical metric s ` i t . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 41 (a) Let m a P N be the order of the conjugacy class of τ ,a : “ ι W p e π i q a q τ a P K . Let e : Σ ˚ C Ñ r , `8q be the energy density function, then for any τ ą , we have lim sup s Ñ`8 e ´ τma s e p s, t q ă `8 . (b) There exist a trivialization of ¯ P | U ˚ a , η a P k and x a P X W X µ ´ p q satisfying theconditions of Corollary 5.8. Moreover, if we write A “ d ` φds ` ψdt and u as amap u a : U ˚ a Ñ X , then for any τ ą s Ñ`8 ´ | φ | ` | ∇ φ | ` | ψ ´ η a | ` | ∇ ψ | ¯ e ´ τma s ă `8 ; if we write x a p t q “ e p i q a ` η a q t x a and u a p s, t q “ exp x a p t q ξ a p s, t q , then lim sup s Ñ`8 ´ } ξ a } ` } ∇ ξ a } ¯ e ´ τma s ă `8 . One can obtain a corresponding results for a converging sequence of solutions, byapplying Theorem 4.8. However the case with converging sequences usually involve withdegeneration of curves, which we would like to postpone until we need.Another consequence of Corollary 5.8 is that a bounded solution represents an equi-variant homology class in X . This generalizes the case we discussed in Subsection 3.3 forclosed non-orbifold surfaces, and the class represented by a solution is denoted by¯ B P H ¯ K p X ; Q q whose rational coefficients explains the orbifold feature of the problem.Moreover, using the notations of Corollary 5.8 and Theorem 5.9, for every marking z a ,denote ¯ τ a : “ e π i q a τ a P ¯ K, τ ,a : “ ι W p e π i q a q τ a P K. Then denote the twisted sector of ¯ X W corresponding to the conjugacy class r τ ,a s by¯ X r ¯ τ a s W Ă I ¯ X W . Denote r ¯ τ s “ pr ¯ τ s , . . . , r ¯ τ n sq . Then one can use the topological information ¯ B and r ¯ τ s to label moduli spaces. Definition 5.10. (Moduli spaces) Given g ě n ě g ` n ě
3. Given ¯ B P H ¯ K p X ; Q q and r ¯ τ s “ pr ¯ τ s , . . . , r ¯ τ n sq P r ¯ K s n , let˜ M rg,n p X, G, W, µ ; ¯ B, r ¯ τ sq be the set of pairs p C , v q where C is a smooth r -spin curve of type p g, n q and v “ p P, A, u q is a smooth bounded solution to the gauged Witten equation over C which represents theclass ¯ B and whose evaluations at the n punctures of C lie in the twisted sectors of ¯ X W labelled by r ¯ τ s , . . . , r ¯ τ n s respectively.Furthermore, the exponential decay property of solutions given in Theorem 5.9 allowsa definition of Banach manifolds of gauged maps over the surface Σ C , where the norm ismodelled on certain weighted Sobolev norm. We will define the Banach manifolds whenwe construct virtual cycles. Strictly speaking we should consider a category of such pairs.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 42
Uniform C bound. The next major result is the uniform C bound on solutions.This is a crucial step towards the compactness of moduli spaces of gauged Witten equa-tions. Theorem 5.11.
For every equivariant curve class ¯ B P H ¯ K p X ; Q q , there exists a ¯ K -invariant compact subset N “ N ¯ B Ă X satisfying the following condition. Given asmooth r -spin curve C and v “ p P, A, u q P ˜ M C p X, G, W, µ ; ¯ B q , one has u p Σ C q Ă ¯ P ˆ ¯ K N .Moreover, if C has at least one marked point, then N can be made independent of ¯ B .Proof. Consider the function F W “ µ ¨ ξ W given by Hypothesis 3.3. Choose local co-ordinate z “ s ` i t on Σ C . Let ∆ be the standard Laplacian in this coordinate. Letthe volume form be σdsdt and let the curvature form of A C be F A C “ κ C dsdt . We alsochoose local trivializations of P C and P so that the combined connection can be writtenas d ` φds ` ψdt and the section is identified with a map u into X .Then by the vortex equation ˚ F A ` µ p u q “
0, one has D A,s v t ´ D A,t v s “ ∇ s pB t u ` X ψ q ` ∇ B t u ` X ψ X φ ´ ∇ t pB s u ` X φ q ´ ∇ B s u ` X φ X ψ “ ∇ B s u X ψ ` X B s ψ ` ∇ B t u ` X ψ X φ ´ ∇ B t u X φ ´ X B t φ ´ ∇ B s u ` X φ X ψ “ X B s ψ ´ X B t φ ` r X ψ , X φ s“ X κ C ´ λ σ X µ . Moreover, since u is holomorphic with respect to A and F W is ¯ K -invariant, one has∆ F W p u q“ B s x ∇F W p u q , B s u y ` B t x ∇F W p u q , B t u y“ B s x ∇F W p u q , v s y ` B t x ∇F W p u q , v t y“ x D A,s ∇F W p u q , v s y ` x ∇F W p u q , D A,s v s y ` x D A,t ∇F W p u q , v t y ` x ∇F W p u q , D A,t v t y“ x ∇ v s ∇F W , v s y ` x ∇ v t ∇F W , v t y ` x ∇F W , ´ J D
A,s v t ` J D
A,t v s y“ x ∇ v s ∇F W , v s y ` x ∇ v t ∇F W , v t y ` x ∇F W , ´ J X κ C ` λ σJ X µ y . Since u is contained in Y ss W , by Hypothesis 3.3, one has∆ F W p u q “ x ∇ v s ∇F W , v s y ` x ∇ v t ∇F W , v t y ` x ∇F W , ´ J X ι W p κ C q ` λ σ X µ y . By Lemma 2.14, which says that F A C is uniformly bounded, one can cover Σ C by coordi-nate charts such that for certain τ R ą C and v , one always have | κ C | ď τ R . Notice that the image of ι W in Z p k q is one-dimensional. By Hypothesis 3.5, whenever F W p u q ě max c W p τ R i q , c W p´ τ R i q ( , one has∆ F W p u q ě . Therefore, F W p u q is subharmonic whenever it is greater than max c W p τ R i q , c W p´ τ R i q ( .Assume that sup Σ C F W p u q ą max c W p τ R i q , c W p´ τ R i q ( . Then F W has to be a large constant over Σ C . If C has at least one puncture, then it isimpossible since the limits of F W p u q at punctures are uniformly bounded by Theorem5.9 and the compactness of X W X µ ´ p q . If C has no puncture, then the area of Σ C isfinite. By the vortex equation, one has0 “ ˚ F A ¨ ξ W ` µ p u q ¨ ξ W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 43
Integrate this equality over Σ C , one obtain ż Σ C F A ¨ ξ W “ ´ ż Σ C F W p u q σdsdt. Here the left hand side is a quantity that only depends on the class B . So F W p u q cannotbe an arbitrarily large constant over Σ C either. Therefore F W p u q is uniformly boundedby a constant depending on B . By the properness of F W , we have proved Theorem5.11. (cid:3) Energy identity.
In this subsection we prove the following result which will bereferred to as the energy inequality.
Theorem 5.12. (Energy inequality)
For each equivariant curve class ¯ B P H ¯ K p X ; Q q there is a positive constant E p ¯ B q ą satisfying the following condition. Suppose v “p P, A, u q is a bounded solution to the gauged Witten equation over a smooth r -spin curve C which represents the class ¯ B . Then E p P, A, u q ď E p ¯ B q . Proof.
Choose a local coordinate chart U Ă Σ C with coordinates z “ s ` i t . Choose alocal trivialization of P , so that u is identified as a map u p s, t q and ¯ A is identified with d ` ¯ φds ` ¯ ψdt . There is a closed 2-form on U ˆ X which transforms naturally with changeof local trivializations. Define ω ¯ A “ ω X ´ d x ¯ µ, ¯ φds ` ¯ ψdt y P Ω p U ˆ X q where ¯ µ “ p µ R , µ q : X Ñ ¯ k » i R ‘ k is the moment map for the ¯ K -action. It is easyto see that ω ¯ A is a well-defined closed form on the total space Y ˚ C . Suppose locally thevolume form is σdsdt . Since u is holomorphic, one has12 | d A u | “ σ ” |B s u ` X ¯ φ p u q| ` |B t u ` X ¯ ψ p u q| ı “ σ ω X ´ B s u ` X ¯ φ p u q , J pB s u ` X ¯ φ p u qq ¯ “ ˚ ” u ˚ ω X ´ d p µ ¨ p ¯ φds ` ¯ ψdt qq ` ¯ µ p u q ¨ pB s ¯ ψ ´ B t ¯ φ ` r ¯ φ, ¯ ψ sq ı “ ˚ ” u ˚ ω ¯ A ` ¯ µ p u q ¨ F ¯ A ı . Therefore, by definition, the holomorphicity of u , and the vortex equation, one has E p P, A, u q “ ” } d A u } L ` } F A } L ` } µ p u q} L ı “ ż Σ C u ˚ ω ¯ A ` ż Σ C ¯ µ p u q ¨ F ¯ A ` } F A } L ` } µ p u q} L “ ż Σ C u ˚ ω ¯ A ` ż Σ C µ R p u q ¨ F A C ` } ˚ F A ` µ p u q} L “ ż Σ C u ˚ ω ¯ A ` ż Σ C µ R p u q ¨ F A C . (5.5)It is a well-known fact that the integral of u ˚ ω ¯ A over Σ C is equal to the topological pairing xr ω ¯ KX s , ¯ B y . On the other hand, by Lemma 2.14 and Theorem 5.11, the last term of (5.5)is bounded by a constant C ą
0. Hence this theorem holds for E p ¯ B q “ xr ω ¯ KX , ¯ B y ` C . (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 44 Moduli Space of Stable Solutions and Compactness
In the last section we considered solutions of the gauged Witten equation over a smooth r -spin curve. We have see that solutions are contained in the critical locus of W and areholomorphic. Hence in particular they are all solutions to the symplectic vortex equationwith target X . So the topologies on their moduli space can be defined as the same as ifit is a subset of the moduli space of vortices.6.1. Solitons.
Since we assume that X is aspherical, sphere bubbling cannot happen.However, solutions can still bubble off solitons at punctures or nodes. The solitons arebounded solutions to the gauged Witten equation over the infinite cylinder Θ : “ p´8 , `8q ˆ S . Notice that over the infinite cylinder, the log-canonical bundle is trivialized by ds ` i dt where s ` i t is the standard cylindrical coordinate. Then the set of isomorphism classes of r -spin structures over Θ is a group isomorphic to Z r . By our convention (see Subsection2.3) the line bundle L C which is trivial has a flat Hermitian metric with Chern connectionwritten as A C “ d ` i qdt, q P r t , , . . . , r ´ u . Definition 6.1. A q - soliton is a bounded solution to gauged Witten equation over theinfinite cylinder Θ “ R ˆ S equipped with translation invariant metric. More precisely,a soliton is a gauged map v “ p u, φ, ψ q from Θ to X solves the equation B s u ` X φ ` J X pB t u ` X i q ` ψ q “ , B s ψ ´ B t φ ` r φ, ψ s ` µ p u q “ . u p Θ q Ă X ss W . (6.1)The energy of a soliton is E p u, φ, ψ q “ ” } d A u } L ` } F A } L ` } µ p u q} L ı . By Theorem 4.7, one can gauge transform a soliton so that(a) as s Ñ ˘8 , φ is asymptotic to zero and ψ is asymptotic to η ˘ P k . Denote τ ˘ “ e πη ˘ .(b) There exists x ˘ P X W X µ ´ p q such that γ ˘ x ˘ “ x ˘ andlim s Ñ˘8 e λ ˘ t u p s, t q “ x ˘ . (6.2)So as before a soliton represents an equivariant curve class ¯ B P H ¯ K p X ; Q q , and we havethe following energy identity. Proposition 6.2. (Energy identity for solitons)
Given a soliton v “ p u, φ, ψ q that rep-resents an equivariant curve class ¯ B , one has E p u, φ, ψ q “ xr ω ¯ KX s , ¯ B y . Proof.
Using the same calculation of the proof of Theorem 5.12, we see in (5.5) thecurvature term F A C vanishes. Hence we obtain an equality. (cid:3) Moreover, for the purpose of proving compactness, one also need to prove that theenergy of a nontrivial soliton is bounded from below.
Theorem 6.3.
There exists ǫ W ą that only depends on the GLSM space, such that forany soliton v with positive energy, one has E p v q ě ǫ W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 45
Proof.
One can use a standard argument to prove an estimate such as E p v ; r s ` T, s ´ T s ˆ S q ď Ce ´ δT E p v ; r s , s s ˆ S q for constants C, δ ą v , whenever the total energy of v is small enough. Itthen implies that E p v q has a positive lower bound. The details are left to the reader. (cid:3) Decorated dual graphs.
Recall that a marked curve is described by a dual graph.
Definition 6.4. (Dual graph) A dual graph is a tupleΓ “ ´ V p Γ q , E p Γ q , T p Γ q , a , g ¯ where(a) p V p Γ q , E p Γ q , T p Γ qq is a graph where V p Γ q is the set of vertices , E p Γ q is the set of oriented edges , T p Γ q the set of tails . They are all finite sets.(b) a : T p Γ q Ñ V p Γ q and g : V p Γ q Ñ t , , . . . , u are maps.The valence of a vertex v P V p Γ q is d p v q “ e P E p Γ q | s p e q “ v ( ` e P E p Γ q | t p e q “ v ( ` t P T p Γ q | a p t q “ v ( . A vertex v P V p Γ q is stable if 2 g p v q ` d p v q ě . Here ends Definition 6.4.To describe r -spin curves, we need extra structures on dual graphs. Given a dualgraph Γ, let ˜Γ be the dual graph obtained by cutting off all edges (including loops). SoE p ˜Γ q “ H and there is a natural inclusion T p Γ q ã Ñ T p ˜Γ q . Denote˜E p Γ q “ T p ˜Γ q r T p Γ q . Definition 6.5. An r -spin dual graph is a tupleΓ “ ´ V p Γ q , E p Γ q , T p Γ q , a , g , m ¯ where p V p Γ q , E p Γ q , T p Γ q , a , g q is a dual graph in the sense of Definition 6.4, and m : T p ˜Γ q Ñ Z r is a map. The following conditions are required.(a) If ˜ e ´ , ˜ e ` P ˜E p Γ q Ă T p ˜Γ q are obtained by cutting off e P E p Γ q , then m p ˜ e ´ q m p ˜ e ` q “ g p v q “
0, then d p v q ě
2, if g p v q “
1, then d p v q ě g p v q “ d p v q “
2, then the two edges connecting v do not form a loop.(d) If g p v q “ d p v q “
2, then for the two tails t, t P T p ˜Γ q that are attached to v ,we have m p t q m p t q “ Definition 6.6.
A map from an r -spin dual graph Γ to another r -spin dual graph Πconsists of a surjective map ρ V : V p Γ q Ñ V p Π q , a bijective ρ T : T p Γ q Ñ T p Π q and aninjective map ρ ˜E : ˜E p Π q Ñ E p Γ q . They satisfy the following properties.(a) If e P E p Γ q connects two vertices v, v P V p Γ q , then either ρ V p v q “ ρ V p v q , or ρ V p v q and ρ V p v q are connected by an edge in E p Π q .(b) If t P T p Γ q is attached to v P V p Γ q , then ρ T p t q is attached to ρ V p v q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 46 (c) For each e P E p Π q , if e is cut to ˜ e ´ , ˜ e ` P ˜E p Π q , then ρ ˜E p ˜ e ´ q and ρ ˜E p ˜ e ` q arealso obtained by cutting off an edge in Γ. Hence ρ ˜E induces an injective map ρ E : E p Π q Ñ E p Γ q .(d) For each v P V p Π q , we have g Π p v q “ ÿ w P ρ ´ p v q g Γ p w q . (e) The maps ρ T and ρ ˜ E preserve the map m .Here ends Definition 6.6.We define a partial order among r -spin dual graphs. We denote Γ ď Π if there is amap from Γ to Π.Given an r -spin dual graph Γ, there is a notion of stabilization , denoted by Γ st . Noticethat there is no morphism Γ Ñ Γ st in any sense. Lemma 6.7.
Let ρ : Γ Ñ Π be a map between r -spin dual graphs. Then it induces acanonical map ρ st : Γ st Ñ Π st between their stabilizations.Proof. Left to the reader. (cid:3)
The combinatorial type of an r -spin curve can be described by an r -spin dual graph.For a stable r -spin dual graph Γ, let M r Γ Ă M rg,n be the subset of points correspondingto stable r -spin curves that have r -spin dual graph Γ. By using the partial order ď , theclosure of M r Γ can be described as M r Γ “ ğ Π ď Γ , Π “ Π st M r Π . The combinatorial type of such smooth or nodal r -spin curves can be described by decorated dual graphs . Definition 6.8. (Decorated dual graph) A decorated dual graph is a tuple Γ “ ´ Γ , p ¯ B v q v P V p Γ q , pr τ t sq t P T p ˜Γ q ¯ where Γ is an r -spin dual graph, ¯ B v P H ¯ K p X ; Q q is a collection of equivariant curveclasses indexed by all vertices v P V p Γ q , and r ¯ τ t s P r ¯ K s is a collection of twisted sectorsindexed by all tails t P T p ˜Γ q . In addition, we require that, for each edge e P E p Γ q whichcorresponds to a pair of tails ˜ e ´ , ˜ e ` P T p ˜Γ q , we have r ¯ τ ˜ e ´ sr ¯ τ ˜ e ` s “ r s . A decorated dual graph is stable if for each unstable vertex v P V p Γ q , ¯ B v ‰ Γ . There is a natural map Γ ÞÑ Γ by forgettingthe decorations. The partial order can be lifted to a partial order among all decorateddual graphs, which is still denoted by Π ď Γ .6.3. Stable solutions.
In this section we define the notion of stable solutions that in-corporate the phenomenon that energy of solutions over stable r -spin curves may escapefrom nodes or punctures. Definition 6.9. (Stable solutions) Let Γ be a decorated dual graph. A solution to thegauged Witten equation of combinatorial type Γ consists of a smooth or nodal r -spin curve C of combinatorial type Γ, and a collection of objects v : “ ” p v v q v P V p Γ q ı AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 47
Here for each stable vertex v P V p Γ q , v v “ p P v , A v , u v q is a solution to the gaugedWitten equation over the smooth r -spin curve C v ; for each unstable vertex v P V p Γ q , v v “ p P v , A v , u v q is a soliton. They satisfy the following conditions.(a) For each vertex v P V p Γ q , the equivariant curve class represented by v v coincideswith ¯ B v (which is contained in the data Γ ).(b) For each tail t P T p ˜Γ q , if t is attached to v P V p Γ q , then the limiting holonomy of v v at the (node or marking) z t P Σ v is r ¯ τ t s .(c) For each edge e P E p Γ q corresponding to a node in C , let ˜ e ´ and ˜ e ` be the twotails of ˜Γ obtained by cutting off e , which are attached to vertices v ´ and v ` (which could be equal), corresponding to preimages ˜ w ´ and ˜ w ` of w in ˜ C . Let themonodromies of the solution at ˜ w ´ and ˜ w ` be r ¯ τ ´ s and r ¯ τ ` s respectively. Thenthe evaluations of v v ´ at w ´ and v v ` at w ` (which exist by Corollary 5.8) areequal as points in the twisted sector ¯ X r ¯ τ ´ s W » ¯ X r ¯ τ ` s W .The solution is called stable if Γ is stable. The energy (resp. homology class) of asolution v is the sum of energies (resp. homology classes) of each component.One can define an equivalence relation among all stable solutions, and define thecorresponding moduli spaces of equivalence classes. Definition 6.10.
Given a decorated dual graph Γ , let M Γ : “ M Γ p X, G, W, µ q be the set of equivalence classes of stable solutions of combinatorial type Γ .6.4. Topology of the moduli spaces.
As usual, the topology of the moduli spaces M g,n p X, G, W, µ q is induced from a notion of sequential convergence. The reader cancompare with the notion of convergence defined by Venugopalan [Ven15, Definition 3.4].We first define the convergence for a sequence of solutions defined on smooth r -spincurves. Definition 6.11. (Convergence of smooth solutions) Let C k be a sequence of stablesmooth r -spin curves of genus g with n markings. Let v k be a sequence of stable solutionsover C k . Let C be a smooth or nodal r -spin curve of genus g with n markings and v bea stable solution over C . We say that v k converges to v if the following conditions hold.(a) The isomorphism classes of C k in M rg,n converges to the isomorphism class of C st . Choose an arbitrary resolution datum r of C st , which contains a universalunfolding U Ñ V and families of cylindrical coordinates near punctures and nodes.Then the convergence C k Ñ C st implies that there exist embeddings φ k : C k Ñ U ,and for each k , for each marking z a (resp. each node w ) of C st , a specifiedcylindrical end (resp. a long cylinder) with cylindrical coordinates s ` i t .(b) Let the dual graph of C be Γ. For each stable vertex v P V p Γ st q , for any compactsubset Z Ă Σ C ,v , the resolution datum r and the embedding φ k induce a canonicalinclusion ι k : Z Ñ Σ C k as r -spin curves. Then there is a bundle isomorphism g k : P | Z Ñ P k | C k that covers ι k such that g ˚ k v k | Z converges in c.c.t. to v | Z .(c) For each unstable vertex v P V p Γ q , it is mapped under the stabilization map C Ñ C st to either a marking or a node. Then there exists a sequence of points y k “ s k ` i t k in the cylinder satisfying the following condition. For any R ą r s k ´ R, s k ` R s ˆ S Ă C k where we use the cylindricalcoordinates specified in the resolution datum r . Then there is a sequence ofbundle isomorphisms g k : P v | r´ R,R sˆ S Ñ P k | r s k ´ R,s k ` R sˆ S that covers the map y ÞÑ y ` y k such that g ˚ k v k converges in c.c.t. to v v | r´ R,R sˆ S . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 48 (d) There is no energy lost, namely E p v q “ lim k Ñ8 E p v k q . Here ends Definition 6.11.We also define the sequence convergence of solitons. First, a chain of solitons is afinite sequence of solitons v ¨ ¨ ¨ v s satisfying the following conditions.(a) The r -spin structures of each v i are labelled by the same m P Z r .(b) Each v i is a nontrivial soliton, i.e., having positive energy.(c) For each i “ , . . . , s ´
1, the limiting twisted sector of v i at `8 is opposite tothe limiting twisted sector of v i ´ at ´8 , and their evaluations are the same. Definition 6.12. (Convergence of solitons) Let v k be a sequence of solitons definedover a sequence of infinite cylinders C k . We say that v k converges to a chain of solitons v ¨ ¨ ¨ v s if the following conditions hold.(a) Removing finitely many elements in this sequence, the r -spin structures on C k areisomorphic.(b) For b “ , . . . , s , there exist a sequence of points y b,k “ s b,k ` i t b,k P C k and asequence of r -spin isomorphisms φ b,k : C Ñ C k which covers the translation map y ÞÑ y ` y b,k , and a sequence of bundle iso-morphisms g b,k : P Ñ P k that covers φ b,k such that g ˚ b,k v k converges in c.c.t. to v b .(c) The sequences of points y b,k satisfy b ă b ùñ lim k Ñ8 s b ,k ´ s b,k “ `8 . (d) There is no energy lost, namely E p v q “ lim k Ñ8 E p v k q . Here ends Definition 6.12.To define the notion of sequential convergence in the most general case, we need tointroduce a few more notations. Let C k be a sequence of (not necessarily stable) smoothor nodal r -spin curves. Let C be another smooth or nodal r -spin curve. Suppose they havethe same genus g and the same number of marked points n . Assume that the stabilizationof C k converges to the stabilization of C . Then if U α Ñ V α is a local universal unfoldingof C , there exist embeddings φ k : C k Ñ U α by identifying C k with a fibre and the sequenceof fibres converge to the central fibre. Further, if we assume that all C k has the samedual graph Π with stabilization Π st and C has the dual graph Γ with stabilization Γ st ,then the convergence and the stabilization give a diagram of maps of dual graphsΓ (cid:15) (cid:15) ρ / / Π (cid:15) (cid:15) Γ st ρ / / Π st . (6.3)Here ρ may not exist in general, but will be required in the following definition.On the other hand, given a stable solution v over a singular curve C . By cuttingoff edges in certain subset of E p Γ q , one obtains a possibly disconnected graph Γ . Eachconnected component gives a stable solution. Suppose we have a surjective graph map ρ : Γ Ñ Π and Π has only one stable vertex. Then each loop in Π
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 49
Definition 6.13.
Let C k be a sequence of smooth or nodal r -spin curve of type p g, n q and v k be a sequence of stable solutions over C k . Let C be another smooth or nodal r -spincurve of type p g, n q and v be a stable solution over C . We say that p C k , v k q convergesto p C , v q if after removing finitely many elements in the sequence, the sequence can bedivided into finitely many subsequences, and each subsequence satisfies the followingconditions. Without loss of generality, we assume there is only one subsequence, whichis still indexed by k .(a) The underlying r -spin dual graphs of C k are all isomorphic. Denote it by Π.(b) The sequence of stabilizations C st k converges to C st . Denote the underlying r -spindual graph of C by Γ. This implies that, given a local universal unfolding U Ñ V of C st , there are embeddings φ k : C st k Ñ U as described above. It also implies thatthere is a map of r -spin dual graphs ρ st : Γ st Ñ Π st (see Definition 6.6).(c) There is a map of r -spin dual graphs ρ : Γ Ñ Π such that its induced map betweenstabilizations (see Lemma 6.7) is ρ st .(d) For each vertex v P V p Π q , let ˜Π v Ă ˜Π be the connected component of the normal-ization ˜Π that contains v , which corresponds to a component ˜ C k,v of the normal-ization ˜ C k . Let ˜ v k,v be the restriction of v k to ˜ C k,v . The preimage Γ v : “ ρ ´ p ˜Π v q is an r -spin dual graph corresponding to a stable solution p C Γ v , v Γ v q by restriction.We require that ( v is either stable or unstable), the sequence p ˜ C k,v , ˜ v k,v q convergesto p C Γ v , v Γ v q in the sense of Definition 6.11 or Definition 6.12.There are the following immediate consequences of convergence, whose proofs arestandard and are left to the reader. Proposition 6.14.
Suppose a sequence of stable solutions p C k , v k q converges to a stablesolution p C , v q in the sense of Definition 6.13, then the following holds.(a) The limiting holonomies are preserved. More precisely, suppose the combinatorialtype of C is Γ . Then for each tail t a P T p Γ q , which corresponds (for all large k )a tail t k,a P T p Π k q , let r ¯ τ k,a s P r ¯ K s be the limiting holonomy of v k at t k,a and let r ¯ τ a s P r ¯ K s be the limiting holonomy of v at t a . Then lim k Ñ8 r ¯ τ k,a s “ r ¯ τ a s . (b) Suppose for all k the combinatorial type of v k are fixed and denoted by Π . Forany t P T p ˜Π q , which corresponds (via the map ρ : Γ Ñ Π in the definition ofconvergence) to a tail t P T p ˜Γ q , let the limiting holonomy of v k at t be r ¯ τ k,t s andlet the limiting holonomy of v at t be r ¯ τ t s . Then lim k Ñ8 r ¯ τ k,t s “ r ¯ τ t s . (c) The homology classes are preserved. More precisely, let the homology class of p C k , v k q be ¯ B k and let the homology class of p C , v q be ¯ B . Then lim k Ñ8 ¯ B k “ ¯ B P H ¯ K p X ; Q q . Indeed for large k , ¯ B k “ ¯ B .(d) The evaluation maps are continuous. More precisely, using the notation of Item(a), if the evaluations of v k at t k,a are ¯ x k,a P ¯ X r ¯ τ k,a s W and the evaluation of v at t a is ¯ x a P ¯ X r ¯ τ a s W , then lim k Ñ8 ¯ x k,a “ ¯ x a . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 50
In the situation of Item (b), if the evaluations of v k at t are ¯ x k,t P ¯ X r ¯ τ k,t s W and theevaluation of v at t is ¯ x t P ¯ X r ¯ τ t s W , then lim k Ñ8 ¯ x k,t “ ¯ x t . One can also show that the notion of sequential convergence defined by Definition 6.13descends to a notion of sequential convergence in the moduli space M rg,n p X, G, W, µ q .By Proposition 6.14, the subsets M rg,n p X, G, W, µ ; ¯ B q Ă M rg,n p X, G, W, µ q are closedunder the sequential convergence. Given a positive number E ą
0, we also define M rg,n p X, G, W, µ q ď E to be the set of isomorphisms classes of solutions whose energiesare at most E .We explain why the notion of sequential convergence induces a unique topology on themoduli spaces. The reason is similar to the case of Gromov–Witten theory and it hasbeen fully explained in [MS04, Section 5.6] in the context for pseudoholomorphic spheres.In the current setting, since we can regard solutions to the gauged Witten equation asspecial vortices over cylindrical ends with structure group ¯ K , the topology is induced inthe same way as [Ven15, Section 5]. We state the result as the following proposition. Proposition 6.15.
Define a topology on M rg,n p X, G, W, µ q as follows. A subset of M rg,n p X, G, W, µ q is closed if it is closed under the sequential convergence. Then thistopology is Hausdorff, first countable, and the set of converging sequences in this topologycoincides with the set of converging sequences defined by Definition 6.13. The next job is to prove the compactness. It suffices to prove sequential compactness.Thanks to the uniform C bound proved in Theorem 5.11, the problem can be reducedto the situation of [Ven15]. Theorem 6.16. (Compactness) (a) For any E ą , M rg,n p X, G, W, µ q ď E is compact and Hausdorff.(b) For any ¯ B P H ¯ K p X ; Q q , M rg,n p X, G, W, µ ; ¯ B q is compact and Hausdorff.(c) For any stable decorated dual graph Γ with n tails and genus g , M Γ p X, G, W, µ q is a closed subset of M rg,n p X, G, W, µ q .Proof. To prove the first item, one only needs to prove sequential compactness. In-deed this is the same as the case of [Ven15] and we only sketch the key ingredients.Let p C k , v k q be a sequence of stable solutions which represent a sequence of points in M rg,n p X, G, W, µ q ď E . Theorem 5.11 implies that the solutions are all contained in acompact subset of X . Hence using the standard compactness results of vortices, whichcombines both the Gromov compactness of pseudoholomorphic curves and the Uhlen-beck compactness of connections, one can obtain a convergent subsequence. There isno bubbling due to the aspherical condition on X (otherwise one needs to incorporatethe result of Ott [Ott14]). Lastly, near punctures or when nodes are forming, energymay escape and form solitons in the limit. This can be analysed as proving compactnessof Floer trajectories. Theorem 6.3 implies that the number of soliton components inthe limit can only be finitely many. Hence we can prove the sequential compactness of M rg,n p X, G, W, µ q ď E . The second item then follows from the energy inequality (Theo-rem 5.12). Lastly, the closedness of the stratum M Γ p X, G, W, µ q follows from then fromProposition 6.14. (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 51
Part II
In Section 7 we provide a detailed abstract theory of virtual cycles in the topologicalcategory. In Section 8 we define the notion of cohomological field theories and state themain theorem of this paper. In Section 9–11 we construct virtual orbifold atlases andvirtual fundamental cycles on moduli spaces of the gauged Witten equation, and definethe correlation functions. In Section 12 we establish the expected properties of the virtualcycles which imply the axioms of CohFT.7.
Topological Virtual Orbifolds and Virtual Cycles
We recall the framework of constructing virtual fundamental cycles associated to mod-uli problems. Such constructions, usually called “virtual technique”, has a long historysince it first appeared in algebraic Gromov–Witten theory by [LT98a]. The currentmethod is based on the topological approach of [LT98b].Our current approach has been elaborated in [TX16b]. Here we provide more detailsin order to give more general construction and topological results and correct a few errorsin the earlier versions of [TX16b].7.1.
Topological manifolds and transversality.
In this subsection we review theclassical theory about topological manifolds and (microbundle) transversality.
Definition 7.1. (Topological manifolds and embeddings)(a) A topological manifold is a second countable Hausdorff space M which is locallyhomeomorphic to an open subset of R n .(b) A subset S Ă M is a submanifold if S equipped with the subspace topology is atopological manifold.(c) A map f : N Ñ M between two topological manifold is called a topologicalembedding if f is a homeomorphism onto its image.(d) A topological embedding f : N Ñ M is called locally flat if for any p P f p N q ,there is a local coordinate ϕ p : U p Ñ R m where U p Ă R n is an open neighborhoodof p such that ϕ p p f p N q X U p q Ă R n ˆ t u . In this paper, without further clarification, all embeddings of topological manifoldsare assumed to be locally flat. In fact we will always assume (or prove) the existence ofa normal microbundle which implies local flatness.7.1.1.
Microbundles.
The discussion of topological transversality needs the concept ofmicrobundles, which was introduced by Milnor [Mil64].
Definition 7.2. (Microbundles) Let B be a topological space.(a) A microbundle over a B is a triple p E, i, p q where E is a topological space, i : M Ñ E (the zero section map) and p : E Ñ M (the projection) are continuousmaps, satisfying the following conditions.(i) p ˝ i “ Id M .(ii) For each b P B there exist an open neighborhood U Ă B of b and an openneighborhood V Ă E of i p b q with i p U q Ă V , j p V q Ă U , such that there is a AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 52 homeomorphism V » U ˆ R n which makes the following diagram commutes. V (cid:15) (cid:15) p ❍❍❍❍❍❍❍❍❍❍ U i ; ; ✈✈✈✈✈✈✈✈✈✈ i ❍❍❍❍❍❍❍❍❍ UU ˆ R n p ; ; ✈✈✈✈✈✈✈✈✈ . (b) Two microbundles ξ “ p E, i, p q and ξ “ p E , i , p q over B are equivalent if thereare open neighborhoods of the zero sections W Ă E , W Ă E and a home-omorphism ρ : W Ñ W which is compatible with the structures of the twomicrobundles.Vector bundles and disk bundles are particular examples of microbundles. More gen-erally, an R n -bundle over a topological manifold M is a fibre bundle over M whose fibresare R n and whose structure group is the group of homeomorphisms of R n which fix theorigin. Notice that an R n -bundle has a continuous zero section, thus an R n -bundle isnaturally a microbundle. A very useful fact, which was proved by Kister [Kis64] andMazur [Maz64], says that microbundles are essentially R n -bundles. Theorem 7.3. (Kister–Mazur Theorem)
Let B be a topological manifold (or a weakerspace such as a locally finite simplicial complex) and ξ “ p E, i, p q be a microbundle over B . Then ξ is equivalent to an R n -bundle, and the isomorphism class of this R n -bundleis uniquely determined by ξ . However, R n -bundles are essentially different from vector bundles. For example, vectorbundles always contain disk bundles, which is not true for R n -bundles. Definition 7.4. (Normal microbundles) Let f : S Ñ M be a topological embedding.(a) A normal microbundle of f is a pair ξ “ p N, ν q where N Ă M is an open neigh-borhood of f p S q and ν : N Ñ S is a continuous map such that together with thenatural inclusion S ã Ñ N they form a microbundle over N . A normal microbundleis also called a tubular neighborhood .(b) Two normal microbundles ξ “ p N , ν q and ξ “ p N , ν q are equivalent if thereis another normal microbundle p N, ν q with N Ă N X N and ν | N “ ν | N “ ν. An equivalence class is called a germ of normal microbundles (or tubular neigh-borhoods).For example, for a smooth submanifold S Ă M in a smooth manifold, there is alwaysa normal microbundle. Its equivalence class is not unique though, as we need to choosethe projection map.7.1.2. Transversality.
We first recall the notion of microbundle transversality. Let Y bea topological manifold, X Ă Y be a submanifold and ξ “ p N, ν q be a normal microbundleof X . Let f : M Ñ Y be a continuous map. Definition 7.5. (Microbundle transversality) Let Y be a topological manifold, X Ă Y be a submanifold and ξ X be a normal microbundle of X . Let f : M Ñ Y be a continuousmap. We say that f is transverse to ξ if the following conditions are satisfied.(a) f ´ p X q is a submanifold of M . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 53 (b) There is a normal microbundle ξ “ p N , ν q of f ´ p X q Ă M such that the follow-ing diagram commute N (cid:15) (cid:15) f / / N (cid:15) (cid:15) f ´ p X q / / X and the inclusion f : N Ñ N induces an equivalence of microbundles.More generally, if C Ă M is any subset, then we say that f is transverse to X near C ifthe restriction of f to an open neighborhood of C is transverse to X .It is easy to see that the notion of being transverse to ξ only depends on the germ of ξ . Remark . The notion of microbundle transversality looks too restrictive at the firstglance. For example, the line x “ y in R intersects transversely with the x -axis in thesmooth category, however, the line is not transverse to the x -axis with respect to thenatural normal microbundle given by the projection p x, y q Ñ p x, q .The following theorem, which is of significant importance in our virtual cycle construc-tion, shows that one can achieve transversality by arbitrary small perturbations. Theorem 7.7. (Topological transversality theorem)
Let Y be a topological manifold and X Ă Y be a proper submanifold. Let ξ be a normal microbundle of X . Let C Ă D Ă Y be closed sets. Suppose f : M Ñ Y is a continuous map which is microbundle transverseto ξ near C . Then there exists a homotopic map g : M Ñ Y which is transverse to ξ over D such that the homotopy between f and g is supported in an open neighborhood of f ´ pp D r C q X X q .Remark . The theorem was proved by Kirby–Siebenmann [KS77] with a restrictionon the dimensions of M , X and Y . Then Quinn [Qui82] [Qui88] [FQ90] completedthe proof of the remaining cases. Notice that in [Qui88], the transversality theorem isstated for an embedding i : M Ñ Y and the perturbation can be made through anisotopy. This implies the above transversality result for maps as we can identify a map f : M Ñ Y with its graph ˜ f : M Ñ M ˆ Y , and an isotopic embedding of ˜ f , written as˜ g p x q “ p g p x q , g p x qq , can be made transverse to the submanifold ˜ X “ M ˆ X Ă M ˆ Y with respect to the induced normal microbundle ˜ ξ . Then it is easy to see that it isequivalent to g : M Ñ Y being transverse to ξ .In most of the situations of this paper, the notion of transversality is about sectionsof vector bundles. Suppose f : M Ñ R n is a continuous map. The origin 0 P R n has acanonical normal microbundle. Therefore, one can define the notion of transversality for f as a special case of Definition 7.5. Now suppose E Ñ M is an R n -bundle and ϕ U : E | U Ñ U ˆ R n is a local trivialization. Each section s : M Ñ E induces a map s U : U Ñ R n . Thenwe say that s is transverse over U if s U is transverse to the origin of R n . This notion isclearly independent of the choice of local trivializations. Then s is said to be transverseif it is transverse over a sufficiently small neighborhood of every point of M .Notice that the zero section of E has a canonical normal microbundle in the total space,and this notion of transversality for sections never agrees with the notion of transversalityfor graphs of the sections with respect to this canonical normal microbundle. Hence there AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 54 is an issue about whether this transversality notion for sections behaves as well as themicrobundle transversality.
Theorem 7.9.
Let M be a topological manifold and E Ñ M be an R n -bundle. Let C Ă D Ă M be closed subsets. Let s : M Ñ E be a continuous section which istransverse near C . Then there exists another continuous section which is transverse near D and which agrees with s over a small neighborhood of C .Proof. The difficulty is that the graph of s is not microbundle transverse to the zerosection, hence we cannot directly apply the topological transversality theorem (Theorem7.7). Hence we need to use local trivializations view the section locally as a map into R n . For each p P D , choose a precompact open neighborhood U p Ă M of p and a localtrivialization ϕ p : E | U p » U p ˆ R n . All U p form an open cover of D . M is paracompact, so is D . Hence there exists a locallyfinite refinement with induced local trivializations. Moreover, D is Lindel¨of, hence thisrefinement has a countable subcover, denoted by t U i u i “ . Over each U i there is aninduced trivialization of E .We claim that there exists precompact open subsets V i Ă U i such that t V i u i “ stillcover D . We construct V i inductively. Indeed, a topological manifold satisfies the T -axiom, hence we can use open sets to separate the two closed subsets D r ğ i ‰ U i , D r U . This provides a precompact V i Ă U i such that replacing U by V one still has an opencover of D . Suppose we can find V , . . . , V k so that replacing U , . . . , U k by V , . . . , V k stillgives an open cover of D . Then one can obtain V k ` Ă U k ` to continue the induction.We see that t V i u i “ is an open cover of D because every point p P D is contained in atmost finitely many U i .Now take an open neighborhood U C Ă M of C over which s is transverse. Since M isa manifold, one can separate the two closed subsets C and M r U C by a cut-off function ρ C : M Ñ r´ , s such that ρ ´ C p´ q “ C, ρ ´ C p q “ M r U C . Define a sequence of open sets C k “ ρ ´ C pr´ , k ` qq . Similarly, we can choose a sequence of shrinkings V i Ă ¨ ¨ ¨ Ă V k ` i Ă V ki Ă ¨ ¨ ¨ Ă U i . Define W k “ C k Y k ď i “ V ki which is a sequence of open subsets of M .Now we start an inductive construction. First, over U , the section can be identifiedwith a map s : U Ñ R n . By our assumption, s is transverse over U C X U . Thenapply the theorem for the pair of closed subsets C X U Ă p C X U q Y V of U . Thenone can modify it so that it becomes transverse near p C X U q Y V , and the change is AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 55 only supported in a small neighborhood of V r C . This modified section still agreeswith the original section near the boundary of U , hence still defines a section of E . Italso agrees with the original section over a neighborhood of C . Moreover, the modifiedsection is transverse near W : “ C Y V . Now suppose we have modified the section so that it is transverse near W k : “ C k Y k ď i “ V ki , and such that s agrees with the original section over an open neighborhood of C k ` .Then by similar method, one can modify s (via the local trivialization over U k ` ) to asection which agrees with s over a neighborhood of C k ` Y k ď i “ V k ` i and is transverse near W k ` . In particular this section still agrees with the very originalsection over C k ` .We claim that this induction process provides a section s of E which satisfies therequirement. Indeed, since the open cover t U i u i “ of D is locally finite, the value ofthe section becomes stabilized after finitely many steps of the induction, hence defines acontinuous section. Moreover, in each step the value of the section remains unchangedover the open set ρ ´ C pr´ , qq . Transversality also holds by construction. (cid:3) Corollary 7.10.
Let M be a topological manifold with or without boundary, and let s , s : M Ñ E be two transverse sections which are homotopic. Then the two submanifolds (withor without boundary) S : “ s ´ p q and S : “ s ´ p q are cobordant.Remark . In the application of this paper, the target pair p X, Y q in the transversalityproblem is either a smooth submanifold inside a smooth manifold (or orbifolds), or thezero section of a vector bundle. Hence X admits a tubular neighborhood and a uniqueequivalence class of normal microbundle. In fact the normal microbundle is equivalent toa disk bundle of the smooth normal bundle. Hence in the remaining discussions, we makethe stronger assumption that all normal microbundles are disk bundles of some vectorbundle. This does not alter the above discussion. For example, the compositions of twoembeddings with disk bundle neighborhoods is still an embedding with a disk bundleneighborhood.7.2. Topological orbifolds and orbibundles.
We use Satake’s notion of V-manifolds[Sat56] instead of groupoids to treat orbifolds, and only discuss it in the topologicalcategory. However it is necessary to consider non-effective orbifolds. For example, theDeligne–Mumford spaces M , and M , are not effective. Definition 7.12.
Let M be a second countable Hausdorff topological space.(a) Let x P M be a point. A topological orbifold chart (with boundary) of x consistsof a triple p ˜ U x , Γ x , ϕ x q , where ˜ U x is a topological manifold with possibly emptyboundary B ˜ U x , Γ x is a finite group acting continuously on p ˜ U x , B ˜ U x q and ϕ x : ˜ U x { Γ x Ñ M AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 56 is a continuous map which is a homeomorphism onto an open neighborhood of x .Denote the image U x “ ϕ x p ˜ U x { Γ x q Ă M and denote the composition˜ ϕ x : ˜ U x / / ˜ U x { Γ x ϕ x / / M . (b) If p P ˜ U x , take Γ p “ p Γ x q p Ă Γ x the stabilizer of p . Let ˜ U p Ă ˜ U x be a Γ p -invariantneighborhood of p . Then there is an induced chart (which we call a subchart) p ˜ U p , Γ p , ϕ p q , where ϕ p is the composition ϕ p : ˜ U p { Γ p (cid:31) (cid:127) / / ˜ U x { Γ x ϕ p / / M . (c) Two charts p ˜ U x , Γ x , ϕ x q and p ˜ U y , Γ y , ϕ y q are compatible if for any p P ˜ U x and q P ˜ U y with ϕ x p p q “ ϕ y p q q P M , there exist an isomorphism Γ p Ñ Γ q , subcharts˜ U p Q p , ˜ U q Q q and an equivariant homeomorphism ϕ qp : p ˜ U p , B ˜ U p q » p ˜ U q , B ˜ U q q .(d) A topological orbifold atlas of M is a SET tp ˜ U α , Γ α , ϕ α q | α P I u of topologi-cal orbifold charts of M such that M “ Ť α P I U α and for each pair α, β P I , p ˜ U α , Γ α , ϕ α q , p ˜ U β , Γ β , ϕ β q are compatible. Two atlases are equivalent if the unionof them is still an atlas. A structure of topological orbifold (with boundary)is an equivalence class of atlases. A topological orbifold (with boundary) is asecond countable Hausdorff space with a structure of topological orbifold (withboundary).We will often skip the term “topological” in the rest of this paper.Now consider bundles. Let E , B be orbifolds and π : E Ñ B be a continuous map. Definition 7.13.
A vector bundle chart (resp. disk bundle chart) of π : E Ñ B is atuple p ˜ U, F n , Γ , ˆ ϕ, ϕ q where F n “ R n (resp. F n “ D n ), p ˜ U, Γ , ϕ q is a chart of B and p ˜ U ˆ F n , Γ , ˆ ϕ q is a chart of E , where Γ acts on F n via a representation Γ Ñ GL p R n q (resp. Γ Ñ O p n q ). The compatibility condition is required, namely, the following diagramcommutes. ˜ U ˆ F n { Γ ˆ ϕ / / ˜ π (cid:15) (cid:15) E π (cid:15) (cid:15) ˜ U { Γ ϕ / / B . If p ˜ U p , Γ p , ϕ p q is a subchart of p ˜ U, Γ , ϕ q , then one can restrict the bundle chart to ˜ π ´ p ˜ U p q .We can define the notion of compatibility between bundle charts, the notion of orbifoldbundle structures and the notion of orbifold bundles in a similar fashion as in the caseof orbifolds. We skip the details.7.2.1. Embeddings.
Now we consider embeddings for orbifolds and orbifold vector bun-dles. First we consider the case of manifolds. Let S and M be topological manifoldsand E Ñ S , F Ñ M be continuous vector bundles. Let φ : S Ñ M be a topologicalembedding. A bundle embedding covering φ is a continuous map p φ : E Ñ F which makesthe diagram E p φ / / (cid:15) (cid:15) F (cid:15) (cid:15) S φ / / M AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 57 commute and which is fibrewise a linear injective map. Since p φ determines φ , we alsocall p φ : E Ñ F a bundle embedding. Definition 7.14. (Orbifold embedding) Let S , M be orbifolds and f : S Ñ M is acontinuous map which is a homeomorphism onto its image. φ is called an embedding iffor any pair of orbifold charts, p ˜ U, Γ , ϕ q of S and p ˜ V, Π , ψ q of M , any pair of points p P ˜ U , q P ˜ V with φ p ϕ p p qq “ ψ p q q , there are subcharts p ˜ U p , Γ p , ϕ p q Ă p ˜ U, Γ , ϕ q and p ˜ V q , Π q , ψ q q Ăp ˜ V, Π , ψ q , an isomorphism Γ p » Π q and an equivariant locally flat embedding ˜ φ pq : ˜ U p Ñ ˜ V q such that the following diagram commutes.˜ U p ˜ ϕ p (cid:15) (cid:15) ˜ φ pq / / ˜ V q ˜ ψ q (cid:15) (cid:15) S φ / / M Multisections and perturbations.
The equivariant feature of the problem impliesthat transversality can only be achieved by multi-valued perturbations. Here we reviewbasic notions and facts about multisections. Our discussion mainly follows the treatmentof [FO99].
Definition 7.15. (Multimaps) Let A , B be sets, l P N , and S l p B q be the l -fold symmetricproduct of B .(a) An l -multimap f from A to B is a map f : A Ñ S l p B q . For another a P N , thereis a natural map m a : S l p B q Ñ S al p B q (7.1)by repeating each component a times.(b) If both A and B are acted by a finite group Γ , then we say that an l -multimap f : A Ñ S l p B q is Γ -equivariant if it is equivariant with respect to the Γ -actionon A and the induced Γ -action on S l p B q .(c) If A and B are both topological spaces, then an l -multimap f : A Ñ S l p B q iscalled continuous if it is continuous with respect to the topology on S l p B q inducedas a quotient of B l .(d) A continuous l -multimap f : A Ñ S l p B q is liftable if there are continuous maps f , . . . , f l : A Ñ B such that f p x q “ r f p x q , . . . , f l p x qs P S l p B q , @ x P A.f , . . . , f l are called branches of f .(e) An l -multimap f : A Ñ S l p B q and an l -multimap f : A Ñ S l p B q are called equivalent if there exists a common multiple l “ a l “ a l of l and l such that m a ˝ f “ m a ˝ f as l -multimaps from A to B .(f) Being equivalent is clearly reflexive, symmetric and transitive. A multimap from A to B , denoted by f : A m Ñ B , is an element of ˜ğ l ě Map p A, S l p B qq ¸ { „ where „ is the above equivalence relation. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 58
In the discussions in this paper, we often identify an l -multimap with its equivalenceclass as a multimap. Definition 7.16. (Multisections) Let M be a topological orbifold and E Ñ M be avector bundle.(a) A representative of a (continuous) multisection of E is a collection ! p ˜ U α , R k , Γ α , ˆ ϕ α , ϕ α ; s α , l α q | α P I ) where tp ˜ U α , R k , Γ α , ˆ ϕ α , ϕ α q | α P I u is a bundle atlas for E Ñ M and s α : ˜ U α Ñ S l α p R k q is a Γ α -equivariant continuous l α -multimap, satisfying the following com-patibility condition. ‚ For any p P ˜ U α and q P ˜ U β with ϕ α p p q “ ϕ β p q q P M , there exist subcharts˜ U p Ă ˜ U α , ˜ U q Ă ˜ U β , an isomorphism p ˆ ϕ pq , ϕ pq q of subcharts, a commonmultiple l “ a α l α “ a β l β of l α and l β , such thatˆ ϕ pq ˝ m a β ˝ s β | ˜ U q “ m a α ˝ s α | ˜ U p ˝ ϕ pq . (b) Two representatives are equivalent if their union is also a representative. Anequivalence class is called a multisection of E , denoted by s : O m Ñ E. (c) A multisection s : M m Ñ E is called locally liftable if for any p P M , there exists alocal representative p ˜ U p ˆ R k , Γ p , ˆ ϕ p , ϕ p ; s p , l p q such that s p : ˜ U p Ñ S l p p R k q whichis a liftable continuous l p -multimap.(d) A multisection s : M m Ñ E is called transverse if it is locally liftable and for anyliftable local representative s p : ˜ U p Ñ S l p R k q , all branches are transverse to theorigin of R k .The space of continuous multisections of E Ñ M , denoted by C m p M, E q , is acted bythe space of continuous functions C p M q on M by pointwise multiplication. C m p M, E q also has the structure of a commutative monoid, but not an abelian group. The additivestructure is defined as follows. If s , s : M m Ñ E are multisections, then for liftable localrepresentatives with branches s a : ˜ U Ñ R n , 1 ď a ď l , s b : ˜ U Ñ R n , 1 ď b ď k , define p s ` s q ab “ r s a ` s b s ď b ď k ď a ď l . However there is no inverse to this addition: one can only invert the operation of addinga single valued section. It is enough, though, since we have the notion of being transverseto a single valued section which is not necessarily the zero section.We also want to measure the size of multisections. A continuous norm on an orbifoldvector bundle E Ñ M is a continuous function } ¨ } : E Ñ r , `8q which only vanisheson the zero section such that over each local chart, it lifts to an equivariant norm on thefibres. It is easy to construct norms in the relative sense, as one can extend continuousfunctions defined on closed sets.The following lemma, which is a generalization of Theorem 7.9, shows one can achievetransversality for multisections by perturbation relative to a region where transversalityalready holds. Lemma 7.17.
Let M be an orbifold and E Ñ M be an orbifold vector bundle. Let C Ă D Ă M be closed subsets. Let S : M Ñ E be a single-valued continuous functionand t C : M m Ñ E be a multisection such that S ` t C is transverse over a neighborhood of C . Then there exists a multisection t D : M m Ñ E satisfying the following condition. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 59 (a) t C “ t D over a neighborhood of C .(b) S ` t D is transverse over a neighborhood of D .Moreover, if E has a continuous norm } ¨ } , then for any ǫ ą , one can require that } t D } C ď } t C } C ` ǫ. Proof.
Similar to the proof of Theorem 7.9, one can choose a countable locally finite opencover t U i u i “ of D satisfying the following conditions.(a) Each U i is compact.(b) There are a collection of precompact open subsets V i Ă U i such that t V i u i “ isstill an open cover of D .(c) Over each U i there is a local representative of t C , written as p ˜ U i ˆ R k , Γ i , ˆ ϕ i , ϕ i ; t i , l i q where t i : ˜ U i Ñ S l i p R k q is a Γ i -equivariant l i -multimap which is liftable. Write t i p x q “ r t i p x q , . . . , t l i i p x qs . In this chart also write S as a map S i : ˜ U i Ñ R k .The transversality assumption implies that there is an open neighborhood U C Ă M of C such that for each i , each a P t , . . . , l i u , s ai is transverse to the origin over˜ U i,C : “ ϕ ´ i p U C q Ă ˜ U C . Using an inductive construction which is very similar to that in the proof of Theorem7.9, one can construct a valid perturbation. We only sketch the construction for the firstchart. Indeed, one can perturb each t a over ˜ U to a function t a : ˜ U Ñ R k , such that S ` t a is transverse over a neighborhood of the closure of ˜ V : “ ϕ ´ p V q inside ˜ U , but t a “ t a over a neighborhood of ˜ U ,C and the near the boundary of ˜ U . Moreover, given ǫ ą x P ˜ U } t a p x q} ď sup x P ˜ U } t a p x q} ` ǫ . (7.2)Then we obtain a continuous l -multimap t p x q “ r t p x q , . . . , t l p x qs . This multimap may not be Γ -transverse. We reset t p x q : “ r t ab s ď b ď n ď a ď l : “ r g ´ b t a p g b x qs ď b ď n ď a ď l where Γ “ t g , . . . , g n u . It is easy to verify that this is Γ -invariant, and agrees with t over a neighborhood of ˜ U ,C and near the boundary of ˜ U . There still holdssup a,b sup x P ˜ U } t ab p x q} ď sup a sup x P ˜ U } t a p x q} ` ǫ . Therefore, together with the original multisection over the complement of U , t definesa continuous multisection of E . Moreover, it agrees with the original one over a neigh-borhood of C and is transverse near C Y V . In this way we can continue the inductionto perturb over all ˜ U i . At the k -th step of the induction, we replace ǫ by ǫ k in the C bound (7.2). Since U i is locally finite, near each point, the value of the perturbationstabilizes after finitely many steps of this induction. This results in a multisection t D which satisfies our requirement. (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 60
Virtual orbifold atlases.
Now we introduce the notion of virtual orbifold atlases.This notion plays a role as a general structure of moduli spaces we are interested in,and is a type of intermediate objects in concrete constructions. The eventual objects wewould like to construct are good coordinate systems , which are special types of virtualorbifold atlases.
Definition 7.18.
Let X be a compact and Hausdorff space.(a) A virtual orbifold chart (chart for short) is a tuple C : “ p U, E, S, ψ, F q where(i) U is a topological orbifold (with boundary).(ii) E Ñ U is a continuous orbifold vector bundle.(iii) S : U Ñ E is a continuous section.(iv) F Ă X is an open subset.(v) ψ : S ´ p q Ñ F is a homeomorphism. F is call the footprint of this chart C , and the integer dim U ´ rank E is called the virtual dimension of C .(b) Let C “ p U, E, S, ψ, F q be a chart and U Ă U be an open subset. The restriction of C to U is the chart C “ C | U “ p U , E , S , ψ , F q where E “ E | U , S “ S | U , ψ “ ψ | p S q ´ p q , and F “ Im ψ . Any such chart C induced from an open subset U Ă U is called a shrinking of C . A shrinking C “ C | U is called a precompact shrinking if U Ă U , denoted by C Ă C .A very useful lemma about shrinkings is the following, whose proof is left to the reader. Lemma 7.19.
Suppose C “ p U, E, S, ψ, F q is a virtual orbifold atlas and let F Ă F bean open subset. Then there exists a shrinking C of C whose footprints is F . Moreover,if F Ă F , then C can be chosen to be a precompact shrinking. Definition 7.20.
Let C i : “ p U i , E i , S i , ψ i , F i q , i “ , X . An embedding of C into C consists of a bundle embedding p φ satisfying the following conditions.(a) The following diagrams commute; E p φ / / π (cid:15) (cid:15) E π (cid:15) (cid:15) U S C C φ / / U S [ [ S ´ p q φ / / ψ (cid:15) (cid:15) S ´ p q ψ (cid:15) (cid:15) X Id / / X (b) (TANGENT BUNDLE CONDITION) There exists an open neighborhood N Ă U of φ p U q and a subbundle E Ă E | N which extends p φ p E q such that S | N : N Ñ E | N is transverse to E and S ´ p E q “ φ p U q .The following lemma is left to the reader. Lemma 7.21.
The composition of two embeddings is still an embedding.
Definition 7.22.
Let C i “ p U i , E i , S i , ψ i , F i q , p i “ , q be two charts. A coordinatechange from C to C is a triple T “ p U , φ , p φ q , where U Ă U is an open subsetand p φ , p φ q is an embedding from C | U to C . They should satisfy the followingconditions. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 61 (a) ψ p U X S ´ p qq “ F X F .(b) If x k P U converges to x P U and y k “ φ p x k q converges to y P U , then x P U and y “ φ p y q . Lemma 7.23.
Let C i “ p U i , E i , S i , ψ i , F i q , p i “ , q be two charts and let T “p U , p φ q be a coordinate change from C to C . Suppose C i “ C i | U i be a shrinkingof C i . Then the restriction T : “ T | U X φ ´ p U q is a coordinate change from C to C .Proof. Left to the reader. (cid:3)
We call T in the above lemma the induced coordinate change from the shrinking.Now we introduce the notion of atlases. Definition 7.24.
Let X be a compact metrizable space. A virtual orbifold atlas ofvirtual dimension d on X is a collection A : “ ´! C I : “ p U I , E I , S I , ψ I , F I q | I P I ) , ! T JI “ ` U JI , φ JI , p φ JI ˘ | I ď J )¯ , where(a) p I , ď q is a finite, partially ordered set.(b) For each I P I , C I is a virtual orbifold chart of virtual dimension d on X .(c) For I ď J , T JI is a coordinate change from C I to C J .They are subject to the following conditions. ‚ (COVERING CONDITION) X is covered by all the footprints F I . ‚ (COCYCLE CONDITION) For I ď J ď K P I , denote U KJI “ U KI X φ ´ JI p U KJ q Ă U I .Then we require that p φ KI | U KJI “ p φ KJ ˝ p φ JI | U KJI as bundle embeddings. ‚ (OVERLAPPING CONDITION) For
I, J P I , we have F I X F J ‰ H ùñ I ď J or J ď I. All virtual orbifold atlases considered in this paper have definite virtual dimensions,although sometimes we do not explicitly mention it.7.3.1.
Orientations.
Now we discuss orientation. When M is a topological manifold,there is an orientation bundle O M Ñ M which is a double cover of M (or a Z -principalbundle). M is orientable if and only if O M is trivial. If E Ñ M is a continuous vectorbundle, then E also has an orientation bundle O E Ñ M as a double cover. Since Z -principal bundles over a base B are classified by H p B ; Z q , the orientation bundles canbe multiplied. We use b to denote this multiplication. Definition 7.25. (Orientability)(a) A chart C “ p U, E, S, ψ, F q is locally orientable if for any bundle chart p ˜ U, R n , Γ , p ϕ, ϕ q ,if we denote ˜ E “ ˜ U ˆ R n , then for any γ P Γ , the map γ : O ˜ U b O ˜ E ˚ Ñ O ˜ U b O ˜ E ˚ is the identity over all fixed points of γ .(b) If C is locally orientable, then O ˜ U b O ˜ E ˚ for all local charts glue together a doublecover O C Ñ U . If O C is trivial (resp. trivialized), then we say that C is orientable (resp. oriented ). AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 62 (c) A coordinate change T “ p U , φ , p φ q between two oriented charts C “p U , E , S , ψ , F q and C “ p U , E , S , ψ , F q is called oriented if the embed-dings φ and p φ are compatible with the orientations on O C and O C .(d) An atlas A is oriented if all charts are oriented and all coordinate changes areoriented.7.4. Good coordinate systems.
Now we introduce the notion of shrinkings of virtualorbifold atlases. However, to obtain a good coordinate system, it is more convenient touse more general shrinkings, which we call generalized shrinkings.
Definition 7.26.
Let A “ pt C I | I P I u , t T JI | I ď J uq be a virtual orbifold atlas on X .(a) A shrinking of A is another virtual orbifold atlas A “ pt C I | I P I u , t T JI | I ď J uq indexed by elements of the same partially ordered set I such that for each I P I , C I is a shrinking C I | U I of C I and for each I ď J , T JI is the induced shrinking of T JI given by Lemma 7.23.(b) If for every I P I , U I is a precompact subset of U I , then we say that A is a precompact shrinking of A and denote A Ă A .Given a virtual orbifold atlas A “ pt C I | I P I u , t T JI | I ď J uq , we define a relation O on the disjoint union Ů I P I U I as follows. U I Q x O y P U J if one of the following holds.(a) I “ J and x “ y ;(b) I ď J , x P U JI and y “ φ JI p x q ;(c) J ď I , y P U IJ and x “ φ IJ p y q .If A is a shrinking of A , then it is easy to see that the relation O on Ů I P I U I defined asabove is induced from the relation O for A via restriction.For an atlas A , if O is an equivalence relation, we can form the quotient space | A | : “ ´ ğ I P I U I ¯ { O . with the quotient topology. There is a natural injective map X ã Ñ | A | . We call | A | the virtual neighborhood of X associated to the atlas A . Denote the quotientmap by π A : ğ I P I U I Ñ | A | (7.3)which induces continuous injections U I ã Ñ | A | . A point in | A | is denoted by | x | , whichhas certain representative x P U I for some I . Definition 7.27.
A virtual orbifold atlas A on X is called a good coordinate system ifthe following conditions are satisfied.(a) O is an equivalence relation.(b) The virtual neighborhood | A | is a Hausdorff space.(c) For all I P I , the natural maps U I Ñ | A | are homeomorphisms onto their images.The conditions for good coordinate systems are very useful for later constructions (thisis the same as in the Kuranishi approach, see [FOOO16]), for example, the construction ofsuitable multisection perturbations. In these constructions, the above conditions are oftenimplicitly used without explicit reference. Therefore, an important step is to constructgood coordinate systems. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 63
Theorem 7.28. (Constructing good coordinate system)
Let A be a virtual orbifold atlason X with the collection of footprints F I indexed by I P I . Let F ˝ I Ă F I for all I P I bea collection of precompact open subsets such that X “ ď I P I F ˝ I . Then there exists a generalized shrinking A of A such that the collection of shrunk foot-prints F I contains F ˝ I for all I P I and A is a good coordinate system.Moreover, if A is already a good coordinate system, then any shrinking of A remainsa good coordinate system. We give a proof of Theorem 7.28 in the next subsection. A similar result is used inthe Kuranishi approach while our argument potentially differs from that of [FOOO16].
Remark . If A is a good coordinate system, and A is a shrinking of A , then theshrinking induces a natural map | A | ã Ñ | A | . If we equip both | A | and | A | with the quotient topologies, then the natural map iscontinuous. However there is another topology on | A | by viewing it as a subset of | A | .We denote this topology by } A } and call it the subspace topology . In most cases, thequotient topology is strictly stronger than the subspace topology. Hence it is necessaryto distinguish the two different topologies.7.5. Shrinking good coordinate systems.
In this subsection we prove Theorem 7.28.First we show that by precompact shrinkings one can make the relation O an equivalencerelation. Lemma 7.30.
Let A be a virtual orbifold atlas on X with the collection of footprints t F I | I P I u . Let F ˝ I Ă F I be precompact open subsets such that X “ ď I P I F ˝ I . Then there exists a precompact shrinking A of A whose collection of footprints F I con-tains F ˝ I for all I P I such that the relation O on A is an equivalence relation.Proof. By definition, the relation O is reflexive and symmetric. By the comments above,any shrinking of A will preserve reflexiveness and symmetry. Hence we only need toshrink the atlas to make the induced relation transitive.For any subset I P I , we say that O is transitive in I if for x, y, z P Ů I P I U I , x O y , y O z imply x O z . Being transitive in any subset I is a condition that is preserved undershrinking. Hence it suffices to construct shrinkings such that O is transitive in t I, J, K u for any three distinct elements I, J, K P I . Let x P U I , y P U J , z P U K be generalelements.Since U I (resp. U J resp. U K ) is an orbifold and hence metrizable, we can choosea sequence of precompact open subsets U nI Ă U I (resp. U nJ Ă U J resp. U nK Ă U K )containing F ˝ I (resp. F ˝ J resp. F ˝ K ) such that U n ` I Ă U nI , U n ` J Ă U nJ , U n ` K Ă U nK , and č n U nI “ ψ ´ I p F ˝ I q , č n U nJ “ ψ ´ J p F ˝ J q , č n U nK “ ψ ´ K p F ˝ K q . Then for each n , U nI , U nJ , U nK induce a shrinking of the atlas A , denoted by A n . Let theinduced binary relation on U nI \ U nJ \ U nK still by O . We claim that for n large enough, O AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 64 is an equivalence relation on this triple disjoint union. Denote the domains of the shrunkcoordinate changes by U nJI , U nKJ and U nKI respectively.If this is not true, then without loss of generality, we may assume that for all large n ,there exist points x n P U nI , y n P U nJ , z n P U nK such that x n O y n , y n O z n , but p x n , z n q R O ; (7.4)Then for some subsequence (still indexed by n ), x n , y n and z n converge to x P ψ ´ I p F ˝ I q Ă U I , y P ψ ´ J p F ˝ J q Ă U J and z P ψ ´ K p F ˝ K q Ă U K respectively. Thenby the definition of coordinate changes (Definition 7.22), one has x O y , y O z ùñ ψ I p x q “ ψ J p y q “ ψ K p z q P F ˝ I X F ˝ J X F ˝ K . By the (OVERLAPPING CONDITION) of Definition 7.24, t I, J, K u is totally ordered. Sincethe roles of K and I are symmetric, we may assume that I ď K . Then since x P ψ ´ I ` F ˝ I X F ˝ K ˘ Ă ψ ´ I p F I X F K q “ ψ ´ I p F KI q Ă U KI and U KI Ă U I is an open set, for n large enough one has x n P U KI . (a) If I ď J ď K , then by (COCYCLE CONDITION) of Definition 7.24, φ KI p x n q “ φ KJ p φ JI p x n qq “ φ KJ p y n q “ z n . So x n O z n , which contradicts (7.4).(b) If J ď I ď K , then (COCYCLE CONDITION) of Definition 7.24, z n “ φ KJ p y n q “ φ KI ` φ IJ p y n q ˘ “ φ KI p x n q . So x n O z n , which contradicts (7.4).(c) If I ď K ď J , then since φ KI p x q P ψ ´ K p F J X F K q Ă U JK , for large n , x n P φ ´ KI p U JK q . Then by (COCYCLE CONDITION) of Definition 7.24, φ JK p z n q “ y n “ φ JI p x n q “ φ JK ` φ KI p x n q ˘ . Since φ JK is an embedding, we have φ KI p x n q “ z n . Therefore, x n O z n , whichcontradicts (7.4).Therefore, O n is an equivalence relation on U nI \ U nJ \ U nK for large enough n . Wecan perform the shrinking for any triple of elements of I , which eventually makes O anequivalence relation. By the construction the shrunk footprints F I still contain F ˝ I . (cid:3) Lemma 7.31.
Suppose A is a virtual orbifold atlas on X such that the relation O isan equivalence relation. Suppose there is a collection of precompact subsets F ˝ I Ă F I offootprints of A such that X “ ď I P I F ˝ I . Then there exists a precompact shrinking A Ă A satisfying(a) The shrunk footprints F I all contain F ˝ I .(b) The virtual neighborhood | A | is a Hausdorff space. Before proving Lemma 7.31, we need some preparations. Order the finite set I as t I , . . . , I m u such that for k “ , . . . , m , I k ď J ùñ J P t I k , I k ` , . . . , I m u . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 65
For each k , O induces an equivalence relation on Ů i ě k U I i and denote the quotient spaceby | A k | . Then the map π A of (7.3) induces a natural continuous map π k : ğ i ě k U I i Ñ | A k | . Lemma 7.32.
For k “ , . . . , m , if | A k | is Hausdorff and A is a shrinking of A , then | A | is also Hausdorff.Proof. Left to the reader. A general fact is that the quotient topology is always strongerthan (or homeomorphic to) the subspace topology (see Remark 7.29). (cid:3)
Lemma 7.33.
The natural map | A k ` | Ñ | A k | (7.5) is a homeomorphism onto an open subset.Proof. The map is clearly continuous and injective. To show that it is a homeomorphismonto an open set, consider any open subset O k ` of its domain. Its preimage under thequotient map U I k ` \ ¨ ¨ ¨ \ U I m Ñ | A k ` | is denoted by ˜ O k ` “ O I k ` \ ¨ ¨ ¨ \ O I m , where O I k ` , . . . , O I m are open subsets of U I k ` , . . . , U I m respectively. Define O I k : “ ď i ě k ` , I i ě I k φ ´ I i I k p O I i q . This is an open subset of U I k . Then the image of O k ` under the map (7.5), denoted by O k , is the quotient of˜ O k : “ O I k \ O I k ` \ ¨ ¨ ¨ \ O I m Ă U I k \ ¨ ¨ ¨ \ U I m On the other hand, ˜ O k is exactly the preimage of O k under the quotient map. Hence bythe definition of the quotient topology, O k is open. This show that (7.5) is a homeomor-phism onto an open subset. (cid:3) Proof of Lemma 7.31.
For each k , we would like to construct shrinkings U I i Ă U I i for all i ě k such that | A k | is Hausdorff and the shrunk footprints F I i contains F ˝ I i for all i ě k .Our construction is based on a top-down induction. First, for k “ m , | A m | » U I m andhence is Hausdorff. Suppose after shrinking | A k ` | is already Hausdorff.Choose open subsets F I i Ă F I i for all i ě k such that F ˝ I i Ă F I i , X “ ď i ě k F I i Y ď i ď k ´ F I i . Choose precompact open subsets U I i Ă U I i for all i ě k such that ψ I i p U I i X S ´ I i p qq “ F I i , ψ I i p U I i X S ´ I i p qq “ F I i . Then U I i for i ě k and U I i for i ă k provide a shrinking A of A . We claim that | A k | isHausdorff.Indeed, pick any two different points | x | , | y | P | A k | . If | x | , | y | P | A k ` | Ă | A k | , thenby the induction hypothesis and Lemma 7.32, | x | and | y | can be separated by two opensubsets in | A k ` | . Then by Lemma 7.33, these two open sets are also open sets in | A k | .Hence we assume that one or both of | x | and | y | are in | A k | r | A k ` | . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 66
Case 1.
Suppose | x | and | y | are represented by x, y P U I k . Choose a distance function on U I k which induces the same topology. Let O ǫx and O ǫy be the open ǫ -balls in U I k centeredat x and y respectively. Then for ǫ small enough, O ǫx X O ǫy “ H . Claim.
For ǫ sufficiently small, for all I k ď I i and I k ď I j , one has π k ` ` φ I i I k p O ǫx X U I i I k q ˘ X π k ` ` φ I j I k p O ǫy X U I j I k q ˘ “ H . Here the closures are the closures in U I i and U I j respectively. Proof of the claim.
Suppose it is not the case, then there exist a sequence ǫ n Ñ I k ď I i , I k ď I j , and a sequence of points | z n | P π k ` ` φ I i I k p O ǫ n x X U I i I k q ˘ X π k ` ` φ I j I k p O ǫy X U I j I k q ˘ Ă | A k ` | . Then | z n | has its representative p n P φ I i I k p O ǫ n x X U I k I i q Ă U I i and its representative q n P φ I j I k p O ǫ n y X U I j I k q Ă U I j . Then p n O q n and without loss of generality, assume that I i ď I j . Then p n P U I j I i and q n “ φ I j I i p p n q .Choose distance functions d i on U I i and d j on U I j which induce the same topologies.Then one can choose x n P O ǫ n x X U I i I k and y n P O ǫ n y X U I j I k such that d i p p n , φ I i I k p x n qq ď ǫ n , d p q n , φ I i I k p y n qq ď ǫ n . (7.6)Since U I i and U I j are compact and p n P U I i , q n P U I j , for some subsequence (still indexedby n ), p n converges to some p P U I i and q n converges to some q P U I j . Then p O q .Moreover, by (7.6), one haslim n Ñ8 φ I i I k p x n q “ p , lim n Ñ8 φ I j I k p y n q “ q . On the other hand, x n converges to x and y n converges to y . By the property of coordinatechanges, one has that x P U I i I k , y P U I j I k and x O p O q O y. Since O is an equivalence relation and it remains an equivalence relation after shrinking, x O y , which contradicts x ‰ y . End of the proof of the claim.
Now choose such an ǫ and abbreviate O x “ O ǫx , O y “ O ǫy . Denote P I i “ φ I i I k p O x X U I i I k q Ă U I i , Q I i “ φ I i I k p O y X U I i I k q Ă U I i (which could be empty). They are all compact, hence P k ` : “ π k ` p ğ i ě k ` P I i q Ă | A k ` | , Q k ` : “ π k ` p ğ i ě k ` Q I i q Ă | A k ` | are both compact. The above claim implies that P k ` X Q k ` “ H . Then by theinduction hypothesis which says that | A k ` | is Hausdorff, they can be separated by opensets V k ` , W k ` Ă | A k ` | . Write π ´ k ` ` V k ` ˘ “ ğ i ě k ` V I i , π ´ k ` ` W k ` ˘ “ ğ i ě k ` W I i . Define V I i “ V I i X U I i , W I i “ W I i X U I i and V I k : “ O x Y ď I k ď I i φ ´ I i I k p V I i q X U I k , W I k : “ O y Y ď I k ď I i φ ´ I i I k p W I i q X U I k AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 67 and V k : “ π k ` ğ i ě k V I i ˘ Ă | A k | , W k : “ π k ` ğ i ě k W I i ˘ Ă | A k | . It is easy to check that V k and W k are disjoint open subsets containing | x | and | y | respectively. Therefore | x | and | y | are separated in | A k | . Case 2.
Now suppose | x | is represented by x P U I k and | y | P | A k ` | . Similar to Case 1 ,we claim that for ǫ sufficiently small, for all I k ď I i , one has | y | R π k ` ` φ I i I k p O ǫx q X U I i I k ˘ “ : P I i Ă U I i . The proof is similar and is omitted. Then choose such an ǫ and abbreviate O x “ O ǫx . P I i are compact sets and so is P k ` : “ π k ` ` ğ i ě k ` P I i ˘ Ă | A k ` | . The above claim implies that | y | R P k ` . Then by the induction hypothesis, | y | and P k ` can be separated by open sets V k ` and W k ` of | A k ` | . By similar procedure as in Case 1 above, one can produce two open subsets of | A k | which separate | x | and | y | .Therefore, we can finish the induction and construct a shrinking such that | A | isHausdorff. Eventually the shrunk footprints still contain F ˝ I . (cid:3) Now we can finish proving Theorem 7.28. Suppose | A | is Hausdorff. By the definitionof the quotient topology, the natural map U I ã Ñ | A | is continuous. Since U I is locallycompact and | A | is Hausdorff, a further shrinking can make this map a homeomorphismonto its image. This uses the fact that a continuous bijection from a compact spaceto a Hausdorff space is necessarily a homeomorphism. Hence the third condition for agood coordinate system is satisfied by a precompact shrinking of A , and this conditionis preserved for any further shrinking. This establishes Theorem 7.28.7.6. Perturbations.
Now we define the notion of perturbations.
Definition 7.34.
Let A be a good coordinate system on X .(a) A multi-valued perturbation of A , simply called a perturbation , denoted by t , con-sists of a collection of multi-valued continuous sections t I : U I m Ñ E I satisfying (as multisections) t J ˝ φ JI “ p φ JI ˝ t I | U JI . (b) Given a multi-valued perturbation t , the object˜ s “ ´ ˜ s I “ S I ` t I : U I m Ñ E I ¯ satisfies the same compatibility condition with respect to coordinate changes. Theperturbation t is called transverse if every ˜ s I is a transverse multisection.(c) Suppose A is infinitesimally thickened by N “ tp N JI , E I ; J q | I ď J u . We say that t is N -normal if for all I ď J , one has t J p N JI q Ă E I ; J | N JI . (7.7) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 68 (d) The zero locus of a perturbed ˜ s gives objects in various different categories. De-note Z “ ğ I P I ˜ s ´ I p q . It is naturally equipped with the topology induced from the disjoint union of U I .Denote by | Z | : “ Z { O . the quotient of Z , which is equipped with the quotient topology. Furthermore,there is a natural injection | Z | ã Ñ | A | . Denote by } Z } the same set as | Z | butequipped with the topology as a subspace of | A | .In order to construct suitable perturbations of a good coordinate system, we needcertain tubular neighborhood structures with respect to coordinate changes. In ourtopological situation, it is sufficient to have some weaker structure near the embeddingimages. Definition 7.35.
Let A be a good coordinate system with charts indexed by elementsin a finite partially ordered set p I , ď q and coordinate changes indexed by pairs I ď J P I .A thickening of A is a collection of objects p N JI , E I ; J q | I ď J ( where N JI Ă U J is an open neighborhood of φ JI p U JI q and E I ; J is a subbundle of E J | N JI .They are required to satisfy the following conditions.(a) If I ď K , J ď K but there is no partial order relation between I and J , then N KI X N KJ “ H . (7.8)(b) For all triples I ď J ď K , E I ; J | φ ´ KJ p N KI qX N JI “ p φ ´ KJ p E I ; K q| φ ´ KJ p N KI qX N JI . (c) For all triples I ď J ď K , one has E I ; K | N KI X N KJ Ă E J ; K | N KI X N KJ . (d) Each p N JI , E I ; J q satisfies the (TANGENT BUNDLE CONDITION) of Definition 7.20. Remark . The above setting is slightly more general than what we need in our appli-cation in this paper and the companion [TX16b]. In this paper we will see the followingsituation in the concrete cases.(a) The index set I consists of certain nonempty subsets of a finite set t , . . . , m u ,which has a natural partial order given by inclusions.(b) For each i P I , Γ i is a finite group and Γ I “ Π i P I Γ i . U I “ ˜ U I { Γ I where ˜ U I is atopological manifold acted by Γ I . Moreover, E , . . . , E m are vector spaces actedby Γ i and the orbifold bundle E I Ñ U I is the quotient E I : “ p ˜ U I ˆ E I q{ Γ I , where E I : “ à i P I E i . (c) For I ď J , U JI “ ˜ U JI { Γ I where ˜ U JI Ă ˜ U I is a Γ I -invariant open subset and thecoordinate change is induced from the following diagram˜ V JI / / (cid:15) (cid:15) ˜ U J ˜ U JI (7.9) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 69
Here ˜ V JI Ñ ˜ U JI is a covering space with group of deck transformations identicalto Γ J ´ I “ Π j P J ´ I Γ j ; then Γ J acts on ˜ V JI and ˜ V JI Ñ ˜ U J is a Γ J -equivariantembedding of manifolds, which induces an orbifold embedding U JI Ñ U J and anorbibundle embedding E I | U JI Ñ E J .In this situation, one naturally has subbundles E I ; J Ă E J for all pairs I ď J . Hence athickening of such a good coordinate system is essentially only a collection of neighbor-hoods N JI of φ JI p U JI q which satisfy (7.8) and S ´ J p E I ; J q X N JI “ φ JI p U JI q . Theorem 7.37.
Let X be a compact Hausdorff space and have a good coordinate system A “ ´ C I “ p U I , E I , S I , ψ I , F I q | I P I ( , T JI | I ď J (¯ . Let A Ă A be any precompact shrinking. Let N “ tp N JI , E I ; J qu be a thickening of A and N JI Ă U J be a collection of open neighborhoods of φ JI p U JI q such that N JI Ă N JI . Thenthey induce a thickening N of A by restriction. Let d I : U I ˆ U I Ñ r , `8q be a distancefunction on U I which induces the same topology as U I . Let ǫ ą be a constant. Thenthere exist a collection of multisections t I : U I m Ñ E I satisfying the following conditions.(a) For each I P I , ˜ s I : “ S I ` t I is transverse.(b) For each I P I , d I ` ˜ s ´ I p q X U I , S ´ I p q X U I ˘ ď ǫ. (7.10) (c) For each pair I ď J , we have p φ JI ˝ t I | U JI “ t J ˝ φ JI | U JI . Hence the collection of restrictions t I : “ t I | U I defines a perturbation t of A .(d) t is N -normal.Proof. To simplify the proof, we assume that we are in the situation described by Remark7.36. The general case requires minor modifications in a few places. Then the subbundles E I b ,I a are naturally define over U I b .We use the inductive construction. Order the set I as I , . . . , I m such that I k ď I l ùñ k ď l. For each k, l with k ă l , define open sets N ´ I l ,k by N ´ I l ,k “ ď a ď k,I a ă I l N I l I a . Define open sets N ` I l ,k inductively. First N ` I m ,k “ H . Then N ` I l ,k : “ ď I l ă I b φ ´ I b I l p N I b ,k q , N I l ,k “ N ´ I l ,k Y N ` I l ,k . Replacing N JI by N JI in the above definitions, we obtain N I l ,k Ă U I l with N I l ,k Ă N I l ,k . If k ě l , define N I l ,k “ U I l , N I l ,k “ U I l . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 70
On the other hand, it is not hard to inductively choose a system of continuous normson E I such that, for all pairs I a ď I b , the bundle embedding p φ I b I a is isometric. Givensuch a collection of norms, choose δ ą I , d I ` x, S ´ I p q X U I ˘ ą ǫ, x P U I ùñ } S I p x q} ě p m ` q δ. (7.11)Now we reformulate the problem in an inductive fashion. We would like to verify thefollowing induction hypothesis. Induction Hypothesis.
For a, k “ , . . . , m , there exists an open subset O I a ,k Ă N I a ,k which contains N I a ,k and multisections t I a ,k : O I a ,k m Ñ E I a . They satisfy the following conditions.(a) For all pairs I a ď I b , over a neighborhood of the compact subset N I a ,k X φ ´ I b I a p N I b ,k q Ă U I b I a Ă U I b I a one has t I b ,k ˝ φ I b I a “ p φ I b I a ˝ t I a ,k . (7.12)(b) In a neighborhood of N I b I a , the value of t I b ,k is contained in the subbundle E I a ; I b .(c) S I a ` t I a ,k is transverse.(d) } t I a ,k } C ď kδ .It is easy to see that the k “ m case implies this theorem. Indeed, (7.10) followsfrom (7.11) and the bound } t I l ,m } ď mδ . Now we verify the conditions of the inductionhypothesis. For the base case, apply Lemma 7.17 to M “ U I , C “ H , D “ U I , we can construct a multisection t I , : U I m Ñ E I making S I ` t I transverse with } t I } ď δ . Now we construct t I a , for a “ , . . . , m via a backward induction. Define O I a , “ N I a , “ N I a I Y ď I a ă I b φ ´ I b I a p N I b , q , a “ , . . . , m. Then (7.12) determines the value of t I m , over the set φ I m I p U I m I q Ă N I m , “ N I m I . It is a closed subset of N I m , , hence one can extend it to a continuous section of E I ; I m which can be made satisfy the bound } t I m , } ď ˆ ` m ˙ δ. Suppose we have constructed t I a , : O I a , m Ñ E I a for all a ě l ` t I , they satisfy the induction hypothesis for k “ } t I a , } ď ˆ ` m ´ lm ˙ δ. Then we construct t I l , : O I l , m Ñ E I l as follows. Given z I l P φ I l I p U I l I q Y N ` I l , “ φ I l I p U I l I q Y ď I l ă I b φ ´ I b I l p N I b , q , AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 71 if z I l is in the first component, then define t I l , p z I l q by the formula (7.12) for b “ l , a “ z I l P N I b I l X φ ´ I b I l p N I b , q for some b , then define t I l , p z I l q “ p φ ´ I b I l p t I b , p φ I b I l p z I l qqq . It is easy to verify using the (COCYCLE CONDITION) that these definitions agree over someclosed neighborhood of φ I l I p U I l I q Y ď I l ă I b φ ´ I b I l p N I b I l q . Then one can extend it to a continuous multisection of E I ; I l satisfying the bound } t I l , } ď ˆ ` m ´ l ` m ˙ δ. Then the induction can be carried on and stops until l “
2, for which one has the bound } t I , } ď ˆ ` m ´ m ˙ δ ď δ. The transversality of S I a ` t I a , for a ě t I a , takes value in E I ; I a , the fact that S I a | N IaI intersects with E I ; I a transversely along φ I a I p U I a I q , andthe fact that S I ` t I , is transverse. Hence we have verified the k “ k ´
1. For all a ď k ´
1, define O I a ,k “ O I a ,k ´ , t I a ,k “ t I a ,k ´ . The induction hypothesis implies that we have a multisection t I k ,k ´ : O I k ,k ´ m Ñ E I k such that S I k ` t I k ,k ´ is transverse and } t I k ,k ´ } ď p k ´ q δ . Then apply Lemma7.17, one can obtain a multisection t I k ,k defined over a neighborhood of U I k “ N I k ,k contained in U I k “ N I k ,k such that S I k ` t I k ,k is transverse, } t I k ,k } ď p k ´ q δ , and t I k ,k “ t I k ,k ´ over a neighborhood of N I k ,k ´ which is smaller than O I k ,k ´ . Then bythe similar backward induction as before, using the extension property of continuousmulti-valued functions, one can construct perturbations with desired properties. Theremaining details are left to the reader. (cid:3) In our argument, condition (7.10) is crucial in establishing the compactness of theperturbed zero locus. In the situation of Theorem 7.37, suppose a perturbation t isconstructed over the shrinking A Ă A . Then for a further precompact shrinking A Ă A , (7.10) remains true (with U I replaced by U I ). Proposition 7.38.
Let A be a good coordinate system on X and let A Ă A be a pre-compact shrinking. Let N be a thickening of A . Equip each chart U I a distance function d I which induces the same topology. Then there exists ǫ ą satisfying the followingconditions. Let t be a multi-valued perturbation of s which is N -normal. Suppose d I ` ˜ s ´ I p q X U I , S ´ I p q X U I ˘ ď ǫ, @ I P I . (7.13) Then the zero locus }p ˜ s q ´ p q} is sequentially compact with respect to the subspace topol-ogy induced from | A | . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 72
Proof.
Now for each x P | A | , define I x P I to be the minimal element for which x canbe represented by a point ˜ x P U I x . Claim.
Given I , there exists ǫ ą s that satisfy (7.13),if x i P }p ˜ s q ´ p q} with I x i “ I for all i , then x i has a convergent subsequence. Proof of the claim.
Suppose this is not true, then there exist a sequence ǫ k ą t k satisfying d I ` ˜ s ´ k,J p q X U J , S ´ J p q X U J ˘ ď ǫ k , @ J P I (7.14)and sequences of points ˜ x k,i P ˜ s ´ k,I p q X U I such that the sequence t x k,i “ π A p ˜ x k,i qu i “ does not have a convergent subsequence. Since ˜ x k,i P U I which is a compact subset of U I ,for all k the sequence ˜ x k,i has subsequential limits, denoted by ˜ x k P ˜ s ´ k,I p q X U I . Thensince ǫ k Ñ
0, the sequence ˜ x k has a subsequential limit ˜ x P U I X S ´ I p q . Denote x “ ψ I p ˜ x q P ψ I p U I X S ´ I p qq “ F I Ă F I Ă X. Since all F I cover X , there exists J P I such that x P F J . Then by the (OVERLAPPINGCONDITION) of the atlas A , we have either I ď J or J ď I but J ‰ I . We claim thatthe latter is impossible. Indeed, if x “ ψ J p ˜ y q with ˜ y P U J , then we have ˜ y P U IJ and ˜ x P ϕ IJ p U IJ q . Then for k sufficiently large, we have ˜ x k in N JI . Fix such a large k , then for i sufficiently large, we have ˜ x k,i P N JI . However, since the perturbation t is N -normal, it follows that ˜ x k,i P ϕ IJ p U IJ q . This contradicts the assumption that I x k,i “ I .Therefore I ď J . Then there is a unique ˜ y P U J X S ´ J p q such that ψ J p ˜ y q “ x and˜ y “ ϕ JI p ˜ x q . Then ˜ x P U JI . Therefore, for k sufficiently large, we have ˜ x k P U JI andwe have the convergence ˜ y k : “ ϕ JI p ˜ x k q Ñ ˜ y since ϕ JI is continuous. Since ˜ y P U J which is an open subset of U J , for k sufficientlylarge, we have ˜ y k P U J . Fix such a large k . Then for i sufficiently large, we have˜ x k,i P U JI X ϕ ´ JI p U J q X U I . Hence we have ˜ y k,i : “ ϕ JI p ˜ x k,i q P U J andlim i Ñ8 ˜ y k,i “ ˜ y k . Since the map U J Ñ | A | is continuous, we have the convergencelim i Ñ8 x k,i “ lim i Ñ8 π A p ˜ y k,i q “ π A p ˜ y q . This contradicts the assumption that x k,i does not converge for all k . Hence the claim isprove. End of the proof of the claim.
Now for all I P I , choose the smallest ǫ such that the condition of the above claimhold. We claim this ǫ satisfies the condition of this proposition. Indeed, let t be sucha perturbation and let x k be a sequence of points in }p ˜ s q ´ p q} . Then there exists an I P I and a subsequence (still indexed by k ) with I x k “ I . Then by the above claim, x k has a subsequential limit. Therefore }p ˜ s q ´ p q} is sequentially compact. (cid:3) Branched manifolds and cobordism.
Now given an oriented good coordinatesystem A on a compact Hausdorff space X , and a transverse perturbation t making theperturbed zero locus compact, one can define a fundamental class of the perturbed zerolocus r X s vir P H d p| Z t | ; Q q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 73
Here d is the virtual dimension of A . We call r X s vir the virtual fundamental class of A ,or more roughly, the virtual fundamental class of X . The precise way of constructingthis class is to either use a triangulation as in [FO99, Section 4], or to use the notion ofweighted branched manifolds as in [MW15, Appendix]. Note that there is no difficulty toextend from the smooth category in the above two references to the current topologicalcategory.More generally, consider an oriented good coordinate system with boundary A on acompact Hausdorff space X . Let the charts of A be C I “ p U I , E I , S I , ψ I , F I q . Thendefine B I “ t I P I | B U I ‰ Hu which is equipped with the induced partial order ď . For each I P B I , define B C I : “ ` B U I , E I | B U I , S I | B U I , ψ I | S ´ I p qXB U I , ψ I p S ´ I p q X B U I q ˘ . For each pair I ď J , I, J
P B I , define B T JI : “ pB U JI , p φ JI | B U JI , p π JI | B U JI q where B U JI could be empty. Define B F I “ ψ I p S ´ I p q X B U I q . Then the image of thenatural map ğ I PB I B F I Ñ | A | is a compact subset of X . We denote it by B X . Then it follows immediately that B A “ ´ tB C I | I P B I u , tB T JI | I ď J P B I u ¯ is a good coordinate system on B X . It is also easy to verify that the natural map |B A | Ñ | A | is a homeomorphism onto a closed set.Now consider a transverse perturbation t on A that has a compact perturbed zero locus | Z | . t can be restricted to B A such that for the perturbed zero locus of the restriction,denoted by |B Z | Ă |B A | , one has the relation |B Z | “ | Z | X |B A | . The restricted perturbation B t is also transverse. Moreover, if A is oriented, then B A hasan induced orientation. Lemma 7.39.
A transverse perturbation t on A with compact perturbed zero locus definesa relative virtual fundamental class r X s vir P H d p| Z | , |B Z | ; Q q such that via the map B : H d p| Z | , |B Z | ; Q q Ñ H d ´ p|B Z | ; Q q one has Br X s vir “ rB X s vir . Lemma 7.39 can be used to prove that the virtual fundamental class is independentof the choice of perturbation. Indeed, let A be an oriented good coordinate system(without boundary) on X . Then there is a natural oriented good coordinate system˜ A on X ˆ r , s . There are also “boundary restrictions” ˜ A | X ˆt u and ˜ A | X ˆt u of ˜ A .Let A Ă A be a precompact shrinking, with the product ˜ A Ă ˜ A . Suppose t , t aretransverse perturbations on A . Then one can construct a transverse perturbation ˜ t on˜ A with compact perturbed zero locus, such that˜ t | X ˆt u “ t , ˜ t | X ˆt u “ t . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 74
Then one obtains a relative cycle in | ˜ A | whose boundary is the difference of the two cyclesdefined by t and t .7.8. Strongly continuous maps.
Let X be a compact Hausdorff space and suppose X is equipped with an oriented, d -dimensional topological virtual orbifold atlas withoutboundary A “ ´ p C I q I P I , p T JI q I ď J ¯ . On the other hand, let Y be a topological manifold or orbifold. Definition 7.40. A strongly continuous map from p X, A q to Y , denoted by e : A Ñ Y ,consists of a family of continuous maps e I : U I Ñ Y such that for all pairs I ď J , e J ˝ φ JI “ e I | U JI . Suppose X is a compact Hausdorff space equipped with an oriented topological virtualorbifold atlas A of dimension d . Suppose e : A Ñ Y is a strongly continuous map. Sup-pose after shrinking A one obtains a good coordinate system. The strongly continuousmap restricts to a strongly continuous map on the shrunk atlas. Choosing certain trans-verse multi-valued perturbations, one obtains the perturbed zero locus as an orientedweighted branched manifold Z of dimension d . Then obviously e induces a continuousmap e : Z Ñ Y . Then we can push forward the fundamental class to Y , namely e ˚ r Z s P H d p Y ; Q q . a priori this homology class depends on various choices, notably on the original atlas A , the shrinking to a good coordinate system, and the perturbation. However, we willalways be in the situation that the pushforward e ˚ r Z s does not depend on these choices.We simply denote it as e ˚ r X s vir P H d p Y ; Q q as if there is a virtual fundamental class supported in X itself. When we prove the splitting axioms of the GLSM correlation function, we need toconstrain the moduli space by certain cycles in the target space of the evaluation map.
Definition 7.41.
Let X be a compact Hausdorff space, A be a virtual orbifold atlaswith charts indexed by elements of a partially ordered set I . Let f : A Ñ Y be a stronglycontinuous map where Y is an orbifold. Let Z Ă Y be a closed suborbifold equippedwith a germ of tubular neighborhoods. We say that f is transverse to Z if for every I P I ,the map f I : U I Ñ Y is transverse to Z .Suppose we are in the situation of the above definition. Define X Z : “ x P X | f X p x q P Z ( . Then X Z is a compact Hausdorff space. Indeed the atlas A induces an atlas on X Z . Let N Z Ñ X Z be the normal bundle, which is an orbifold vector bundle. Then for each I P I ,define C Z,I “ p U Z,I , E
Z,I , S
Z,I , ψ
Z,I , F
Z,I q where U Z,I “ f ´ I p Z q , E Z,I “ E I | U Z,I , S Z,I “ S I | U Z,I , ψ Z,I “ ψ I | S ´ Z,I p q , and F Z,I “ X Z X F I . The transversality condition guarantees that U Z,I is still a topological orbifold.Hence C Z,I is a chart on X Z . Moreover, by restricting the coordinate changes from U I Indeed there is a well-defined class in the ˇCech homology of X . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 75 to U Z,I , one obtains a virtual orbifold atlas on X Z , denoted by A Z . Further, if A is agood coordinate system, so is A Z . Proposition 7.42.
Let ι Z : Z Ñ Y be a closed oriented suborbifold. Then p ι Z q ˚ p e Z q ˚ r X Z s vir “ e ˚ r X s vir X r Z s . (7.15) Proof.
First, one constructs a transverse perturbation t Z of A Z , consisting of chartwiseperturbations t Z,I : U Z,I m Ñ E Z,I . Then one can extend t Z,I to a multisection t I : U I m Ñ E I such that all t I gives a transverse perturbation t of s . Then on the zero locus | ˜ s ´ p q| as aweighted branched manifold, the restriction of f is transverse to Z , and the preimage of Z is exactly the zero locus | ˜ s ´ Z p q| . Then (7.15) follows from basic algebraic topology. (cid:3) Virtual Cycles and Cohomological Field Theory
In this section we define the cohomological field theory associated to a GLSM space.The definition relies on the existence and various properties of the virtual cycles on themoduli spaces of gauged Witten equation. We also state the desired properties of thevirtual cycle as Theorem 8.1 and prove the axioms of CohFT using these properties.Lastly, we give a brief argument showing the relation between the GLSM CohFT andthe orbifold Gromov–Witten theory of the classical vacuum ¯ X W .8.1. Cohomological field theories.
Let Γ be a stable decorated dual graph with n tails(not necessarily connected). Then we have defined the moduli space of gauge equivalenceclasses of solutions to the gauged Witten equation with combinatorial type Γ , and itscompactification, denoted by M Γ : “ M Γ p X, G, W, µ q . Γ contains the information of conjugacy classes r ¯ τ a s for a “ , . . . , n , hence identifies acollection of twisted sectors ¯ X r ¯ τ s W , . . . , ¯ X r ¯ τ n s W . Denote ¯ X Γ W “ n ź a “ ¯ X r ¯ τ a s W . Then there are natural evaluation maps ev : M Γ Ñ ¯X Γ W . There is another map, called the forgetful map or the stabilization map , denoted byst : M Γ Ñ M Γ Ă M g,n . Here Γ is the underlying dual graph of Γ , where unstable rational components are con-tracted, and st is defined by forgetting the r -spin structure, forgetting the gauged maps,and stabilization. Denote ev : “ p ev , st q : M Γ Ñ ¯X Γ W ˆ M Γ . (8.1)Notice that both ¯ X Γ W and M Γ are compact complex orbifolds, hence have Poincar´e dualand homological intersection pairings.Now we state the main result about the virtual cycles. Recall the knowledge in Sub-section 7.8 which says that if a compact Hausdorff space X has an oriented topologicalvirtual orbifold atlas A equipped with a strongly continuous map to an orbifold Y , thenthere is a well-defined pushforward virtual fundamental cycle in H ˚ p Y ; Q q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 76
Theorem 8.1.
For each stable decorated dual graph Γ with n tails (not necessarily con-nected), there exists an oriented topological virtual orbifold atlas A Γ on the compactifiedmoduli space M Γ , and a strongly continuous map ev : A Γ Ñ ¯ X Γ W ˆ M Γ . which extends the map (8.1) . They further satisfy the following properties. All homologyand cohomology groups are of rational coefficients.(a) (DIMENSION) If Γ is connected, then the virtual dimension of the atlas is “ dim C ¯ X ´ ‰ p ´ g q ` x c ¯ K p T X q , ¯ B y ` n ´ n ÿ a “ p ι r ¯ τ n s ` rank C q ´ p Γ q . Here ¯ B P H ¯ K p X q is the equivariant curve class associated to Γ .(b) (DISCONNECTED GRAPH) Let Γ , . . . , Γ k be connected decorated dual graphs. Let Γ “ Γ \ ¨ ¨ ¨ \ Γ k be the disjoint union. Then one has p ev q ˚ “ M Γ ‰ vir “ k â α “ p ev q ˚ “ M Γ α ‰ vir . Here the last equality makes sense with respect to the K¨unneth decomposition H ˚ ` ¯ X Γ W ˆ M Γ ˘ » k â α “ H ˚ ` ¯ X Γ α W ˆ M Γ α ˘ . (c) (CUTTING EDGE) Let Γ be the decorated dual graph and let Π be the decorated dualgraph obtained from Γ by shrinking a loops. Then one has p ι Π q ˚ ev ˚ “ M Π ‰ vir “ ev ˚ “ M Γ ‰ vir X “ M Π ‰ ; Notice that ¯ X Γ W “ ¯ X Π W , and the map p ι Π q ˚ : H ˚ ` ¯ X Γ W ˘ b H ˚ ` M Π ˘ Ñ H ˚ ` ¯ X Γ W ˘ b H ˚ ` M Γ ˘ is induced from the inclusion ι Π : M Π ã Ñ M Γ .(d) (COMPOSITION) Let Π be a decorated dual graph with one edge labelled by the pairof twisted sectors r ¯ τ s , r ¯ τ s ´ . Let ˜ Π be the decorated dual graph obtained from Π bynormalization. Consider the fibre product F : “ M ˜Π ˆ M Π M Π . Let pr : F Ñ M Π be the projection onto the second factor, and q : F Ñ M ˜ Π bethe natural map. On the other hand, let ∆ r ¯ τ s be the diagonal of the twisted sector ¯ X r ¯ τ s W » ¯ X r ¯ τ s ´ W . Then one has p q q q ˚ pr ˚ “ M Π ‰ vir “ “ M ˜ Π ‰ vir r PD “ ∆ r ¯ τ s ‰ . Here we use the slant product r : H ˚ ` ¯ X r ¯ τ s W ˆ ¯ X r ¯ τ s ´ W ˘ b H ˚ ` ¯ X ˜ Π W ˘ b H ˚ ` M ˜Π ˘ Ñ H ˚ ` ¯ X Π W ˘ b H ˚ ` M ˜Π ˘ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 77
From this theorem one can define a cohomological field theory on the Chen–Ruan coho-mology of ¯ X W as follows. Indeed it is completely parallel to the case of orbifold Gromov–Witten invariants [CR02] [AGV02]. Recall that the coefficient field is the Novikov fieldof formal Laurent seriesΛ : “ ! ÿ i “ a i T λ i | a i P Q , lim i Ñ8 λ i “ `8 ) where T is a formal variable. For each g, n with 2 g ` n ě
3, and a collection of twistedsectors r ¯ τ s “ ` r ¯ τ s , . . . , r ¯ τ n s ˘ P C p K q n , consider the set of decorated dual graphs with only one vertex and no edges of genus g with n tails labelled by r ¯ τ s , . . . , r ¯ τ n s respectively. Such decorated dual graphs aredetermined by an equivariant curve class ¯ B . Denote the corresponding moduli space by M rg,n pr ¯ τ s ; ¯ B q . Then for classes α a P H ˚ ` ¯ X r ¯ τ a s W ; Q ˘ , β P H ˚ ` M g,n ; Q ˘ , define x α b ¨ ¨ ¨ b α n ; β y g,n “ r g deg p st q ÿ ¯ B A ev ˚ “ M rg,n pr ¯ τ s ; ¯ B q ‰ vir , α b ¨ ¨ ¨ b α n b β E T xr ω ¯ KX s , ¯ B y . (8.2)Here we implicitly used the K¨unneth formula for the cohomology of products.Before verifying the axioms of CohFT, one needs to show that the right hand side of(8.2) is always an element of the Novikov field Λ. This follows from the compactness(Theorem 6.16). Indeed, below any energy level E , if there are infinitely different classes¯ B k with xr ω ¯ KX s , ¯ B k y ď E which contribute to the expression (i.e., the moduli spaces arenonempty), then there is a sequence of solutions representing ¯ B k . Then Theorem 6.16implies that a subsequence converges modulo gauge transformation to a stable solution.However, Proposition 6.14 says that the sequence of homology classes ¯ B k should stabilizefor large k . This is a contradiction. Hence (8.2) is well-defined.Then the correlation can extends to classes with Λ-coefficients, and extends as a mul-tilinear function on the Chen–Ruan cohomology. The axioms of CohFT follows from theproperties of the virtual cycles and the proof is similar to the case of Gromov–Wittentheory. Notice that the fractional factors in the (COMPOSITION) property and in (8.2)are similar to the case of FJRW invariants (see [FJR13]).8.2. Relation with orbifold Gromov–Witten theory.
An interesting and importantquestion to ask is what the relation between the CohFT defined here and the CohFTdefined by orbifold Gromov–Witten invariants. Without such a relation, it is hard to seeif our CohFT is nontrivial.For convergence concern, take the subring of the Novikov fieldΛ ` “ ! ÿ i “ a i T λ i P Λ | λ i ą ) . For α P H ˚ CR p ¯ X W ; Λ ` q , we have the GLSM potential ˜ τ g p α q “ ÿ n ě x α b ¨ ¨ ¨ b α ; 1 y GLSM g,n
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 78 and the
Gromov–Witten potential τ g p α q “ ÿ n ě x α b ¨ ¨ ¨ b α ; 1 y GW g,n . The corresponding invariants are just derivatives of ˜ τ g or τ g at α “
0, namely x α , ¨ ¨ ¨ , α n y GW g,n “ B n τ g B α ¨ ¨ ¨ B α n p q , x α , . . . , α n y GLSM g,n “ B n ˜ τ g p α qB α ¨ ¨ ¨ B α n p q . Recall that Gromov–Witten invariants can be deformed by inserting extra classes,which corresponds to partial derivatives of τ g at a nonzero point . In the case of La-grangian Floer theory such interior insertions give rise to the so-called bulk deformation (see [FOOO09, FOOO11]). Then the relation between GLSM invariants and Gromov–Witten invariants can be stated as the following conjecture (see also [TX17]). Conjecture 8.2.
There is an element c P H ˚ CR p ¯ X W ; Λ ` q such that B n ˜ τ g B α ¨ ¨ ¨ B α n p q “ B n τ g B α ¨ ¨ ¨ B α n p c q . Here the class c stands for “correction,” and it should count point-like instantons,i.e. solutions to the gauged Witten equation over the complex plane. We give a verybrief sketch of the proof here and the details will appear in the forthcoming [TX]. Theargument is based on an extension of the adiabatic limit argument which originated fromthe work of Gaio–Salamon [GS05]. Recall that we have chosen a fibrewise area form σ g,n on the universal curve over M g,n of cylindrical type such that over long cylinders theperimeters are a fixed constant (say 1). This choice is used to define the gauged Wittenequation. Replace σ g,n by ǫ ´ σ g,n and let ǫ to to zero. For the rescaled metric, the energyfunctional of a gauged map p P, A, u q reads E ǫ p P, A, u q “ ż Σ C ´ } d A u } ` ǫ } F A } ` ǫ ´ } µ p u q} ` ǫ ´ } ∇ W p u q} ¯ vol Σ C . Here the integral and the norms are still taken with respect to the unscaled metrics, andwe also rescale W to ǫ ´ W . Then the gauged Witten equation becomes p d A u q , ` ǫ ´ ∇ W p u q “ , ˚ F A ` ǫ ´ µ p u q “ . Then one can prove a theorem about compactness in the adiabatic limit, saying thatfor a given sequence of ǫ k Ñ p P k , A k , u k q with a uniformenergy bound, a subsequence converges in a suitable sense to a holomorphic curve to ¯ X W away from finitely many points in the domain. Moreover, near each one of the finitelymany points where the convergence does not hold, by rescaling the sequence convergesto a point-like instanton , namely a solution to the gauged Witten equation p d A u q , ` ∇ W p u q “ , ˚ F A ` µ p u q “ C . These instantons are generalizations of affine vortices studiedin the context of vortex equation in [GS05] [Zil09, Zil05, Zil14] [VW16] [VXne]. Countingsuch point-like instantons gives the class c , in the same way as the correction termconstructed by the second-named author with Woodward in [WX] for Lagrangian Floertheory of GIT quotients. Moreover, since the Gromov–Witten theory is conformal, theinsertion at the original cylindrical ends and the insertion of the class c are treatedequally. However this is not true for GLSM.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 79
To prove Conjecture 8.2 one also needs to prove an inverse of the compactness theorem.The c -deformed Gromov–Witten invariants of ¯ X W can be thought of as counting holo-morphic curves in ¯ X W with ambient bubble trees, which is a singular object in the ǫ “ ǫ “ ǫ , in the virtual sense.Such a gluing construction can be generalized from that in [Xu], although the analysiscould be much more involved.8.3. Relation with the mirror symmetry.
Mirror symmetry predicts that the A-twisted topological string theory of a Calabi–Yau manifold M is isomorphic to the B-twisted topological string theory of its mirror Calabi–Yau manifold M _ , in particularthere is an identification between the A-side moduli (K¨ahler parameters) and the B-sidemoduli (complex paramters). The identification is generally called the mirror map . Inthe case that M is realized as the classical vacuum of a gauged linear sigma model p X, G, W, µ q (when X is indeed a linear space and G acts linearly, for example, when M is a Calabi–Yau hypersurface in a projective space), many evidences indicate thatthe mirror map can be understood purely from the A-side, and it counts the correctionbetween the GLSM and the low-energy nonlinear sigma model (the Gromov–Wittentheory) (see [MP95]). A reasonable explanation of this phenomenon was given by Hori–Vafa [HV00], which says that the mirror symmetry for vector spaces is trivial and theonly nontrivial effect for the compact Calabi–Yau comes from the point-like instantoncorrection. From such a principle, there have been many A-side calculations of themirror map, for example, [Giv96] [CLLT17] [GW] [CK]. In particular, the counting ofaffine vortices as done in [GW], provides the A-side mirror map interpretation for gaugedlinear sigma model spaces with zero superpotential. Our forthcoming work [TX] will thengive the foundation of the general case with nonzero superpotential, following the ideaof [MP95].9. Constructing the Virtual Cycle. I. Fredholm Theory
In this section we construct the virtual atlas in the “infinitesimal” level. Namely, weprove that the deformation complex of the moduli space of gauged Witten equation isFredholm, and calculate the Fredholm index.9.1.
Moduli spaces with prescribed asymptotic constrains.
We first fix the topo-logical data. Let n ě g ě m , . . . , m n P Z r be a collection of monodromies of r -spin structures over agenus g surface. Then there exists the component M rg,n p m , . . . , m n q Ă M rg,n . Another part of the topological data consists of asymptotic constrains of solutions. Let τ , . . . , τ n P K be elements of finite orders, which are the monodromies of the K -connections. Denote τ a “ τ a ι W p exp p π i m a { r qq P K and r ¯ τ a s its conjugacy class. With the above data fixed, let X a Ă X be the fixed pointset of ¯ τ a and ¯ X a Ă ¯ X be the corresponding twisted sector¯ X a : “ “ KX a X µ ´ p q ‰ { K. Definition 9.1.
Given a collection r ¯ τ s “ pr ¯ τ s , . . . , r ¯ τ n sq of twisted sectors. Let ˜ M r ¯ τ s g,n be the set of triples p C , v q , where(a) C represents a point of M r, m g,n . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 80 (b) v “ p P, A, u q is a smooth bounded solution to the gauged Witten equation (3.9)over C such that the limiting monodromy of A at the a -th marked point is r τ a s .Again one can define an equivalence relation among elements of ˜ M r ¯ τ s g,n . Denote thecorresponding quotient by M r ¯ τ s g,n . Here ends Definition 9.1.One can define a natural topology on M g,n pr ¯ τ sq which is Hausdorff and locally compact.One can decompose the moduli space into connected components: M r ¯ τ s g,n “ ğ ¯ B P H ¯ K p X ; Q q M r ¯ τ s g,n p ¯ B q . Meanwhile, we have a well-defined continuous evaluation mapev : M r ¯ τ s g,n Ñ n ź a “ ¯ X r ¯ τ a s W . (9.1)We would like to realize M r ¯ τ s g,n as the zero set of certain Fredholm section, and calculatethe Fredholm index. In particular, we prove the following formal statement. Proposition 9.2.
The expected dimension of M r ¯ τ s g,n p ¯ B q is equal to dim C ¯ X p ´ g ´ n q ` dim R M g,n ` n dim C ¯ X W ` x c ¯ K p T X q , ¯ B y ´ n ÿ a “ ι r ¯ τ a s . More generally, the compactified moduli space M r ¯ τ s g,n has lower strata indexed by dec-orated dual graphs. Proposition 9.3.
Given a decorated dual graph Γ , the expected dimension of the modulispace M Γ p X, G, W, µ q is equal to dim C ¯ X p ´ g ´ n q ` dim R M g,n ` n dim C ¯ X W ` x c ¯ K p T X q , ¯ B y ´ n ÿ a “ ι r ¯ τ a s ´ E p Γ q . This result follows easily (also formally) from Proposition 9.2 and the property of thedegree shifting numbers ι r τ s (see (3.6)). Remark . We remark that the expected dimension is the same as the expected di-mension of the moduli space of holomorphic curves into the classical vacuum ¯ X W withselected twisted sectors at punctures.9.2. The Banach manifold and the Banach bundle.
We first set up the Banachmanifold B and the Banach vector bundle E Ñ B for this problem. We fix the followingdata and notations.(a) Fix a smooth r -spin curve C . The only case for an unstable C is when it is aninfinite cylinder. Let Σ be the underlying punctured Riemann surface.(b) Fix a cylindrical metric on Σ is fixed. Let P R Ñ Σ be the U p q -bundle comingfrom the Hermitian metric on the orbifold line bundle L C Ñ C .(c) Fix a smooth K -bundle P Ñ Σ. Let ¯ P Ñ Σ be the ¯ K -bundle defined by P R and P .(d) Fix trivializations of ¯ P over certain cylindrical ends, and fix cylindrical coordinateson these ends.(e) Fix conjugacy classes r τ s , . . . , r τ n s P C p K q all of which have finite orders. Let r ¯ τ s , . . . , r ¯ τ n s be the conjugacy classes in ¯ K combined with the monodromies ofthe r -spin structure on C . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 81
When the above data are understood from the context, we only use the pair p C , P q torepresent them.To define various norms, we also need a Riemannian metric on X whose exponentialmap has good properties. The original K¨ahler metric doesn’t satisfy our need, for exam-ple, the level set of the moment map µ ´ p q is not totally geodesic. We choose a metric h X on X that satisfies the following conditions. ‚ h X is invariant under the ¯ K -action. ‚ h X “ g X near the unstable locus of the G -action. ‚ X ss W is totally geodesic near µ ´ p q . ‚ X ss W X µ ´ p q is totally geodesic. ‚ J : T X Ñ T X is isometric.We leave to the reader to verify the existence of such Riemannian metrics.Let exp be the exponential map associated to h X . If ¯ P is a principal ¯ K -bundle and Y Ñ Σ is the associated bundle with fibre X , then h X induces a fibrewise metric on Y and we also use exp to denote the exponential map in the vertical directions.Now we can define the Banach manifolds. First, the cylindrical metric on Σ allows usto define the weighted Sobolev spaces W k,p,w p Σ q , @ k ě , p ą , w P R . Now define A k,p,w C ,P to be the space of connections A on P of regularity W k,ploc satisfying the following condi-tions. Over each cylindrical end U a , with respect to the fixed trivializations of P | U a , A “ d ` λ a dt ` α, where α P W k,p,w p U a , T ˚ U a b k q where λ a P k such that e π i λ a “ r τ a s . On the other hand, define S k,p,w C ,P to be the space of sections u : ¯ P Ñ X of regularity W k,ploc , satisfying the followingcondition. Over each cylindrical end U a , under the fixed trivializations of P | U a , we canidentify u as a map u : U a Ñ X . Then we require that there exists a normalized criticalloop x a “ p x a p t q , λ a q in µ ´ p q X X W , such that e π i λ a P r τ a s and u p s, t q “ exp x a p t q ξ a p s, t q , where ξ a P W k,p,w p U a , x ˚ a T X q . (Here to differentiate ξ a , we use the Levi–Civita connection of certain Riemannian metricon X . The resulting Banach manifold is independent of choosing the metric.) We alsodefine a Banach manifold of gauge transformations. Let G k,p,w C ,P be the space of gauge transformations g : P Ñ K of regularity W k,ploc , such that over thecylindrical end U a , with respect to the trivialization of P | U a , g is identified with a map g “ g a e h , where g a P K, h P W k,p,w p U a , k q . Then we define B k,p,w C ,P Ă A k,p,w C ,P ˆ S k,p,w C ,P to be the subset of pairs p A, u q such that over each U a , p ¯ A, u q converges to a normalizedcritical loop x a “ p x a p t q , ¯ λ a q belong to the twisted sector ¯ X r ¯ τ a s W . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 82
In most cases, we will take k “ k from thenotations. There is a smooth action by the group G p,w C ,P on B p,w C ,P . Further, notice that forany p A, u q P B p,w C ,P , u is continuous. Hence an element represents an equivariant curveclass in H ¯ K p X ; Q q . Then we can decompose the Banach manifold by the equivariantcurve classes, i.e., B p,w C ,P “ ğ ¯ B P H ¯ K p X ; Q q B p,w C ,P p ¯ B q . Each component is invariant with respect to the action by G p,w C ,P . Next we define a Banachvector bundle E p,w C ,P Ñ B p,w C ,P . Over each point p A, u q of the Banach manifold, we have the vector bundle u ˚ T vert Y Ñ Σwhose transition functions are of class W ,ploc . It also has an induced Hermitian metric.Fix a metric connection on u ˚ T vert Y which is trivialized over cylindrical ends, we define E p,w C ,P | v “ L p,w p Σ , Λ , b u ˚ T vert Y q ‘ L p,w p Σ , ad P q . Then the union over all v P B p,w C ,P gives a smooth Banach space bundle. An action bythe group G p,w C ,P makes it an equivariant bundle. It is straightforward to verify that thegauged Witten equation (3.9) defines a smooth section F p,w C ,P : B p,w C ,P Ñ E p,w C ,P . Moreover, this section is equivariant with respect to the action of G p,w C ,P . Hence we define˜ M p,w C ,P : “ ´ F p,w C ,P ¯ ´ p q . It has the topology induced as a subset of the Banach manifold B p,w C ,P . Then define thequotient which is equipped with the quotient topology: M p,w C ,P : “ ˜ M p,w C ,P { G p,w C ,P . Proposition 9.5.
Let p ą and w ą be constants.(a) B p,w C ,P is a Banach manifold and E p,w C ,P is a Banach vector bundle over it.(b) Suppose v “ p A, u q P B p,w C ,P and it is asymptotic to the normalized critical loop x a “ p x a p t q , ¯ λ a q at the a -th puncture, then the tangent space of B p,w C ,P at v “ p A, u q can be identified with the Banach space W ,p,w p Σ , Λ b k ‘ u ˚ T X q ‘ n à a “ T x a p q ´ K ` X ¯ τ a W X µ ´ p q ˘¯ . (c) For any smooth r -spin curve C and any smooth solution v “ p P, A, u q to thegauged Witten equation over C , there exist ¯ B P H ¯ K p X ; Q q and a smooth gaugetransformation g on P such that g ˚ p A, u q P B p,w C ,P p ¯ B q .(d) There is a natural homeomorphism M p,w C ,P p ¯ B q » M C ,P p ¯ B q where the latter is equipped with the c.c.t. topology. Such homeomorphisms fordifferent p and δ are compatible in a natural sense. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 83
Remark . Notice that we only used underlying smooth structure of Σ and the cylin-drical metric to define the spaces A k,p,w C ,P , S k,p,w C ,P , G k,p,w C ,P . Moreover, if we modify thecylindrical metric over a compact subset of the punctured surface Σ, then the resultingBanach manifolds are the same.9.3. Gauge fixing.
In this subsection we take a digress onto the discussion of gaugefixing. This discussion is not only relevant to the index calculation, but is also useful inthe gluing construction. Let C , P be as above and we have defined the Banach manifold B p,w C ,P , the Banach space bundle E p,w C ,P and the section F p,w C ,P . Abbreviate the underlyingpunctured surface of C by Σ.First we would like to define a local distance function on B p,w C ,P . Fix a reference point v “ p A, u q P B p,w C ,P , for another p A , u q P B p,w C ,P , suppose u and u are C close and u “ exp u ξ, where ξ P W ,ploc p Σ C , u ˚ T vert Y q . Then we definedist p v , v q “ } A ´ A } L p,w p Σ q ` } D A p A ´ A q} L p,w p Σ q ` } ξ } L p Σ q ` } D A ξ } L p,w p Σ q ` } dµ p u q ξ } L p,w p Σ q ` } dµ p u q J ξ } L p,w p Σ q . (9.2)As before, D A ξ is actually the covariant derivative with respect to ¯ A . This is gaugeinvariant in the sense that if g P G p,w C ,P , then dist p g ˚ v , g ˚ v q “ dist p v , v q . This functiongives a way to specify a natural open neighborhood of v . We do not need to know whetherdist p v , v q is equal to dist p v , v q or not. For ǫ ą B ǫ v “ ! v P B p,w C ,P | dist p v , v q ă ǫ ) . Definition 9.7.
Given p ą w ą ǫ ą
0. Suppose v , v P B p,w C ,P such thatdist p v , v q ă ǫ , and u “ exp u ξ . We say that v is in Coulomb gauge relative to v , if d ˚ A p A ´ A q ` dµ p u mid ξ q J ξ “ . (9.3)Here u mid ξ is defined by u mid ξ “ exp u ˆ ξ ˙ and the ξ in (9.3) is identified via a tangent vector along u mid ξ .If P is trivialized locally so that A and A are written as A “ d ` φds ` ψdt, A “ d ` φ ds ` ψ dt and u , u are identified with genuine maps into X , then (9.3) reads B s p φ ´ φ q ` r φ, φ ´ φ s ` B t p ψ ´ ψ q ` r ψ, ψ ´ ψ s ` dµ p u mid ξ q J ξ “ B s p φ ´ φ q ` r φ, φ s ` B t p ψ ´ ψ q ` r ψ, ψ s ` dµ p u mid ξ q J ξ “ . It clearly changes sign if we switch p A, u q with p A , u q . Hence we obtain the followingfact. Lemma 9.8. v is on the Coulomb slice through v if and only if v is on the Coulombslice through v . Using the implicit function theorem it is routine to prove the following results (cf.[CGMS02]) for the case where the domain curve is compact).
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 84
Lemma 9.9. (Local slice theorem)
Given v “ p A, u q P B p,w C ,P . Then there exist ǫ ą , δ ą and C ą satisfying the following condition. For any v “ p A , u q P B ǫ v , thereexists a unique gauge transformation g “ e h P G p,w C ,P satisfying } h } W ,p,δ ď δ, d ˚ A p A h ´ A q ` dµ p u mid ξ h q ¨ J ξ h “ , where A h : “ p e h q ˚ A and ξ h is defined by exp u ξ h “ u h : “ p e h q ˚ u . Moreover, } h } W ,p,w ď C dist p v , v q . Proof.
Define R A p h q “ d ˚ A p A h ´ A q ` dµ p u mid ξ h q ¨ J ξ h P L p,w p Σ C , ad P q . We look for zeroes of this map via the implicit function theorem. A crucial step is toshow the invertibility of the linearization of R . When A “ A we have L A : “ D R A p h q “ d ˚ A d A h ` dµ p u q ¨ J X h . This defines a Fredholm operator L A : W ,p,w p Σ , ad P q Ñ L p,w p Σ , ad P q and it is easy toshow the positivity of this operator. Hence L A is invertible and bounded from below bya constant C ą v ). On the other hand, there exists ǫ ą p v , v q ď ǫ , then } L A ´ L A } ď C . Hence L A is invertible and its inverse is uniformly bounded. Then this lemma followsfrom the implicit function theorem and the smoothness of the map R A . (cid:3) Lastly we consider the gauge fixing problem over the infinite cylinder, which has anadditional subtlety. Let v be a soliton. Notice that by reprarametrization, one obtains acontinuous family of solitons, locally parametrized by w P C . For w “ S ` i T which issmall, let v w “ p u w , φ w , ψ w q be the reparametrized soliton, namely u w p s, t q “ u p s ` S, t ` T q , φ w p s, t q “ φ p s ` S, t ` T q , ψ w “ ψ p s ` S, t ` T q . For another v which is close to v , the gauge fixing conditions relative to any v w belongingto this family are different.In particular, when w is small, by Lemma 9.9, one can gauge transform v w to theCoulomb slice through v . Such gauge transformation is of the form e h w with h w P W ,p,w p Θ , k q small and unique. The following lemma will be useful. Lemma 9.10.
Suppose v “ p u, φ, ψ q , then B h w B s ˇˇˇˇ w “ “ φ, B h w B t ˇˇˇˇ w “ “ ψ. Proof.
The implicit function theorem follows from the method of Newton iteration. For w small enough, h w is close to the unique solution to∆ ˚ A h ` dµ p u q ¨ J X h “ ´ d ˚ A ` p φ w ´ φ q ds ` p ψ w ´ ψ q dt ˘ ` dµ p u mid w q J ξ w , where ξ w and u mid w are defined byexp u ξ w “ u w , u mid w “ exp u ˆ ξ w ˙ . The difference from h w and the actual solution h is of higher order in w . Hence differen-tiating in w in the s -direction, one has∆ ˚ A B s h w ` dµ p u q J X B s h w “ d ˚ A ` B s φds ` B s ψdt ˘ ´ dµ p u q ¨ J B s u. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 85
We only verify that B s h w “ φ solves the above equation; the case for the t -derivative issimilar. Indeed, the difference between the left hand side and the right hand side reads ´ pB s ` ad φ qpB s φ q ´ pB t ` ad ψ qpB t φ ` r ψ, φ sq ` dµ p u q ¨ J X φ ` pB s ` ad φ qpB s φ q ` pB t ` ad ψ qpB s ψ q ` dµ p u q ¨ J B s u “ pB t ` ad ψ qpB s ψ ´ B t φ ` r φ, ψ sq ` dµ p u q ¨ J pB s u ` X φ q“ pB t ` ad ψ q ` B s ψ ´ B t φ ` r φ, ψ s ` µ p u q ˘ . The last line vanishes by the vortex equation. (cid:3)
The index formula.
Now we form the deformation complex of our problem. Fixthe smooth r -spin curve C and a principal K -bundle P Ñ C . Recall P contains theinformation of limiting holonomies. Given v “ p A, u q P M ,p,δ C ,P p ¯ B q . Then we have thefollowing deformation complex . C v : E v B v / / E v D v / / E v . Here E v “ W ,p,δ p Σ , ad P q ‘ à j k parametrizes infinitesimal gauge transformations h : Σ Ñ ad P . E v is the tangent spaceof B p,δ C ,P at v . E v is the fibre of E p,δ C ,P at v . The map D v is the linearization of the gaugedWitten equation at v , while B v is the infinitesimal gauge transformation given by B v p h q “ p d A h, X h p u qq . Since the gauged Witten equation is gauge invariant, D v ˝ B v “ Theorem 9.11.
The deformation complex C v is Fredholm. Moreover, its Euler charac-teristic is given by dim C ¯ X ` ´ g ´ n ˘ ` n dim C ¯ X W ` x c ¯ K p T X q , ¯ B y ´ n ÿ j “ ι p γ j q . In other words, this is the expected dimension of the moduli space.
Proposition 9.2 follows immediately since there are additional 6 g ´ ` n parametersfor the variation of the conformal structures.To prove Theorem 9.11, one needs to introduce certain gauge fixing conditions. Define C v p α, ξ q : “ ´ d ˚ A α ` dµ p u q ¨ J X ξ. Define D v : T v B C Ñ E C by˜ D v p α, ξ q “ »——– D , A ξ ` ∇ ξ ∇W ` X , α p u q´ d ˚ A K α ` dµ p u q ¨ J X ξ ˚ d A K α ` dµ p u q ¨ ξ fiffiffifl . (9.4)Also consider the operator C v ˝ B v : Lie G C Ñ E v given by C v ˝ B v h “ ´ d ˚ A d A h ` dµ p u q ¨ J X h By a standard result in functional analysis (see [TX15, Lemma 5.7]), we know that thedeformation complex of v is Fredholm if both D v and C v ˝ B v are both Fredholm, and inthat case χ p v q “ ind ˜ D v ´ ind pC v ˝ B v q . It is easy to see that C v ˝ B v is Fredholm of indexzero. Hence χ p v q “ ind D v . Therefore, the proof of Theorem 9.11 reduces to the proofof the following proposition. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 86
Proposition 9.12. ˜ D v is Fredholm and ind ˜ D v “ dim C ¯ X p ´ g ´ n q ` n dim C ¯ X W ` x c ¯ K p T X q , ¯ B y ´ n ÿ j “ ´ ι p γ j q ` dim R ¯ X W, p γ j q ¯ . Riemann–Roch.
Consider a smooth complex line bundle L Ñ Σ ˚ C . A smoothconnection A P A p L q is called a meromorphic connection if there exist trivializationsof L near the punctures, i.e., ξ j : U ˚ j ˆ C Ñ L U ˚ j such that with respect to ξ j , A “ d ` α j ` ν j dθ , where α j extends to a continuous 1-form over the puncture. We assumethat ν j “ i n j { r is rational. Then L and the collection of ν j determine an orbifold linebundle L Ñ C , whose isomorphism class is independent of the trivializations ξ i . On theother hand, using ξ “ p ξ , . . . , ξ k q one defines a smooth line bundle L ξ Ñ Σ C . We define deg L “ deg L ξ ` k ÿ j “ n j r P Q , t L u “ deg L ξ ` k ÿ j “ t n j r u P Z . Both deg L and t L u are independent of the choice of ξ . We call the rational number ι j p L q : “ n j r ´ t n j r u the degree shifting . It is convenient to use it in the orbifold Riemann–Roch formula.Recall that in the usual Riemann–Roch formula for smooth line bundles there appearsthe degree of the line bundle. In the orbifold case the degree is in general a rationalnumber while the index must be an integer.Now consider a real linear first order differential operator D : Ω p L q Ñ Ω , p L q be areal linear, first-order differential operator. D is called an admissible Cauchy–Riemannoperator if the following conditions are satisfied.(a) There is a meromorphic connection A on L such that D and B A differ by a zerothorder operator.(b) For each cylindrical end U ˚ j , there are real numbers λ j , τ j such that ´ ξ j ˝ D ˝ ξ ´ j ¯ f “ B f B s ` i ” B f B t ` λ j f ı ` τ j ¯ f. Lemma 9.13.
Suppose p ą and δ P R . If for all j , ´ δ ` λ j ˘ τ j R Z , then D extendsto a Fredholm operator D : W ,p,δ p Σ C , L q Ñ L p,δ p Σ C , Λ , b L q . Moreover, if we define b δ p L , D q : “ k ÿ j “ ! t ´ δ ` λ j ` τ j u ` t ´ δ ` λ j ´ τ j u ) , then the real index of D is ind D “ ´ g ` deg “ L ξ ‰ ` b δ p L , D q . Proof.
Because the Riemann–Roch formula satisfies cut-and-paste property, it sufficesto prove this lemma for the case when Σ C is a sphere with only one puncture. Using acylindrical coordinate near the puncture, we see that the operator D is (up to a compactoperator) is BB s ` L where L “ J ˜ BB t ` « λ j ´ τ j ´ τ j λ j ff¸ ` « ´ δ ´ δ ff When τ j “ λ j “ δ is slightly bigger than 0, we know that the index is equal to 2 t L u ,while in this case b δ p L , D q “
2. So we proved the Riemann–Roch formula in this special
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 87 case. In general, the self-adjoint operator L has spectrum Z ` λ j ˘ τ j ´ δ . From the abovespecial case, the spectrum crosses zero for b δ p L , D q times from negative to positive. Bythe relation between spectral flow and index, we obtain the index formula. (cid:3) Corollary 9.14.
Given p ą and δ ą sufficiently small. Let D K : W ,p,δ ´ Σ C , g ‘ ` Λ , b p ad P C q ˘¯ Ñ L p,δ ´ Σ C , ` Λ , b g ˘ ‘ ad P C ¯ be a differential operator such that over each cylindrical end U ˚ j , using the cylindricalcoordinate w “ ds ` i dt and a trivialization of ad P , for certain λ j P k , one has D K « fhdw ff “ »—– ´ B f B w ` r i λ j , f s ` h ¯ dw B h B w ` r i λ j , h s ` f fiffifl . Moreover, D K differs from B ‘ B : by a zeroth order operator. Then D K is Fredholm and ind D K “ . Proof.
Let the collection of trivializations of ad P C | U ˚ j be ξ . Then ad P C and ξ togetherinduces a continuous complex vector bundle E ξ Ñ Σ C . On the other hand, the trivial-ization and the cylindrical coordinates induce an identification g ‘ p Λ , b ad P C q » g ‘ g . Define a real linear map ρ p a, b q “ p a ` b, a ´ b q . Then ρ gives a different trivialization of g ‘ p Λ , b ad P C q over U ˚ j . These trivializations can be extended to a splitting ρ : g ‘ Λ , b ad P C » R ` ‘ R ´ . Near the punctures, one has ρ D K ρ ´ « f ` f ´ ff “ « B f ` {B w ` r i λ j , f ` s ` f ` B f ´ {B w ` r i λ j , f ´ s ´ f ´ ff . (9.5)If we regard f ˘ “ a ˘ ` i b ˘ , then the two components of (9.5) can be written as « a ˘ b ˘ ff ÞÑ ˆ BB s ` J BB t ˙ « a ˘ b ˘ ff ` « ˘ Id ´ ad λ j ad λ j ¯ Id ff « a ˘ b ˘ ff . It is easy to see that the last 2 ˆ ind D K “ K p ´ g q ` deg ` g ‘ ` Λ , b E ξ ˘˘ ´ n dim K “ dim K p ´ g q ` K deg p K log q ´ n dim K “ . Here the term ´ n dim K comes from the limiting behavior of the operator (9.5), and weused deg E ξ “ K is in SU p g q . (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 88
The splitting.
We would like to construct a splitting of u ˚ T vert Y ‘ p Λ b E q intoline bundles so that we can apply the Riemann–Roch formula (Lemma 9.13) for these linebundles. Suppose at the puncture z j , the section u (with respect to certain trivializationof P ) converges to a loop e ´ λ j t x j where x j P X j “ X γ j and γ j “ exp p πλ j q . One cansplit the tangent space T x j X as T x j X » G X ‘ H X » G X ‘ ´ H X X N x j X j ¯ ‘ ´ H X X T x j X j ¯ » G X ‘ ´ H X X N x j X j ¯ ‘ ´ H X X T x j X j X N x j X W ¯ ‘ ´ H X X T x j X j X T x j X W ¯ . (9.6)Here G X is the subspace generated by X a and J X a for all a P k , and H X is the orthogonalcomplement of G X .The decomposition (9.6) is γ j -equivariant. We can split the first factor G X as eigenspacesof γ j , denoted by G X » n à α “ L p q α . We also decompose the second factor H X X N x j X j into irreducible representations of thecyclic group x γ j y » Z { a j Z , as H x j X X N x j X j » n à α “ L p q α , where n “ codim C ` ¯ X r γ j s , ¯ X ˘ . Moreover, the Hessian of W restricted to the third factor H X X T x j X j X N x j X W isnondegenerate. Since W is holomorphic, one can decompose the third factor as H X X T x j X j X N x j X W » n à α “ L p q α , where n “ codim C ` ¯ X W, p γ j q , ¯ X p γ j q ˘ . Lastly, we can decompose the fourth component of (9.6) into H x j X X T x j X j X T x j X W » n à α “ L p q α , where n “ dim C ¯ X W, p γ j q . For each cylindrical end U ˚ j , we can identify u ˚ T vert Y | U ˚ j with U ˚ j ˆ T x j X . Accordingto the above decomposition, we can trivialize u ˚ T vert Y | U ˚ j as u ˚ T vert Y | U ˚ j » à ď s ď ď α ď ns L p s q α , so that up to compact operators, the Cauchy–Riemann operator B A is equivalent to thedirect sum of admissible Cauchy–Riemann operators D α on L α such that(a) For each L p q α , z j is a type I puncture.(b) For each L p q α , z j is a type II puncture.(c) For each L p q α , z j is a type II puncture.There is no topological obstruction to extend the trivializations of u ˚ T vert Y | U ˚ j to Σ C .Hence we can assume that u ˚ T vert Y is decomposed over Σ C into complex line bundles u ˚ T vert Y » à ď s ď ď α ď ns L p s q α , such that the direct sum of all L p q α is isomorphic to the trivial bundle with fibre g . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 89
Lemma 9.15.
Let B P H K p X W ; Q q be the equivariant curve class represented by v , then ď α ď n s ÿ ď s ď t L p s q α u “ x c K p T X q , B y ´ k ÿ j “ ι p γ j q . (9.7) Proof.
The case that Σ C has no puncture is easy to prove. Hence we assume that Σ C hasat least one puncture. Then the bundle P ˚ C is trivial. Take a trivialization of P over Σ C which identifies the section u with a map u : Σ C Ñ X and induces identificationsad P C » Σ C ˆ g , u ˚ T vert Y » u ˚ T X.
With respect to the first identification we can write A over each cylindrical end U ˚ j as A “ d ` α ` λ j dt. Moreover, we can trivialize u ˚ T X | U ˚ j » U ˚ j ˆ T x j X . This completes u ˚ T X to a smoothvector bundle p u ˚ T X q ξ Ñ Σ C . Then it is easy to see that x c ¯ K p T X q , ¯ B y “ deg p u ˚ T X q ξ ` n ÿ j “ ´ i e p λ j q . Here e p λ j q P i Q is the sum of eigenvalues of ∇X λ j on T x j X . Moreover, since ad λ j istraceless on g , e p λ j q is equal to the sum of eigenvalues of ∇X λ j on H X . Hence we obtain(9.7) as ´ ι p γ j q is exactly the sum of these eigenvalues. (cid:3) By the expression (9.4) and the splitting (9.6), as well as Corollary 9.14 and Lemma9.15, one obtains ind D v “ ind D K ` ď α ď n s ÿ ď s ď ind D p s q α “ ď α ď n s ÿ ď s ď ´ p ´ g q ´ b p L i , D i q ` t L p s q α u ¯ “ dim C ¯ X p ´ g q ` dim C ¯ X ÿ i “ t L i u ´ n ÿ j “ ´ dim R ¯ X W, p γ j q ´ rank C ∇ W | T xj X j ¯ “ dim C ¯ X p ´ g ´ n q ` n dim C ¯ X W ` x c ¯ K p T X q , ¯ B y ´ n ÿ j “ ´ ι p γ j q ` dim R ¯ X W, p γ j q ¯ . This finishes the proof of Proposition 9.12.10.
Constructing the Virtual Cycle. II. Gluing
In this section we construct a local virtual orbifold chart for every point of the com-pactified moduli space M rg,n p X, G, W, µ ; B q . The main result of this section is statedas Corollary 10.17, while a crucial step is Proposition 10.16, in which via the gluingconstruction we obtain local charts for the thickened moduli space . The construction ofthe thickened moduli space, which is similar to the case of Gromov–Witten theory (see[LT98b] [FO99] [Par16]), depends on many choices.The construction depends on a number of choices we have to make. We will use theenvironment Choice to explicitly declare them.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 90
Stabilizers of solutions.
An automorphism of a solution a priori consists of twoparts, one coming from the domain symmetry, i.e., the underlying r -spin curve has anautomorphism, the other coming from the gauge symmetry. To construct a local model,we first show that any stable solution has finite automorphism group. Definition 10.1. (Isomorphisms of stable solutions) Let C , C be smooth or nodal r -spin curves and v P ˜ M C , v P ˜ M C . An isomorphism from p C , v q to p C , v q is a pair p ρ, g q , where ρ : C Ñ C is an isomorphism of r -spin curves, and g : P Ñ P is a smoothisomorphism of principal K -bundles that covers ρ . We require that p ρ, g q satisfies thefollowing condition. ‚ For each irreducible component α of C , the isomorphism ρ of r -spin curvesprovides a corresponding irreducible component α of C and a biholomorphicmap γ α : Σ ˚ α Ñ Σ ˚ α that preserves the punctures. Moreover, there is a bundleisomorphism g α : P | Σ ˚ α Ñ P | Σ ˚ α that covers γ α . We require that A | Σ ˚ α “ g ˚ α A | Σ ˚ α , u | P | Σ ˚ α “ u | P | Σ ˚ α ˝ g α . In particular, a gauge transformation on P induces an isomorphism from p C , v q to p C , g ˚ v q . Isomorphisms can be composed. For any pair p C , v q , denote by Aut p C , v q theautomorphism (self-isomorphism) group.Then we show that automorphism groups of stable solutions are finite. A key pointis that when n ě
1, a solution converges at punctures to points in µ ´ p q where K -stabilizers are finite. When the curve has no punctures, we need to require that the areais sufficiently large (compare to the energy) so that solutions are also close to µ ´ p q . Lemma 10.2. (Finiteness of automorphism groups) (a) Suppose n ě . Then for every λ ą , every B P H K p X ss W ; Q q , every stable solu-tion v to the gauged Witten equation over a smooth or nodal curve C representinga point in M rg,n p X, G, W, µ ; B q λ , the automorphism group Γ v is finite.(b) For every B P H K p X ss W ; Q q there exists λ B ą such that for every λ ě λ B , everystable solution v to the gauged Witten equation over a smooth C representing apoint in M rg, p X, G, W, µ ; B q λ , the automorphism group Γ v is finite.Proof. Consider p C , v q and its automorphism group Aut p C , v q . Let Aut p C , v q Ă Aut p C , v q be the subgroup of automorphisms p ρ, g q which fix every irreducible component of C .Since there are at most finitely many permutations among components, it suffices toprove that Aut p C , v q is finite. Hence we only need to prove the finiteness for a smooth r -spin curve C .First consider the case that C is stable. Then ρ is an automorphism of the stable r -spincurve, which is finite. Then it suffices to consider automorphisms p ρ, g q such that ρ “ Id C .Since g fixes the connection, g is a covariantly constant section of P ˆ Ad K . Hence such g corresponds to a subgroup of K . On the other hand, if C has a marking, at which thesection u approaches to µ ´ p q . Since g has to fix the section, and stabilizers of µ ´ p q are at most finite. Hence there are at most finitely many gauge transformations that fix v . If C is stable but has no marking, then Σ C has finite volume. By the energy inequality(Theorem 5.12), one has C ` x ω ¯ KX , ¯ B y “ E p v q ě λ ż Σ C | µ p u q| . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 91
Hence for λ B being sufficiently large and when λ ě λ B , there is a nonempty subset ofΣ C whose images under u is in the region of X where the K -action has no continuousstabilizer. Hence for the same reason as the above case the automorphism group is finite.Lastly we consider the case when C is unstable, namely v is a soliton. In this case C “ R ˆ S . Let w “ s ` i t be the cylindrical coordinate. We can view the r -spinstructure p L C , φ C q as a trivial one. Namely, the log-canonical bundle ω C is trivializedby dw , L C is the trivial bundle and φ C : L b C Ñ ω C is the identity of the trivial bundle.Then for an automorphism p ρ, g q of v , the curve isomorphism ρ is a translation of thecylinder, namely, p s, t q ÞÑ p s ` S, t ` T q for certain constants S, T . Then ρ ˚ L C and L C areboth canonically identified with the trivial bundle. Therefore the bundle isomorphismcontained in ρ , L C Ñ ρ ˚ L C » canonical L C is essentially an element of Z r , and there is a group homomorphism Aut v Ñ Z r . Henceto prove that Aut v is finite, it suffices to consider the subgroup of automorphism whoseimages in Z r are trivial. Furthermore, by the finite energy condition, it is easy to seethat S must be zero, namely ρ is a rotation of the infinite cylinder. Claim. T is a rational multiple of 2 π . Proof of the claim.
We can transform v into temporal gauge, namely A “ d ` ψdt. Hence we can abbreviate v by a pair p u, ψ q . Since g preserves A upto a rotation, g mustonly depend on t . Furthermore, we can also fix the gauge at ´8 so thatlim s Ñ´8 p ψ p s, t q , u p s, t qq “ p η ´ , x ´ p t qq where η ´ P k and x ´ p t q “ p x ´ p t q , η ´ q is a critical loop, i.e. x p t q ` X η ´ p x ´ p t qq “ . Then e πη ´ x ´ p q “ x ´ p π q “ x ´ p q “ : x ´ , x ´ p t q “ e ´ η ´ t x ´ . Since x ´ P µ ´ p q which has finite stabilizer, e πη ´ is of finite order in K , say order k .Then we can lift v to k -fold covering of the infinite cylinder, denoted by p ˜ u, ˜ ψ q , which isdefined by ˜ u p ˜ s, ˜ t q “ u p ˜ s, ˜ t q , ˜ ψ p ˜ s, ˜ t q “ ψ p ˜ s, ˜ t q . The automorphism is also lifted to the k -fold cover by ˜ ρ p ˜ s, ˜ t q “ p ˜ s, ˜ t ` T q , ˜ g p ˜ s, ˜ t q “ g p ˜ s, ˜ t q “ g p ˜ t q . Then using the gauge transformation˜ h p ˜ s, ˜ t q “ e ´ η ´ ˜ t the pair ˜ v is transformed to ˜ v “ p ˜ u , ˜ ψ q such thatlim s Ñ´8 ˜ u p ˜ s, ˜ t q “ x ´ , lim s Ñ´8 ˜ ψ p ˜ s, ˜ t q “ . The automorphism is transformed to the T -rotation and˜ g “ ˜ h ´ ˜ g ˜ h. Therefore, ˜ g fixes the constant loop x ´ . So ˜ g p ˜ t q is a constant element γ ´ of the finitestabilizer of x ´ . Then it follows that g p t q “ e ´ η ´ t γ ´ e η ´ t . (10.1) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 92
Since the T -rotation and g p t q altogether fixes the loop x ´ p t q “ e ´ η ´ t x ´ , we see e ´ η ´ t γ ´ ´ e η ´ t ¨ e ´ η ´ p t ` T q x ´ “ e ´ η ´ t x ´ ùñ γ ´ ´ e ´ η ´ T x ´ “ x ´ . Therefore e η ´ T is a stabilizer of x ´ . Since there is no continuous stabilizer of x ´ , either T is a rational multiple of 2 π or η ´ “ η ´ “ T still has to be a rational multiple of 2 π .Indeed, if η ´ “
0, then (10.1) implies that g p t q ” γ ´ . Since γ k ´ “
1, we see that the kT -rotation is an automorphism of p C , v q . If T is an irrational multiple of 2 π , it followsthat v is independent of t . Then the gauged Witten equation is reduced to the ODE forpairs p u, ψ q : R Ñ X ˆ k : u p s q ` J X ψ p s q p u p s qq “ , ψ p s q ` µ p u p s qq “ . Indeed this is the negative gradient flow equation of the Morse–Bott function p x, η q ÞÑ x µ p x q , η y . whose critical submanifold is µ ´ p q ˆ t u and which has only one critical value 0. Itfollows that v must be trivial, which contradicts the stability condition. Therefore T isa rational multiple of 2 π whether η ´ “ (cid:3) Therefore, we can only consider automorphisms of p C , v q whose underlying automor-phism of the cylinder is the identity. They are given by gauge transformations of theform (10.1), hence only depends on a stabilizer γ ´ of x ´ . Therefore there are only finitelymany such automorphisms. (cid:3) Thickening data.
Thickening data.
Recall the notion of generalized r -spin curves in Subsection 2.3and resolution data in Subsection 2.2. Definition 10.3. (cf. [Par16, Definition 9.2.1]) A thickening datum α is a tuple p C α , v α , y ˚ α , r α , E α , ι α , H α q where(a) C α is a smooth or nodal r -spin curve with n punctures.(b) v α is a stable smooth solution over C α .(c) y ˚ α is an unordered list of l α points on C α that makes p C α , y ˚ α q a stable generalized r -spin curve of type p n, , l α q . Moreover, we require the following condition. Let Γ α be the automorphism group of p C α , v α q , then the image of the natural map Γ α Ñ Aut p C α q lies in the finite subgroup Aut p C α , y ˚ α q .(d) r α is a resolution data for p C α , y ˚ α q (see Subsection 2.2), which induces an explicituniversal unfolding of p C α , y ˚ α q , denoted by π α : U α Ñ V α “ V α, def ˆ V α, res . The resolution data induces a smooth (orbifold) fibre bundle Y α Ñ V α . Noticethat the automorphism Γ α acts on V α via the map Γ α Ñ Aut p C α , y ˚ α q , and alsoacts on the total space Y α .(e) E α is a finite dimensional representation of Γ α .(f) ι α is a Γ α -equivariant linear map ι α : E α Ñ C ` Y α , Ω , Y α { V α b C T vert Y α ˘ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 93
We also require that for all e α P E α , ι α p e α q is supported away from the nodalneighborhoods. Moreover, the restriction of the image ι α p E α q to the centralfibre of U α Ñ V α is transverse to the image of D α : T ˝ v α B α Ñ E α .(g) H α Ă X is a compact smooth embedded codimension two submanifold withboundary (not necessarily K -invariant) that satisfies the following condition. Theresolution data r α induces a thin-thick decomposition of C α and a trivializationof the K -bundle over the thin part C thin α , hence the matter field u α restricted tothe thin part is identified with an ordinary map u thin α : C thin α Ñ X . We requirethe following conditions.(i) u thin α intersects with H α transversely in the interior exactly at y ˚ α .(ii) There is no intersection on either B C thin α or B H α .(iii) H α is transverse to the complex line spanned by B s u thin α ` X φ thin α and B t u thin α ` X ψ thin α at each point of y ˚ α .Here ends Definition 10.3.The pregluing construction (i.e. the definition of approximate solutions) depends onchoosing a gauge. The last condition on gauge implies the following convenient fact: thefamily of approximate solutions have inherited symmetry.10.2.2. Thickened moduli space within the same stratum.
Fix the thickening datum (seeDefinition 10.3) α “ p C α , v α , y ˚ α , r α , E α , ι α , H α q . We would like to construct a family of gauged maps over the fibres of U α Ñ V α whichhave the same combinatorial type as C α . First we need to setup a Banach manifold,a Banach space bundle and a Fredholm section. Recall that in the resolution data, V a “ V α, def ˆ V α, res , and the resolution data r α contains a smooth trivialization U α | V α, def ˆt u » V α, def ˆ C α . Let the coordinates of V α, def and V α, res be η and ζ respectively. This trivialization inducesa smooth trivialization of the universal log-canonical bundle (viewed as the cotangentbundle of punctured surfaces), and hence a smooth trivialization the U p q -bundle P C α,η over each fibre. On the other hand, r α contains trivializations of P α Ñ V α, def . Thendefine B α “ V α, def ˆ B p,w C α ,P α . Here B p,w C α ,P α is the Banach manifold defined in Subsection 9.2. Moreover, including vectorsin the obstruction space, defineˆ B α “ E α ˆ V α, def ˆ B p,w C α ,P α . Let the Banach space bundle ˆ E α Ñ ˆ B α to be the pull-back of E p,w C α ,P α Ñ B p,w C α ,P α . Then,using the linear map ι α , one can define operator associated to the E α -perturbed gaugedWitten equation ˆ F α p e α , η, v q “ F α,η p v q ` ι α p e α qp C α,η , v q . (10.2)Here F α,η includes both the gauged Witten equation over the fibre C α,η and the gaugefixing condition relative to the gauged map v α .Let the linearization of the operator ˆ F α beˆ D α : E α ‘ V α, def ‘ T v α B α Ñ E α . Notice that we do not require the obstruction space E α to be gauge invariant. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 94
When the deformation parameter is turned off, one obtains the linear map D α : E α ‘ T v α B α Ñ E α . (10.3)By the transversality assumption it is surjective. Define V α, map “ ker D α Ă E α ‘ T v α B α . Denote a general element of V α, map by ξ and add the subscript α whenever necessary.Then we are going to construct a familyˆ v α,ξ,η “ ´ v α,ξ,η , e α,ξ,η ¯ , ξ P V α, map , η P V α, def where v α,ξ,η is a gauged map over the bundle P α,η Ñ C α,η .The following construction needs another choice. We choose a bounded right inverseto the operator D Eα in (10.3) Q α : E α Ñ E α ‘ T v α B α . It induces a right inverse to ˆ D α , denoted byˆ Q α : E α Ñ V α, def ‘ E α ‘ T v α B α . (10.4)Using the exponential map of the Banach manifold B α , we define the approximate solution as ˆ v app α,ξ,η “ ´ v app α,ξ,η , e app α,ξ,η ¯ “ ´ exp v α ξ, e ξ ¯ , ξ P V α, map , η P V α, def , where e ξ P E α is the E α -component of ξ . Lemma 10.4.
There exists a constant C ą such that when } ξ } and } η } are sufficientlysmall, one has ››› ˆ F α ´ e app α,ξ,η , η, v app α,ξ,η ¯››› ď C ´ } ξ } ` } η } ¯ . Proof.
Left to the reader. (cid:3)
Then the implicit function theorem implies that one can correct the approximatesolutions by adding a vector in the image of the right inverse ˆ Q α and the correction isunique. Therefore, we obtain a familyˆ v α,ξ,η “ ´ v α,ξ,η , e α,ξ,η ¯ (10.5)which solves the E α -perturbed gauged Witten equation over the r -spin curve C α,η .10.2.3. Pregluing.
Now we construct approximate solutions by allowing nonzero gluingparameters. The construction relies on choosing certain cut-off functions. We fix a paircut-off functions ρ ˘ : R Ñ r , s satisfyingsupp ρ ´ “ p´8 , s , ρ | p´8 , ´ s “ , supp dρ ´ “ r´ , s , ρ ` p t q “ ρ ´ p´ t q . (10.6)For all T ąą
1, also denote ρ T ´ p t q “ ρ ´ p tT q , ρ T ` p t q “ ρ ` p tT q . Consider a typical node w P Irre C α . Let the monodromies of the r -spin structure atthe two sides of w be γ ´ , γ ` P Z r .Using the resolution data and the functions z ÞÑ ˘ log z we can identify the two sidesof the nodes as semi-infinite cylinders C ´ “ r a, `8q ˆ S , C ` “ p´8 , b s ˆ S . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 95
The bundle L R is then trivialized up to Z r action. Moreover, the resolution datatrivializes P K over C ´ \ C ` . Then we can identify the connection as forms A | C ´ “ d ` α ´ , A | C ` “ d ` α ` . The restrictions of the section u onto C ´ and C ` are identified with maps u ´ : C ´ Ñ X W , u ` : C ` Ñ X W . Then using the cut-off function ρ and the exponential map of X ss W we can define theapproximate solution as in the case of Gromov–Witten theory. For any gluing parameter ζ “ p ζ w q w , introduce ´ log ζ w “ T w ` i θ w . For k P R ` , let Σ kT Ă Σ C be the closed subset obtained by removing the radius e ´ kT w disk around each nodal points (but not punctures). The long cylinder is then identifiedwith N w,kT “ rp k ´ q T, p ´ k q T s ˆ S . Then we have Σ η “ Σ T Y à w N w,T where the intersections of Σ T and N w,T are only their boundaries.Moreover, suppose p A ˘ , u ˘ q converges to the loop p x ˘ , η ˘ q : S Ñ X W X µ ´ p q ˆ k as z approaches ˆ w ˘ . The gauge we choose guarantees that the two loops are identical,denoted by p x w , η w q . Further, using the exponential map of X ss W , we can write u ˘ p s, t q “ exp x w p t q ξ ˘ p s, t q , where ξ ˘ P W ,p,δ p C ˘ , x ˚ w T X ss W q . Definition 10.5. (Central approximate solution) Given a thickening datum α as denotedin Definition 10.3, for any gluing parameter ζ P ˆ V res , we obtain an r -spin curve C α,η,ζ with extra unordered marked points, which represents a point in M rg,n,l α . The centralapproximate solution is the object v α,η “ p A α,η , u α,η q where ‚ p A α,η , u α,η q| Σ T “ p A α , u α q| Σ T . ‚ For each node w with η w ‰
0, we have A α,η | N w,T “ d ` a η “ d ` ρ T ´ a ´ ` ρ T ` a ` , u α,η | N w,T “ exp x w ` ρ T ´ ξ ´ ` ρ T ` ξ ` ˘ . (10.7)We also consider an auxiliary object v η that is defined over the unresolved curve C .Indeed, we can include Σ T ã Ñ Σ C and define v η | Σ T “ v η | Σ T . Further, (10.6) and(10.7) implies that v η is equal to the loop p x w , η w q near the boundary of Σ T . Then overΣ C r Σ T , v η is defined to be the loop p x w , η w q .10.2.4. Thickened moduli spaces.
We postponed the definition of I -thickened modulispaces here because its definition relies on the description of approximate solutions. Definition 10.6. (Thickened solutions) (cf. [Par16, Definition 9.2.3]) For a finite set I of thickening data, an I -thickened solution is a quadruple p C , v , p y α q α P I , p p φ α q α P I , p e α q α P I q where(a) C “ p Σ C , ~ z C , L C , ϕ C q is a smooth or nodal r -spin curve of type p g, n q and for each α P I , y α is an unordered list of marked points y α labelled by α P I . We requirethat for every α P I , p C , y α q is a stable generalized r -spin curve.(b) v “ p P, A, u q is a smooth gauged map over C . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 96 (c) For each α P I , p φ α : P Ñ P α is an inclusion of K -bundles which covers anisomorphism φ α : C » C φ α “ C α,η,ζ for some fibre C φ α Ă U α such that φ α p y α q “ y α,η,ζ “ y φ α . Remember there is an approximate solution v app α,η,ζ on C α,η,ζ . Thereis also a thick-thin decomposition C φ α “ C thick φ α Y C thin φ α . (d) v and p φ α need to satisfy the following condition. For each α and each irreduciblecomponent v , p p φ ´ α q ˚ v v is in Coulomb gauge with respect to v app α,η,ζ | C φα,v . Here C φ α ,v Ă C φ α is the component identified with C v . Moreover, over the intersection C φ α ,v X C thin φ α where the K -bundle is trivialized, p p φ ´ α q ˚ u v is identified with a smoothmap u thin v : C thin φ α ,v Ñ X . We require that y α X Σ v “ u thin v & H α and the intersectionsare transverse and are away from the boundaries.(e) For every α P I , e α P E α .(f) The following I -thickened gauged Witten equation is satisfied: F C p v q ` ÿ α P I ι α p e α qp C , v , y α , p φ α q “ . Two I -thickened solutions p C , v , p y α q , p p φ α q , p e α qq and p C , v , p y α q , p p φ α q , p e α qq are iso-morphic if for every α P I , e α “ e α , and if there is a commutative diagram P p φ α / / (cid:15) (cid:15) p ρ ! ! P α (cid:15) (cid:15) P p φ α o o (cid:15) (cid:15) C φ α / / ρ ? ? U α C φ α o o where ρ is an isomorphism of r -spin curves and p ρ is an isomorphism of bundles such that v “ p ρ ˚ v . Here ends Definition 10.6.The I -thickened solutions give local charts of the moduli spaces (in a formal sense).Indeed, it is straightforward to define a topology on the space of isomorphism classes of I -thickened solutions, denoted by M I . The finite group Γ I “ ź α P I Γ α acts on M I continuously as follows. For any γ “ p γ α q P Γ I , define γ ¨ ´ C , v , p y α q , p p φ α q , p e α q ¯ “ ´ C , v , p y α q , p γ P α α ˝ p φ α q , p γ α ¨ e α q ¯ . Since all ι α are Γ α -equivariant, this is indeed an action on M I . Furthermore, there ishence a continuous, Γ α -equivariant map˜ S I : M I Ñ E I : “ à α E α sending an I -thickened solution to t e α u P E I . Denote U I “ M I { Γ I , E I “ p M I ˆ E I q{ Γ I . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 97 and the induced section S I : U I Ñ E I . There is the canonical map ψ I : S ´ I p q Ñ M by only remembering the data C and v , and sending p C , v q to its isomorphism class. Theimage of ψ I is denoted by F I and denote C I “ p U I , E I , S I , ψ I , F I q . (10.8)By the definition of local charts (Definition 7.18), C I is a local chart if we can verify thefollowing facts: ‚ M I is a topological manifold. ‚ F I is an open subset of M . ‚ ψ I is a homeomorphism.They will be treated in the next section for general I . In the rest of this section, we willprove the corresponding results for the case that I contains a single thickening datum α .10.2.5. Constructing thickening data.
Now we prove the following lemma.
Lemma 10.7.
Let Γ be a stable decorated dual graph. For any p P M Γ , there exists atransverse thickening datum α centered at p . First, choose a representative v α as a soliton solution to the gauged Witten equationover a smooth or nodal r -spin curve C α . Let Γ α be the automorphism group of p C α , v α q .Then Γ α acts on the set of irreducible components Irre C and the subset Unst C Ă Irre C of unstable components is Γ -invariant. For each Γ -orbit O of unstable components,for each representative v P Unst C of this orbit, by the stability condition, the soliton v v “ p A v , u v q is nonconstant. Notice that we regard the K -bundle over this rationalcomponent is trivialized and u v is regarded as a genuine map into X . Hence the subsetof Σ v of points where the covariant derivative D A v u v is nonzero is an open and densesubset. Hence we can choose a codimension two submanifold H O with boundary whichintersects transversely with u v at some point of Σ v . Choose H O for every Γ -orbit O ofunstable components, and define H α “ ď O P Unst C { Γ H O . Then let y α be the (unordered) set of intersection points between H α and the images ofall unstable components.Now the smooth or nodal r -spin curve C α “ p Σ C α , ~ z C α q together with the collection y α (which is unordered) form a stable generalized r -spin curve. Then one can choose aresolution data r α of p C α , y α q . It contains the following objects(a) A universal unfolding π α : U α Ñ V α , where V α “ V α, def ˆ V α, res .(b) A smooth principal K -bundle P K,α Ñ U α whose restriction to the central fibre isidentified with P K,α Ñ C α .(c) The universal cylindrical metric induces a fibrewise cylindrical metric, and hencea Hermitian metric on the bundle L R,α Ñ U α . This provides a unit circle bundle P R,α . Together with P K,α one obtains a principal ˆ K -bundle P α Ñ U α .(d) A thick-thin decomposition U α “ U thick α Y U thin α .(e) A trivialization of the thick part of the universal unfolding t thick C : U thick α » V α ˆ C thick α . Different H O might intersect. So we actually need maps from the disjoint union of H O into X . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 98 (f) Trivializations of the K -bundles t thin P : P K,α | U thin α » K ˆ U thin α , t thick P : P K,α | U thick α » V α ˆ P K,α | C thick α . Moreover, t thin P extends the trivialization of P K,α over C thin α .(g) P α induces a fibre bundle Y α Ñ U α whose fibres are X . The above trivializationsinduce trivializations of Y α t thin Y : Y α | U thin α » X ˆ U thin α , t thick Y : Y α | U thick α » V α ˆ Y α | C thick α . (10.9)Now we construct the obstruction space E α and the inclusion ι α . For each irreduciblecomponent Σ v Ă Σ C α , the augmented linearization of the gauged Witten equation overΣ v at v v is a Fredholm operator D v : T v v B v Ñ E v . Here T v v B v consists of infinitesimal deformations of the gauged map v v . Inside there isa finite-codimensional subspace T ˝ v v B v Ñ T v v B v consisting of infinitesimal deformations whose values at all punctures and nodes on thiscomponent vanish. Then as in the case of Gromov–Witten theory, one can find a finitedimensional space of sections E α Ă L p,δ p Σ v , Λ , b u ˚ v T vert Y v q satisfying the followingconditions.(a) Elements of E v are smooth sections and are supported in a compact subset O v Ă Σ v that is disjoint from the special points.(b) There is a K -invariant open neighborhood O of µ ´ p q in X ss W such that u v p O v q is contained in P v ˆ K O .(c) E v is transverse to the image D v p T ˝ v v B v q .We may regard each e v P E v as a smooth section of π ˚ Y v Λ , b T vert Y v restricted to thegraph of u v . Then we extend them to global smooth sections of π ˚ Y v Λ , b T vert Y v over Y v , or equivalently, construct a linear map ι v : E v Ñ C p Y v , π ˚ Y v Λ , b T vert Y v q . Then define E α “ à v P Irre C α E v . We have a linear map ι α : E α Ñ C p Y α , π ˚ Y Λ , b T vert Y α q . (10.10)We have not imposed the Γ α equivariance condition. Recall that Γ α acts on Y α andthe bundle π ˚ Y Ω , b T vert Y α . Hence by enlarging E α so that it becomes Γ α -invariantwhile remaining finite-dimensional. Hence the above inclusion is Γ α -equivariant.Lastly we need to construct a Γ α -equivariant linear map ι α : E α Ñ C p Y α , Ω , Y α { V α b T vert Y α q which extends (10.10). Notice that we have a Γ α -equivariant decomposition E α “ E thick α ‘ E thin α and such that ι α “ ι thick α ‘ ι thin α where ι thick α : E thick α Ñ C p Y thick α , π ˚ Y Λ , ‘ T vert Y thick α q , (10.11) ι thin α : E thin α Ñ C p Y thin α , π ˚ Y Λ , b T vert Y thin α q . (10.12) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 99
For the part E thick α which comes from obstructions over stable components, using thetrivialization t thick Y in (10.9), the map (10.11) can be extended to ι thick α : E thick α Ñ C p Y thick α , Ω , Y α { V α b T vert Y thick α q . On the other hand, for the part E thin α which comes from the unstable components, sinceover the thin part the resolution data provides canonical cylindrical coordinates as wellas a trivialization of Y α | U thin α , one has the identification C p Y thin α , π ˚ Y Ω , b T vert Y α q » C p U thin α ˆ X, T X q Then using the trivialization t thin Y in (10.9), one can extend the inclusion (10.12) to alinear map ι thin α : E thin α Ñ C p Y thin α , π ˚ Y Ω , b T vert Y α q . Both ι thin α and ι thick α remain Γ α -equivariant. Hence their direct sum ι α : “ ι thick α ‘ ι thin α provides the last piece of the thickening datum α .This finishes the proof of Lemma 10.7.10.3. Nearby solutions.
Let α be a thickening datum, which contains an r -spin curve C α and stable solution v α to the gauged Witten equation. Then for any gluing parameter ζ , we have constructed an approximate solution v app α,ζ over every curve of the form C α,η,ζ .Notice that the role of the deformation parameter η is only to vary the complex structure(including the position of the markings) on the same curve.We would like to have a quantitative way to measure the distance between an α -thickened solution to the central one. When the α -thickened solution is defined over aresolved domain, we consider the distance from a corresponding approximate solution.Hence we need to define certain weighted Sobolev norm over the approximate solution. Definition 10.8. (Weight function over the resolved domain) Let r α be the resolutiondatum contained in the thickening datum α . Let U ˚ α Ă U α be the complement of thenodes and punctures. Define a function ω α : U ˚ α Ñ R ` as follows. For each component of the thin part U thin α,i Ă U thin α , if it corresponds to amarking with fibrewise cylindrical coordinate s ` i t , then define ω α | U thin α,i “ e s . If the component U thin α,i corresponds to a node, then for each corresponding gluing param-eter ζ i with the long cylinder w ` i w ´ i “ ζ i with | w ˘ i | ď r i , then define ω α | U thin α,i “ | w ` i | ´ , a | ζ i | ď | w ` i | ď r i , | w ´ i | ´ , a | ζ i | ď | w ´ i | ď r i . Then extend ω α | U thin α to a smooth function over the thick part that has positive valueswith minimal value at least 1. For any deformation parameter η P V α, def and gluingparameter ζ P V α, res with p η, ζ q P V α , define ω α,η,ζ : “ ω α | C α,η,ζ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 100
Definition 10.9. (Weighted Sobolev norms over the resolved domain) Let α be a thick-ening data and w α be the weight function defined in Definition 10.8. Then for any p η, ζ q P V α , define the weighted Sobolev norm } f } W k,p,wα,η,ζ “ k ÿ l “ ” ż Σ C α,η,ζ | ∇ l f | p ω pwα,η,ζ ν α,η,ζ . ı p Here ν α,η,ζ P Ω p Σ C α,η,ζ q is the area form of the family of cylindrical metrics specified inSubsection 2.5, ∇ is the covariant derivative associated to the same cylindrical metric,and | ¨ | is the norm on tensors associated to the same cylindrical metric.Now for any p η, ζ q , the object p C α,η,ζ , y α,η,ζ q is a stable generalized r -spin curve. Thereis also the principal K -bundle P α,η,ζ Ñ C α,η,ζ which is contained in the resolution data.Forgetting y α,η,ζ , one can define the Banach manifold B p,wα,η,ζ “ B p,w C α,η,ζ ,P α,η,ζ of gauged maps from C α,η,ζ to X which satisfy the prescribed asymptotic constrain. Thereis also the Banach space bundle E p,wα,η,ζ “ E p,w C α,η,ζ ,P α,η,ζ Ñ B p,wα,η,ζ . On the fibres of E p,wα,η,ζ and the tangent spaces of B p,wα,η,ζ , we define the weighted Sobolevnorms associated to the weight function w α,η,ζ . Notice that for any gauged map v “p A, u q P B p,wα,η,ζ , we use the covariant derivative associated to A in the definition of higherSobolev norms. Definition 10.10. ( ǫ -closedness) Let p C , y q be a stable generalized r -spin curve and v “ p P, A, u q be a smooth gauged map over C . Let ǫ ą p C , v , y q is ǫ -close to α , if there exists an inclusion of bundles P p φ α / / (cid:15) (cid:15) P φ α “ P α,η,ζ (cid:15) (cid:15) C φ α / / C φ α “ C α,η,ζ satisfying the following conditions.(a) η P V ǫα, def and ζ P V ǫα, res .(b) If we view p p φ ´ α q ˚ v as a gauged map over C α,η,ζ which is smoothly identified with C α,ζ , then it is in the ǫ -neighborhood of v app α,ζ in the Banach manifold B p,wα,ζ .We say that p C , v q is ǫ -close to α if there exists a stabilizing list y such that p C , v , y q is ǫ -close to α . Here ends Definition 10.10.We need to prove that the notion of ǫ -closedness defines open neighborhoods of p α in M α . This is necessarily to construct the manifold charts of M α near this point. Lemma 10.11.
For any ǫ ą sufficiently small, there exists an open neighborhood W ǫα of p α in M α such that for any point p P W ǫα , any representative p C , v , y , φ α , g α , e α q of p ,the triple p C , v , y q is ǫ -close to α .Proof. One can estimate the distance between p p φ ´ α q ˚ v and the approximate solution v app α,ζ by utilizing the annulus lemma (Lemma 4.6). The details are left to the reader. (cid:3) Similarly, suppose p C α , v α q represents a point p α in the moduli space M rg,n p X, G, W, µ q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 101
Lemma 10.12.
For any ǫ ą sufficiently small, there exists an open neighborhood W ǫα of p α in M rg,n p X, G, W, µ q such that for any point p P W ǫα , any representative p C , v q of p is ǫ -close to α .Proof. Left to the reader. (cid:3)
Lemma 10.13.
Given a thickening datum α , there exists ǫ α ą satisfying the followingcondition. Let p C , v q be ǫ α -close to α . Then there exist y α and p φ α satisfying the followingconditions.(a) y α stabilizes C as a generalized r -spin curve and p C , v , y α q is ǫ α -close to α .(b) p φ α is an isomorphism p φ α : p C , y α , P q » p C α,η,ζ , y α,η,ζ , P α,η,ζ q such that p p φ ´ α q ˚ v is in the ǫ α -neighborhood of v app α,ζ .(c) p p φ ´ α q ˚ v is in the Coulomb slice through v app α,ζ .(d) If we write p p φ ´ α q ˚ v “ p A, u q , then y α “ u thin & H α . In particular, if p C , v q is a stable solution to the gauged Witten equation, then ` C , v , y α , p φ α , α ˘ is an α -thickened solution.Proof. By definition, there exist y , p φ α and η, ζ satisfying the first two conditions above.Without loss of generality, we assume that C “ C α,η,ζ and v “ p P α,η,ζ , A, u q is defined over C α,η,ζ . One can choose an order among points of y , say y , . . . , y l . A small deformationof y can be written as y ` w , . . . , y l ` w l , w , . . . , w l P C since the resolution datum r α provides cylindrical coordinates near y , . . . , y l . Such asmall deformation y “ y ` w induces a small deformation of η and ζ , denoted by η w and ζ w . Then there exists a canonical isomorphism C α,η,ζ » C α,η w ,ζ w . One can then identify the approximate solution v app α,ζ w as defined over C α,η,ζ .Now we consider a nonlinear equation on variables w and h α P W ,p,wα,η,ζ p Σ C , ad P q . Foreach small w , there is a unique small gauge transformation k w “ e h w making k ˚ w v in the Coulomb slice of v app α,ζ w . Regard H α as defined by the local vanishing locus of f , . . . , f l : X Ñ C . Then define F v p w , h q “ »———– f p k ˚ w u thin w p y qq¨ ¨ ¨ f l p k ˚ w u thin w p y l qq h ´ k w fiffiffiffifl . (10.13)We would like to use the implicit function theorem to prove the existence of a zero. Firstconsider the case that v “ v app α,ζ . At w “
0, we claim that B k ˚ w u thin w B w i p y i q „ B s u thin α p y i q ` X φ thin α p y i q , B k ˚ w u thin w B w i p y i q „ B t u thin α p y i q ` X ψ thin α p y i q . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 102
The error can be controlled by | ζ | . Here w i “ w i ` i w i is the complex coordinates of w i .Indeed, the derivatives of u thin w in w i and w i at y i are clearly B s u thin α and B t u thin α . Thethe fact that the derivatives of h w are very close to φ thin α and ψ thin α follows from a gluingargument and Lemma 9.10. The proof is left to the reader.By the last condition on the hypersurface H α in the definition of thickening data (seeDefinition 10.3), the derivative of the first l coordinates of (10.13) is surjective. On theother hand, the derivative of h ´ h w on h is uniformly invertible. Hence, when ǫ α is smallenough and p p φ ´ α q ˚ v is in the ǫ α -neighborhood of v app α,ζ , the implicit function theoremimplies the existence and uniqueness of a solution F v p w , h q “ (cid:3) We also need to show the uniqueness of p y α , p φ α q up to automorphisms of p C α , v α q . Lemma 10.14.
Let α be a thickening datum. There exists ǫ α ą satisfying the followingcondition. Let p C , v q be a stable solution to the gauged Witten equation which is ǫ α -closeto α , and such that ` C , v , y α , p φ α , α ˘ , ` C , v , y α , p φ α , α ˘ are both α -thickened solution with dist p v app α,ζ , p p φ ´ α q ˚ v q ď ǫ α , dist p v app α,ζ , pp p φ α q ´ q ˚ v q ď ǫ α . Then y α “ y α and there exists γ P Aut p C α , v α q such that γ P α ˝ p φ α “ p φ α . Proof.
Let ǫ i be a sequence of positive numbers converging to zero and let p C i , v i q bea sequence of stable solutions to the gauged Witten equation which are ǫ i -close to α .Suppose we have two sequences ` C i , v i , y i , p φ i , α ˘ , ` C i , v i , y i , p φ i , α ˘ which are α -thickened solutions satisfyingdist ` v app α,ζ i , p p φ ´ i q ˚ v i ˘ ď ǫ i , dist ` v app α,ζ i , pp p φ i q ´ q ˚ v i ˘ ď ǫ i . If we can show that for large i , y i “ y i and there exists γ i P Aut p C α , v α q such that γ P α i ˝ p φ i “ p φ i then the lemma follows. Suppose this is not the case. Then by taking a subsequence, wemay assume that all C i has the same topological type, and that the maps p φ i , p φ i : C i Ñ C α,η i ,ζ i have their combinatorial types independent of i . More precisely, let the r -spindual graphs of C i be Π and the r -spin dual graph for C α be Γ. Then there are two fixedmaps ρ, ρ : Γ Ñ Π modelling the maps p φ i and p φ i . Let v P V p Π q be an arbitrary vertex,which corresponds to a subtree Γ v Ă Γ and Γ v Ă Γ. Then consider the isomorphisms p φ i,v : C i,v » C α,η i ,ζ i ,v , p φ i,v : C i,v » C α,η i ,ζ i ,v . We claim that there is a subsequence (still indexed by i ) for which p φ i,v ˝ p φ ´ i,v converges to an isomorphism γ v : p C α,v , P α,v q » p C α,v , P α,v q . Here C α,v , C α,v Ă C α are the r -spin curves corresponding to the subtrees Γ v and Γ v .Moreover, one has γ ˚ v v α,v “ v α,v . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 103
The proof of the claim is left to the reader.By the above claim, γ v induces a family of isomorphisms p γ U α v , γ P α v q : p C α,η v ,ζ v , P α,η v ,ζ v q » p C α,γ v η v ,γ v ζ v , P α,γ v η v ,γ v ζ v q . Here p η v , ζ v q denote the deformation and gluing parameters on the subtree Γ v , and p γ v η v , γ v ζ v q are the transformed deformation and gluing parameters on the subtree Γ v .We can see that for i sufficiently large, one has that | γ v η i ´ η i | Ñ , | log p γ v ζ i,v q ´ log ζ i,v | Ñ . It means that there is a (canonical) isomorphism ϑ i,v : C α,γ v η i,v ,γ v ζ i,v » C α,η i,v ,ζ i,v and that the extra markings γ v y i,η v ,ζ v differ from y i,η v ,ζ v by a small shift. Moreover,there is a bundle isomorphism p ϑ i,v lifting ϑ i,v such that we can write e h i,v “ k i,v “ p φ i,v p p φ i,v q ´ p ϑ i,v ˝ γ P α v : P α,η i,v ,ζ i,v Ñ P α,η i,v ,ζ i,v , and } h i,v } W ,p,wηi,v,ζi,v Ñ . Now using p φ i,v we may regard v i restricted to the component v (denoted by v i,v ) isdefined over C α,η i,v ,ζ i,v with extra marked points equal to y α,η i,v ,ζ i,v and we regard v app α,ζ i,v defined on the same curve with possibly different extra markings y α,η i,v ,ζ i,v . But thetwo sets of markings are very close to each other (distance measured via the cylindricalmetric). Moreover, we have the condition that v i,v is in Coulomb gauge relative to v app α,ζ i,v while k ˚ i,v v i,v is in the Coulomb slice through v app α,ζ i,v , and u thin i,v p y α,η i,v ,ζ i,v q Ă H α , k ˚ i,v u thin p y α,η i,v ,ζ i,v q Ă H α . Then by the uniqueness part of the implicit function theorem and Lemma 10.13, we havethat for i large, k i,v is the identity and y α,η v ,ζ v “ y α,η v ,ζ v . Hence it follows that for big i ,one has γ P α i,v ˝ p φ i,v “ p φ i,v Putting all components of C i together, one obtains an automorphism γ P Aut p C α , v α q and obtains that for large i , γ P α p φ i “ p φ i and y i “ y i ðñ γ U α y α,η i,v ,ζ i,v “ y α,η i,v ,ζ i,v . Hence we finished the proof. (cid:3)
Thickened moduli space and gluing.
In this subsection we construct a chartof topological manifold for the moduli space of α -thickened solutions. This induces avirtual orbifold chart of the moduli space M . We first show that the α -thickened modulispace M α is a topological manifold near p α . Proposition 10.15.
There is a Γ α -invariant neighborhood of p α which is a topologicalmanifold. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 104
It suffices to construct a chart, and the construction is based on gluing. For eachirreducible component v of C α , there is the augmented linearization map D v : T v v B v Ñ E v which is a Fredholm operator. The domain of the total linearization is the subspace ofdirect sum of T v v B v under the constrain given by the matching condition. Let T v v B v Ă T v v B v be the finite-codimensional subspace consisting of infinitesimal deformations whosevalues at special points are zero. Then set T v α B α “ à v T v v B v The restriction D α : T v α B α Ñ E α is still Fredholm.The following is the main gluing theorem for the α -thickened moduli space. Let M ` α be the moduli space of the same type of objects as M α without imposing the conditionthat y α “ u thin & H α . Then M ` α contains M α as an Γ α -invariant subset. Proposition 10.16.
Abbreviate ξ “ p ξ, η, ζ q . Let α be a thickening datum. Then for ǫ ą sufficiently small, there exist a family of objects ˆ v α, ξ “ ` v α, ξ , e α, ξ ˘ , ξ “ p ξ, η, ζ q P V ǫα, map ˆ V ǫα, def ˆ V ǫα, res (10.14) where v α, ξ is a gauged map over P α,η,ζ Ñ C α,η,ζ and e α, ξ P E α . They satisfy the followingconditions.(a) When ξ “ p ξ, η, q , ˆ v α, ξ coincides with ˆ v α,ξ,η of (10.5) .(b) The family if Γ α -equivariant in the sense that for any γ α P Γ α , p v α,γ α ξ , e α,γ α ξ q “ p v α, ξ ˝ γ ´ α , γ α e α, ξ q . (c) There holds ˆ F α,η,ζ ` ˆ v α, ξ ˘ “ . (d) The natural map V ǫα, map ˆ V ǫα, def ˆ V ǫα, res Ñ M ` α defined by ξ ÞÑ “ C α,η,ζ , v α, ξ , y α,η,ζ , p φ α,η,ζ , e α, ξ ‰ (10.15) is a Γ α -equivariant homeomorphism onto an open neighborhood of p α . Here y α,η,ζ and p φ α,η,ζ are defined tautologically. As a consequence one obtains a local chart of the original moduli space. Namely, defineˆ U ǫα “ ! ξ P V ǫα, map ˆ V ǫα, def ˆ V ǫα, res | u thin α, ξ & H α “ y α,η,ζ ) . In the following notations, we omit the dependence on ǫ . This is a topological manifoldacted continuously by Γ α . Define U α “ ˆ U α { Γ α , E α “ p ˆ U α ˆ E α q{ Γ α . (10.16)Then E α Ñ U α is an orbifold vector bundle. The natural map ˆ S α : M α Ñ E α induces asection S α : U α Ñ E α (10.17)and there is an induced map ψ α : S ´ α p q Ñ M Γ ψ α “ ˆ v α, ξ ‰ “ “ C α,η,ζ , v α, ξ ‰ . (10.18)Let the image of ψ α be F α . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 105
Corollary 10.17.
When ǫ is small enough, the 5-tuple C α “ p U α , E α , S α , ψ α , F α q (10.19) is a topological virtual orbifold chart around of M Γ around p α .Proof. According to Definition 7.18, one only needs to prove that ψ α is a homeomorphismonto F α and F α is an open neighborhood of p α . Since U α is a topological orbifoldwhich is locally compact, while the moduli space M Γ is Hausdorff, to prove that ψ α is a homeomorphism, one only needs to verify it is one-to-one. Suppose there are twoisomorphism classes of α -thickened solutions p “ ” C , v , y α , p φ α , α ı , p “ ” C , v , y α , p φ α , α ı whose Γ α -orbits are mapped to the same point in M Γ . Then by definition, p C , v q isisomorphic to p C , v q , meaning that there is an isomorphism ρ : C Ñ C as r -spin curvesand an isomorphism h : P Ñ P as K -bundles that cover ρ such that h ˚ v “ v . Thereforewe may regard the two α -thickened solutions are defined on the same r -spin curve C withthe same gauged map v . Their differences are the stabilizing points y α , y α and the bundleinclusions p φ α , p φ α . Moreover, by Proposition 10.16, they are isomorphic as α -thickenedsolutions to two specific exact solutions, and hence in particular ǫ α -close to α . Then byLemma 10.14, y α “ y α , and p φ α and p φ α differ by an action of Γ α . It means that the twopoints in M α are on the same Γ α -orbit, from which injectivity of ψ α follows.It remains to show the local surjectivity, namely F α contains an open neighborhood of p α . Suppose on the contrary that this is not true. Then there exists a sequence of stablesolutions p C i , v i q which converge modulo gauge in c.c.t. to p C α , v α q . Then by Lemma10.12, given any ǫ ą
0, for sufficiently large i , p C i , v i q will be ǫ -close to α . Then byLemma 10.13, for large i one can upgrade p C i , v i q to an α -thickened solution, and theisomorphism classes of the sequence of α -thickened solutions converge in the topology of M α to p α . Hence for large i , the isomorphism class p i of p C i , v i q is indeed in the imageof ψ α , which is a contradiction. (cid:3) Proof of Proposition 10.16.
The strategy of the proof is standard. However weneed to set up the Fredholm theory more carefully.First we construct a family of approximate solutions using the pregluing constructionas in Definition 10.5. More precisely, we have obtained a family of α -thickened solutions ´ C α,ξ,η , y α,ξ,η , v α,ξ,η , φ α,ξ,η , g α,ξ,η , e α,ξ,η ¯ . Notice that the domains C α,ξ,η are smoothly identified with the domain C α where theidentification is contained in the thickening datum α . Hence we only need to repeatthe pregluing construction of Definition 10.5 while replacing the central element v α by v α,ξ,η . Then for each small gluing parameter ζ P V ǫα, res , denoting ξ “ p ξ, η, ζ q , we obtaina gauged map v app α, ξ : “ v app α,ξ,η,ζ . Define e app α, ξ “ e app α,ξ,η,ζ “ e α,ξ,η and ˆ v app α, ξ “ p v app α, ξ , e app α, ξ q . Our next task is to “estimate the error” of the approximate solution.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 106
Lemma 10.18.
There exist C α ą and ǫ α ą such that for | ζ | ď ǫ α , we have } ˆ F α,η,ζ p ˆ v app α, ξ q} W ,p,wα,η,ζ ď C α | ζ | ´ τ ´ w . Proof.
For every irreducible component Σ v Ă Σ α,η,ζ , the gauged Witten equation plus thegauge fixing condition relative to the central approximate solution has three components,denoted by ˆ F , ˆ F , ˆ F .For a “ , ,
3, we first estimate ˆ F a p ˆ v app α, ξ q over Σ T . Indeed, over Σ T Ă Σ α,η,ζ , we haveˆ v app α,ξ,η,ζ “ ˆ v α,ξ,η . Hence we haveˆ F α,η,ζ p ˆ v app ξ q “ F α,η,ζ p ˜ v η,η q` ι α,η,ζ p e α,ξ,η , u α,ξ,η q “ ι α,η,ζ p e α,ξ,η , u α,ξ,η q´ ι α,η, p e α,ξ,η , u α,ξ,η q . Since the inclusion ι α is smooth in ζ , and the supports of the images of ι α,η,ζ are containedin a region where the weight function is uniformly bounded, there is C α ą } ˆ F α,η,ζ p ˆ v app α, η q} W ,p,wα,η,ζ p Σ T q ď C α | ζ | . Then look at the neck regions, where the domain complex structure is fixed and theobstruction vanishes. Hence ˆ F a p ˆ v app α, ξ q “ F a p v app α, ξ q . Recall how the approximate solution is defined in Definition 10.5. Here we see over theinterval C ˘ T “ ˘r , T s the approximate solution is u app α, ξ “ exp x w p t q p ρ T ˘ ξ ˘ q , φ app “ ρ T ˘ β ˘ ,s , ψ app “ ρ T ˘ β ˘ ,t ` η w p t q . Here p x p t q , η p t qq is the limiting critical loop at the node w which satisfies x p t q ` X η p t q p x p t qq “ . Using the splitting
T X “ H X ‘ G X , we can decompose ξ ˘ “ ξ H ˘ ` ξ G ˘ . Then by the exponential decay property (see Theorem 5.9), one hassup C ˘ T ” } ξ ˘ p s, t q} ` }B s ξ ˘ p s, t q} ` } ∇ t ξ ˘ p s, t q ` ∇ ξ ˘ X η } ı ď e ´ τT , (10.20)and for a “ s, t ,sup C ˘ T ” } β ˘ ,a p s, t q} ` }B s β ˘ ,a p s, t q} ` }B t β ˘ ,a p s, t q ` r η p t q , β ˘ ,a p s, t qs} ı ď e ´ τT . (10.21)Moreover, denote the derivative of the exponential map by two maps E , E , namely d exp x ξ “ E p x, exp x ξ q dx ` E p x, exp x ξ q ∇ ξ. Then because the metric is ¯ K -invariant, for all ¯ a P ¯ k , one has (see [GS05, Lemma C.1]) X ¯ a p exp x v q “ E p x, exp x v q X ¯ a p x q ` E p x, exp x v q ∇ v X ¯ a p x q . (10.22)Moreover, the bundle is trivialized over the neck region. Hence we can identify theconnections with 1-forms and sections with maps. –Estimate F over the neck region. One has F p v app α, ξ q “ B s u app α, ξ ` X φ app α, ξ p u app α, ξ q ` J ´ B t u app α, ξ ` X ψ app α, ξ p u app q ¯ ` ∇ W p u app α, ξ q . We estimate each term above as follows. First, by the definition of E , E , one has ››› B s u app α, ξ ››› “ ››› B s ` exp x w p t q ρ T ˘ ξ ˘ ˘››› “ ››› E ` B s ρ T ˘ ξ ˘ ` ρ T ˘ B s ξ ˘ ˘››› ď C ´ } ξ ˘ } ` }B s ξ ˘ } ¯ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 107
Further by (10.22), one has ››› X φ app α, ξ p u app α, ξ q ››› “ ››› X ρ T ˘ β ˘ ,s ` exp x w p t q ρ T ˘ ξ ˘ ˘››› “ ρ T ˘ ››› E ` X β ˘ ,s p x w p t qq ˘ ` E ` ∇ ρ T ˘ ξ ˘ X β ˘ ,s ˘››› ď C ›› β ˘ ,s ›› . Also by (10.22), one has B t u app α, ξ ` X ψ app α, ξ p u app α, ξ q“ B t ` exp x w p t q ρ T ˘ ξ ˘ ˘ ` X ψ app α, ξ ` exp x w p t q ρ T ˘ ξ ˘ ˘ “ E x w p t q ` ρ T ˘ E p ∇ t ξ ˘ q ` E X ψ app α, ξ ` E ∇ ρ T ˘ ξ ˘ X ψ app α, ξ “ ρ T ˘ E p ∇ t ξ ˘ q ` E X ρ T ˘ β ˘ ,t ` ρ T ˘ E ∇ ξ ˘ X ψ app α, ξ “ ρ T ˘ E ` ∇ t ξ ˘ ` ∇ ξ ˘ X η w ˘ ` ρ T ˘ E p X β ˘ t q ` p ρ T ˘ q E ∇ ξ ˘ X β ˘ ,t Hence ››› B t u app α, ξ ` X ψ app α, ξ p u app α, ξ q ››› ď C ´ } ξ ˘ } ` } ∇ t ξ ˘ ` ∇ ξ ˘ X η w } ` } β ˘ ,t } ¯ . Lastly because ∇ W p x w p t qq ”
0, one has ››› ∇ W p u app α, ξ q ››› “ ››› ∇ W p exp x w p t q ρ T ˘ ξ ˘ q ››› ď C } ξ ˘ } . In all the above estimates, the constant C can be made independent of ξ . Then using(10.20), (10.21), one has ››› F p v app α, ξ q ››› L p,w p C ˘ ,T q ď Ce ´ τT ” ż C ˘ ,T e pws dsdt ı p ď Ce ´p τ ´ w q T . (10.23) –Estimate F over the neck region. Over the neck region one has F p v app α, ξ q “ B s ψ app α, ξ ´ B t φ app α, ξ ` ” φ app α, ξ , ψ app α, ξ ı ` µ p u app α, ξ q“ B s p ρ T ˘ β ˘ ,t q ´ B t p ρ T ˘ β ˘ ,s q ` ρ T ˘ “ β ˘ ,s , η w p t q ` ρ T ˘ β ˘ ,t ‰ ` µ ´ exp x w p t q ρ T ˘ ξ ˘ ¯ . Using (10.21), one has ››› B s p ρ T ˘ β ˘ ,t q ´ B t p ρ T ˘ β ˘ ,s q ` ρ T ˘ r β ˘ ,s , η w p t q ` ρ T ˘ β ˘ ,t s ››› L p,w p C ˘ T q ď Ce ´p τ ´ w q T . On the other hand, we know µ p x w p t qq ”
0. Hence by (10.20), one has ››› µ ´ exp x w p t q ρ T ˘ ξ ˘ ¯››› L p,wα,ζ p C ˘ T q ď C ›› ρ T ˘ ξ G ˘ ›› L p,w p C ˘ T q ď Ce ´p τ ´ w q T . Hence we have ››› F p v app α, ξ q ››› L p,wα,ζ p C ˘ T q ď Ce ´p τ ´ w q T . (10.24) –Estimate F over the neck region. The estimate of F is slightly different becausethe gauge fixing is relative to the approximate solution. Let the two sides of the originalsingular solutions be p u ˘ α , φ ˘ α , ψ ˘ α q , p u ˘ α,ξ,η , φ ˘ α,ξ,η , ψ ˘ α,ξ,η q . The central approximate solution is p u app α,ζ , φ app α,ζ , ψ app α,ζ q . Over the semi-infinite cylinders r T, `8q ˆ S we write u ˘ α,ξ,η “ exp u ˘ α v ˘ α,ξ,η . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 108
Over the long cylinder r´ T, T s ˆ S we write u app α, ξ “ exp u app α,ζ v α, ξ . Then F p v app α, ξ q “ B s p φ app α, ξ ´ φ app α,ζ q ` “ φ app α,ζ , φ app α, ξ ´ φ app α,ζ ‰ ` B t ` ψ app α, ξ ´ ψ app α,ζ ˘ ` “ ψ app α,ζ , ψ app α, ξ ´ ψ app α,ζ ‰ ` dµ p u app α,ζ q ¨ J v α, ξ We still estimate each term separately. ››› B s ` φ app α, ξ ´ φ app α,ζ ˘››› “ ››› B s ` ρ T ˘ p φ ˘ α,ξ,η ´ φ ˘ α q ˘››› ď C ´ } φ ˘ α,ξ,η ´ φ ˘ α } ` }B s p φ ˘ α,ξ,η ´ φ ˘ α q} ¯ . Lastly we estimate dµ ¨ J v α, ξ . We know thatsup r T, `8qˆ S ››› dµ p u ˘ α q ¨ J v ˘ α,ξ,η ››› ď Ce ´ τT . Moreover, by (10.20), over C ˘ T , the C distance between u app α,ζ and u ˘ α is controlled by e ´ τT , while the C distance between u app α, ξ and u ˘ α,ξ,η is controlled by e ´ τT . Hence we stillhave sup C ˘ T ››› dµ p u app α,ζ q ¨ J v α, ξ ››› ď Ce ´ τT . (cid:3) Estimate the variation of the linear operator.
Consider the E α -perturbed, aug-mented operator ˆ F α,ζ : E α ˆ B p,wα,ζ Ñ E p,wα,ζ . We identify a neighborhood of ˆ v app α,ζ in E α ˆ B p,wα,ζ with a neighborhood ˆ O α,ζ of the originin the tangent space. Then an element ˆ v near ˆ v app α,ζ is identified with a point ˆ x P ˆ O α,ζ .We also trivialize the bundle E p,wα,ζ . Then the map ˆ F α,η,ζ can be viewed as a nonlinearoperator between two Banach spaces. To apply the implicit function theorem one needsto have a quadratic estimate. Lemma 10.19. (Quadratic estimate)
There exist δ α ą , ǫ α ą and C α ą such thatfor all | ζ | ď ǫ α and ˆ x , ˆ x P ˆ O α,ζ corresponding to ˆ v , ˆ v with } ˆ x } , } ˆ x } ď δ α , we have ››› ˆ D ˆ v ´ ˆ D ˆ v ››› ď C α ´ } ˆ x } ` } ˆ x } ¯››› ˆ x ´ ˆ x ››› . Proof.
It can be proved by straightforward calculation. Notice that because the weightfunctions defining the weighted Sobolev norms are uniformly bounded from below, wealways have the Sobolev embedding W ,p,wα,η,ζ ã Ñ C with a uniform constant. The details are left to the reader. (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 109
Constructing the right inverse.
We construct a right inverse to the linear operatorˆ D α,ζ . Choose a constant ~ P p , q . Define χ ~ T ˘ : R Ñ r , s such that supp χ ~ T ´ Ă p´8 , ~ T s , supp χ ~ T ` Ă r´ ~ T, `8q ; χ ~ T ´ | p´8 , s ” χ ~ T ` | r , `8q ” , sup | ∇ χ ~ T ´ | , sup | ∇ χ ~ T ` | ď ~ T . (10.25)Now we compare the approximate solution v app α,ζ with v α . For each irreducible compo-nent v P IrreΣ C α , we can identify Σ v, T with a compact subset of Σ α,ζ andΣ α,ζ “ ď v Σ v, T . Further the intersections of the pieces on the right hand side are circles. Furthermore,by the construction of the approximate solution, we know that the distancedist ´ u app α,ζ | Σ v, T , u α | Σ v, T ¯ is sufficiently small. Then we can define the parallel transport Par v : u ˚ α T X | Σ v, T Ñ p u app α,ζ q ˚ T X | Σ v, T . The union over all components v is denoted by Par . Further, the bundle ad P C .We define two maps, Cut : E v app α,ζ Ñ E v α , and Glue : E α ‘ T v α B p,wα Ñ E α ‘ T v app α,ζ B p,wα,ζ . –Definition of Cut . Given ς P E v app α,ζ and each irreducible component v of C α , restrict ς to Σ v, T , and use the inverse of Par v to obtain an element of E v α over the component v .This is well-defined as an element of E is only required to have L p,δ -regularity. –Definition of Glue . Glue maps E α identically to E α . For ξ P T v α B p,wα and eachirreducible component v of C α , restrict v η to Σ v, p ` ~ q T .We estimate the norms of Cut and
Glue . Lemma 10.20.
There is a universal constant C ą such that } Cut } ď C, } Glue } ď C. Proof.
By the way we define the norms (see Subsection 9.2 and Definition 10.8, Definition10.9), we see that } Cut } ď
1. To estimate the norm of
Glue , it remains to estimate thenorm of the first order derivatives. Given ξ v P T v α,v , we see that } Glue p ξ v q} W ,p,δ “ } ξ v } W ,p,δ p Σ v, T q ` } Paste p ξ v qq} W ,p,δ pr T, p ` ~ q T sˆ S q . The last term can be controlled as follows. } Paste p ξ v q} W ,p,δ pr T, p ` ~ q T sˆ S q ď ´ ` ~ T ¯ } ξ v } W ,p,δ ď } ξ v } W ,p,δ . This gives a bound on the norm of
Paste . (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 110
Now we can define the approximate right inverse. Recall that we have chosen a rightinverse to the linear operator ˆ D α :ˆ Q α : E p,wα | v α Ñ E α ‘ T v α B p,wα . Then define ˆ Q app α,ζ “ Glue ˝ ˆ Q α ˝ Cut . (10.26)By Lemma 10.20, there exist C α ą ǫ α ą } ˆ Q app α,ζ } ď C α , @ ζ P p , ǫ α q . (10.27)Next we show that the family of operators ˆ Q app α,ζ are approximate right inverses to ˆ D α,ζ . Lemma 10.21.
For T sufficiently large, one has ››› ˆ D α,ζ ˝ ˆ Q app α,ζ ´ Id ››› ď . (10.28) Proof.
We need to estimate ››› ˆ D α,ζ ` ˆ Q app α,ζ p ς q ˘ ´ ς ››› L p,wα,ζ for any ς P E α,ζ | v app α,ζ . Since the regularity of ς is only L p and Σ α,ζ is the union of Σ v, T for all irreducible components v of Σ α , it suffices to consider the case that ς ζ is supportedin Σ v, T for one component v . Without loss of generality, assume that v has only onecylindrical end and let the cut-off function be denoted by χ ~ Tv . Denote the paralleltransport ς α “ Par ´ ς P L p,w p Σ α,v , u ˚ α T vert Y ‘ ad P ‘ ad P q . Then by definition, over Σ v, T where χ ~ Tv ”
1, one hasˆ D α,ζ ` ˆ Q app α,ζ p ς q ˘ ´ ς “ ˆ D α,ζ ` Par ` ˆ Q α p ς α q ˘˘ ´ Par ` ˆ D α ` ˆ Q α p ς α q ˘˘ . (10.29)Over the complement of Σ v, T where ς ”
0, one hasˆ D α,ζ ` ˆ Q app α,ζ p ς q ˘ ´ ς “ ˆ D α,ζ ` Par ` χ ~ Tv ` ˆ Q α p ς α q ˘˘˘ ´ Par D α Q α ξ α “ “ ˆ D α,ζ , χ ~ Tv ‰ Par ` ˆ Q α p ς α q ˘ ` χ ~ Tv ` ˆ D α,ζ Par ´ Par ˆ D α ˘` ˆ Q α p ς α q ˘ . Hence the last line coincides with (10.29). Since ˆ D α,ζ is a first order operator, by (10.25), ›››“ ˆ D α,ζ , χ ~ Tv ‰ Par ` ˆ Q α p ς α q ˘››› L p,wα,ζ ď sup ˇˇ ∇ χ ~ Tv ˇˇ›› ˆ Q α p ς α q ›› L p,wα,ζ ď C α ~ T } ς } L p,wα,ζ . (10.30)The last coefficient can be made arbitrarily small when T is large. On the other hand,to estimate ››› χ ~ Tv ` ˆ D α,ζ Par ´ Par ˆ D α ˘` ˆ Q α p ς α q ˘››› we need to estimate the variation of the linear maps. Indeed, using the same method asproving Lemma 10.19, one can show that when ζ is small, one has ››› χ ~ Tv ` ˆ D α,ζ Par ´ Par ˆ D α ˘` ˆ Q α p ς α q ˘››› ď C α } ς } L p,wα,ζ } ζ } (10.31)for some abusively used C α ą
0. Combining (10.30) and (10.31), one obtains (10.28). (cid:3)
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 111
It follows from Lemma 10.21 that when | ζ | is sufficiently small, one has the followingexact right inverse to ˆ D α,ζ : ˆ Q α,ζ : “ ˆ Q app α,ζ ˝ ` ˆ D α,ζ ˝ ˆ Q app α,ζ ˘ ´ . (10.27) and Lemma 10.21 further imply that the norm of ˆ Q α,ζ is uniformly bounded forall ζ by a constant C α ą Lemma 10.22. (Implicit function theorem) [MS04, Proposition A.3.4]
Let X , Y beBanach spaces, U Ă X be an open subset, and F : U Ñ Y be a C map. Let x ˚ P U be such that the differential dF p x ˚ q : X Ñ Y is surjective and has a bounded linearright inverse Q : Y Ñ X . Choose positive r ą and C ą such that } Q } ďď C , B r p x ˚ , X q Ă U , and } x ´ x ˚ } ď r ùñ } dF p x q ´ D } ď C . (10.32)
Suppose that x P X satisfies } F p x q} ă r C , } x ´ x ˚ } ă r . Then there exists a unique x P X such that F p x q “ , x ´ x P Im Q, } x ´ x ˚ } ď r. Moreover, } x ´ x } ď C } F p x q} . Indeed, we identify x ˚ with the central approximate solution ˆ v app α,ζ , identity X “ E α ‘ T v app α,ζ B p,wα,ζ , Y “ E p,wα,ζ | v app α,ζ , Then one can apply the implicit function theorem. More precisely, for each approximatesolution ˆ v app α, ξ , we can write it uniquely asˆ v app α, ξ “ exp ˆ v app α,ζ ˆ x app α, ξ , where ˆ x app α, ξ P ˆ T ˆ v app α,ζ . We summarize the result as the following proposition.
Proposition 10.23.
There exists ǫ α ą , δ α ą , C α ą satisfying the followingproperties. For each ξ P V α, map ˆ V α, def ˆ V α, res with } ξ } ď ǫ α , there exists a unique ˆ x cor α, ξ P ˆ T ˆ v app α,ζ satisfying ˆ F α, ξ ´ exp ˆ v app α,ζ ` ˆ x app α, ξ ` ˆ x cor α, ξ ˘¯ “ , ˆ x cor α, ξ P Im ˆ Q α,ζ , ››› ˆ x app α, ξ ››› ď δ α . Moreover, there holds ››› ˆ x cor α, ξ ››› ď C α ››› ˆ F α, ξ p ˆ v app α, ξ q ››› . Now we denoteˆ v α, ξ : “ ˆ v α,ξ,η,ζ : “ exp ˆ v app α,ζ ` ˆ x α, ξ ˘ : “ exp ˆ v app α,ζ ` ˆ x app α, ξ ` ˆ x cor α, ξ ˘ and call it the exact solution . Then we are ready to prove Proposition 10.16. Indeed, Item(a), (b), (c) of Proposition 10.16 all follow from the construction. The Γ α -equivariance ofItem (d) also follows from the construction. Hence we only need to prove that the map AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 112 (10.15) is a homeomorphism onto its image. Indeed, since the domain is locally compactand the target is Hausdorff, one only needs to prove that it is injective and its imagecontains an open neighborhood of p ` α in M ` α . –Injectivity Given two ξ “ p ξ , η , ζ q , ξ “ p ξ , η , ζ q . Suppose the correspondingexact solutions are isomorphic. Then by definition (see Definition 10.6), η “ η and ζ “ ζ . Hence ˆ v α, ξ and ˆ v α, ξ are in the same Banach manifold E α ˆ V α, def ˆ B p,wα,ζ .Then ˆ v α, ξ “ ˆ v α, ξ follows from the implicit function theorem. –Surjectivity We need to prove the following fact: for any sequence of points in M ` α represented by α -thickened solutions (without the requirement at the markings y α ) ´ C n , v n , y n , p φ n , e n ¯ that converge to p ` α , for sufficiently large n , ` C n , v n , y n , p φ n , e n q is isomorphic to a memberof the family of exact solutions of (10.14). Indeed, by Lemma 10.11, given any ǫ ą
0, for n sufficiently large, p C n , v n , y n , p φ n , e n q is ǫ -close to α . Then we can identify p C n , y n q withthe fibre p C α,φ n , y φ n q and regard v n as a gauged map over P α,φ n Ñ C α,φ n . Suppose C α,φ n corresponds to deformation parameter η n and gluing parameter ζ n . Then the ǫ -closednessimplies that dist p v n , v app α,ζ n q ď ǫ. Then v n belonging to the family (10.14) is a fact that follows from the implicit functiontheorem. This finishes the proof of Proposition 10.16.10.6. Last modification.
The construction of this section provides for each thickeningdatum α and a sufficiently small number ǫ α ą C α of the modulispace M Γ for any stable decorated dual graph. In particular, one can construct a chartof the top stratum M rg,n p X, G, W, µ ; ¯ B q . Further, one can shrink the set V ǫα, res of gluingparameters to obtain subcharts. Indeed, V α, res has canonical coordinates correspondingto the nodes. Choose a vector ~ǫ “ p ǫ , ¨ ¨ ¨ , ǫ m q with 0 ă ǫ i ă ǫ α corresponding to howmuch we can resolve the i -th node. We require that V ~ǫα, res is Γ α -invariant. From now on,a thickening datum α also contains a small positive number ǫ α ą ~ǫ .Each such thickening datum provides a chart C α by restricting the previous construction.11. Constructing the Virtual Cycle. III. The Atlas
In the previous section we have shown that we can construct for each stable decorateddual graph Γ and each point p P M Γ , one can construct a local chart C p “ p U p , E p , S p , ψ p , F p q whose footprint is an open neighborhood of p . See Corollary 10.17. The aim of thissection is to construct a good coordinate system out of these charts on the moduli space,which allows one to define the virtual fundamental cycle and the correlation functions.As the first step, we prove the following proposition. Proposition 11.1.
Let Γ be a stable decorated dual graph. Then there exist the followingobjects.(a) A finite collection of topological virtual orbifold charts on M Γ C ‚ p i “ p U ‚ p i , E ‚ p i , S ‚ p i , ψ ‚ p i , F ‚ p i q , i “ , . . . , N. (b) Another collection of topological virtual orbifold charts C ‚ I “ p U ‚ I , E ‚ I , S ‚ I , ψ ‚ I , F ‚ I q , I P I “ t ,...,N u r tHu . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 113 (c) Define the partial order on I by inclusion. Then for each pair I ď J , a weakcoordinate change T ‚ JI : C ‚ I Ñ C ‚ J . They satisfy the following conditions.(a) Each C ‚ p i is constructed by the gluing construction in the last section. In particular,for each i , there is a thickening datum α i which contains an obstruction space E α i ,and a Γ i -invariant open neighborhood ˜ U ‚ p i of the α i -thickened moduli space M α i which is a topological manifold such that U ‚ p i “ ˜ U ‚ p i { Γ i , E ‚ p i “ ` ˜ U ‚ p i ˆ E p i ˘ { Γ i . Moreover, p i is contained in F ‚ p i .(b) For each I P I which is identified with the set of thickening data t α i | i P I u , thereis a Γ I -invariant open subset ˜ U ‚ I Ă M I of the I -thickened moduli space which isa topological manifold, such that U ‚ I “ ˜ U ‚ I { Γ I , E ‚ I “ ` ˜ U ‚ p i ˆ E I ˘ { Γ I . Moreover, F ‚ I “ Ş i P I F ‚ p i .(c) For I “ t i u , C ‚t i u “ C ‚ p i .(d) The charts C ‚ I and the coordinate changes T ‚ JI satisfy the (COVERING CONDITION) property and the (COCYCLE CONDITION) property of Definition 7.24.(e) All charts and all coordinate changes are oriented. The construction of the objects in Proposition 11.1 is done in an inductive way. Fromnow on we abbreviate M : “ M Γ . Moreover, we know that M “ ğ Γ ď Γ M Γ . Then we list all strata Γ indexing the above disjoint union as Γ , . . . , Γ a such that Γ k ď Γ l ùñ k ď l. In each step we also need to make various choices and all relevant choices will be high-lighted as
CHOICE or CHOOSE .11.1.
The inductive construction of charts.
We first state our induction hypothesis.
INDUCTION HYPOTHESIS . For k P t , . . . , a ´ u we have constructed the following objects.(a) A collection of virtual orbifold charts C kp i “ p U kp i , E kp i , S kp i , ψ kp i , F kp i q , i “ , . . . , n k . (b) A map ρ k : t p , . . . , p n k u Ñ t Γ , . . . , Γ k u . (11.1)(c) Define I k “ t ,...,n k u r tHu . For any I P I k , a chart C kI “ p U kI , E kI , S kI , ψ kI , F kI q . They satisfy the following condition.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 114 (a) These charts satisfy Item (a), (b) and (c) of Proposition 11.1 if we replace C ‚ p i by C kp i and C ‚ I by C kI . In particular F kI “ č i P I F kp i . (11.2)(b) For each l ď k , one has M l : “ ď s ď l M Γ s Ă ď s ď l ď ρ k p p i q“ Γ s F kp i . (11.3)(c) For all l ď k and I Ă t , . . . , n l u , we have F kI ‰ H ðñ F kI X M l ‰ H (11.4) END OF THE INDUCTION HYPOTHESIS.
Suppose the
INDUCTION HYPOTHESIS holds for k . We aim at extending the objectsstated in the induction hypothesis to k ` k “ M k is compact. Denote Y k ` : “ M k ` r ď ď i ď n k F kp i which is also compact. Then for each p P Y k ` , we CHOOSE a thickening datum α p at p which provides a chart C (cid:4) p “ p U (cid:4) p , E (cid:4) p , S (cid:4) p , ψ (cid:4) p , F (cid:4) p q of M around p . Then because Y k ` is compact, we can then CHOOSE a finite collection of such charts around points p n k ` , . . . , p n k ` with footprints F (cid:4) p nk ` , . . . , F (cid:4) p nk ` such that Y k ` Ă n k ` ď i “ n k ` F (cid:4) p i . (11.5)Define I k ` “ t ,...,n k ` u r tHu . Extend the map ρ k of (11.1) to a map ρ k ` : t , . . . , n k ` u Ñ t Γ , . . . , Γ k ` u , ρ k ` p i q “ ρ k p i q , i ď n k ; Γ k ` , n k ă i ď n k ` (11.6)By (11.4) of INDUCTION HYPOTHESIS , by shrinking the range of the gluing parameters,one can
CHOOSE shrinkings F kp i Ă F (cid:4) p i for all i P t n k ` , . . . , n k ` u such that (11.5) stillholds with F (cid:4) p i replaced by F kp i and such that for all I P I k ` , we have F kI : “ č i P I F kp i ‰ H ðñ F kI X M k ` ‰ H . Now we are going to construct the charts C k ` I “ p U k ` I , E k ` I , S k ` I , ψ k ` I , F k ` I q , @ I P I k ` . If I P I k , then C k ` I will be obtained later by shrinking C kI . We assume that I P I k ` r I k .If F kI “ H , define C k ` I to be the empty chart. Then assume F kI ‰ H . Fix such an I .By abusing notations, we regard I as a set of thickening data. Recall that one hasdefine the moduli space of I -thickened solutions, denoted by M I . Every point of M I isrepresented by a tuple ` C , v , t y α u , t p φ α u , t e α u ˘ . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 115
It has a Hausdorff and second countable topology, and has a continuous Γ I -action. More-over, there is a Γ I -equivariant map ˜ S I : M I Ñ E I , and a natural map˜ ψ I : ˜ S ´ I p q Ñ M . Proposition 11.2.
For I P I k ` r I k with F kI ‰ H , there is a Γ I -invariant openneighborhood ˜ U (cid:4) I Ă M I of ˜ ψ ´ I p F kI X M k ` q Ă M I which is a topological manifold.Proof. By our assumption, there exists i P t n k ` , . . . , n k ` u X I . Hence F kI X M k ` Ă M Γ k ` . Choose any point p P F kI X M k ` . Then it can be represented by an I -thickenedsolution ˆ v I : “ ` C , v , t y α u , t p φ α u , t α u ˘ . By forgetting the data for α ‰ α i , we obtain an α i -thickened solutionˆ v α i : “ ` C α i , v α i , y α i , p φ α i , α i ˘ . Then we can identify the curve C α i and the bundle P α i Ñ C α i as a fibre over the unfolding U α i Ñ V α i contained in the thickening datum α i . Then ˆ v α i belongs to the family ˆ v α i ,ξ i ,η i (10.5) for α “ α i , where ξ i P V α i , map and η i P V α i , def .Remember that we used the right inverse to the linearization of the α i -thickened gaugedWitten equation ˆ Q α i : E α i Ñ V α i , def ‘ E α i ‘ T v αi B α i chosen in (10.4) for α “ α i . By the inclusion E α i Ă E I , we obtained a right inverseˆ Q I,α i : E α i Ñ V α i , def ‘ E I ‘ T v αi B α i . Notice that the stratum Γ k ` also specifies a stratum of the I -thickened moduli spacewith a point p I represented by the I -thickened solution ˆ v I . This combinatorial typeprovides a Banach manifold of gauged maps B Γ k ` that contains ˆ v I . Then using the rightinverse ˆ Q I,α i and the implicit function theorem, we can construct a family of I -thickenedsolutions of combinatorial type Γ k ` which are close to ˆ v I in the Banach manifold B Γ k ` ,parametrized by a topological manifold acted by Γ I .Then by turning on the gluing parameters, a family of approximate solutions can beconstructed as I -thickened objects. The same gluing construction provides a collection of I -thickened solutions parametrized by a topological manifold of the expected dimension.The gluing construction is the same as that of Section 10 because, here we just enlargedthe space of obstructions from E α i to E I . (cid:3) Then take such a Γ I -invariant neighborhood ˜ U (cid:4) I Ă M I of ˜ ψ ´ I p F kI X M k ` q , we obtaina chart C (cid:4) I “ p U (cid:4) I , E (cid:4) I , S (cid:4) I , ψ (cid:4) I , F (cid:4) I q Notice that we have the inclusion F kI X M k ` Ă F (cid:4) I . Let us summarize the charts we have obtained. For all i P t , . . . , n k ` u , we havecharts C kp i with footprints F kp i . For all I P I k , we have charts C kI with footprints F kI . Forall I P I k ` r I k , we have charts C (cid:4) I with footprints F (cid:4) I . We would like to shrink thesecharts so that their footprints satisfy (11.2) of INDUCTION HYPOTHESIS for k ` C kp i to a chart C k ` p i with footprints F k ` p i which satisfy thefollowing conditions. Define F k ` I “ č i P I F k ` p i . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 116 (a) (11.3) still holds. More precisely, for all l ď k `
1, we have M l Ă ď s ď l ď ρ k ` p p i q“ Γ s F k ` p i (b) (11.4) still holds. More precisely, for all l ď k ` I P I k ` , F k ` I ‰ H ðñ F kI X M l ‰ H . (c) For all I P I k ` r I k , we have F k ` I Ă F (cid:4) I . Then shrink all C kI for I P I k (resp. C (cid:4) I for I P I k ` r I k ) to C k ` I so that the shrunkfootprint is F k ` I . Together with the map ρ k ` defined by (11.6) these charts satisfyconditions for charts listed in INDUCTION HYPOTHESIS for k `
1. Therefore the inductioncan be carried on. After the last step of the induction, denote the charts we constructedby C N I “ p U N I , E N I , S N I , ψ N I , F N I q . They satisfy Item (a), (b) and (c) of Proposition 11.1 if we replace C ‚ I by C N I . Moreover,the footprints of C N I cover M .11.2. Coordinate changes.
Now we start to define the coordinate changes. Since themoduli space M is metrizable, one can find precompact open subsets F ‚ p i Ă F N p i such thatthe union of F ‚ p i still cover M . Define F ‚ I : “ č α i P I F ‚ p i . (11.7)Then F ‚ I Ă č α i F ‚ p i Ă F N I . Consider I ď J P I which corresponds to two sets of thickening data, which are stilldenoted by I and J . Denote E JI “ à α P J r I E α , Γ JI “ ź α P J r I Γ α , where E α are the vector spaces of obstructions which are acted by Γ α . Define˜ S JI : M J Ñ E JI to be the natural map. Then there is a natural map˜ ψ JI : ˜ S ´ JI p q Ñ M I by forgetting y α , p φ α , e α for all α P J r I . This is clearly equivariant with respect to thehomomorphism Γ J Ñ Γ I which annihilates Γ JI , hence descends to a map ψ JI : ˜ S ´ JI p q{ Γ JI Ñ M I . Lemma 11.3.
There is a Γ J -invariant open neighborhood ˜ N N JI Ă M J of ˜ S ´ J p q suchthat the map ψ JI : p ˜ N N JI X ˜ S ´ JI p qq{ Γ JI Ñ M I (11.8) is a homeomorphism onto a Γ I -invariant open neighborhood ˜ U N JI of ˜ ψ ´ I p F ‚ J q Ă M I . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 117
Proof.
First we show the surjectivity. For any p I P ˜ ψ ´ I p F ‚ J q Ă ˜ ψ ´ I p F ‚ I q , it is representedby an I -thickened solution ` C , v , t y α u α P I , t p φ α u α P I , t α u α P I ˘ . By our construction, for any β P J r I , p C , v q is ǫ β -close to β (see Definition 10.10).Hence by Lemma 10.13, there exist y β and p φ β such that ` C , v , y β , p φ β , β ˘ is a β -thickened solution. Moreover, by essentially the same method as in the proof ofLemma 10.13, for any point p I P M I that is sufficiently close to p I , for any representativethat contains a gauged map p C , v q , it is ǫ β -close to β and can be completed to a β -thickened solution. Apply this for all β P J r I , it means that any p I P M I sufficientlyclose to p I is in the image of the map (11.8). Since ˜ ψ ´ I p F ‚ J q is compact, one can choosean open neighborhood ˜ U N JI Ă M I of π ´ I p F J q which is contained in the image of (11.8).Then we could find a Γ J -invariant open subset ˜ N N JI Ă M J such that˜ N N JI X ˜ S ´ JI p q “ ˜ V N JI : “ ˜ ψ ´ JI p ˜ U N JI q . (11.9)On the other hand, Lemma 10.14 says that the map π JI : ˜ V N JI { Γ JI Ñ ˜ U N JI is bijective. Its continuity is obvious. Hence using the fact that a continuous bijectionfrom a compact space to a Hausdorff space is a homeomorphism, and using the factthat the thickened moduli spaces are locally compact and Hausdorff, after a precompactshrinking of ˜ U N JI which still contains ˜ ψ ´ I p F ‚ J q , one proves this lemma. (cid:3) Define U N JI “ ˜ U N JI { Γ I which is an open suborbifold of U N I . Lemma 11.3 implies that the inclusion ˜ V N JI ã Ñ M J induces a map between orbifolds φ N JI : U N JI Ñ U N J . The natural inclusion E I Ñ E J induces a bundle map p φ N JI : E N I | U N JI Ñ E N J which covers φ N JI . Proposition 11.4.
For every I there exists a shrinking C ‚ I of C N I and an open subset U ‚ JI Ă U N JI satisfying the following conditions.(a) The footprint of C ‚ I is F ‚ I (defined previously in (11.7) ).(b) T ‚ JI : “ p U ‚ JI , φ ‚ JI , p φ ‚ JI q : “ p U ‚ JI , φ N JI | U ‚ JI , p φ N JI | U ‚ JI q is a coordinate change from C ‚ I to C ‚ J .Proof. According to the definition of coordinate changes (Definition 7.22), we first showthat the map φ N JI : U N JI Ñ U N J AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 118 is a topological embedding of orbifolds. Indeed, one only needs to show that the map˜ S JI : M J Ñ E JI is transverse over ˜ V N JI . Indeed, choose i P I . For every p J P ˜ V N JI whichdescends to an isomorphism of I -thickened solutions p I “ π JI p p J q , for any representative ` C , v , t y α u , t p φ α u , t e α u ˘ of p I , since p I is very close to ˜ ψ ´ I p F ‚ J q Ă ˜ ψ ´ I p F ‚ I q , the linearization of the I -thickenedequation ˆ D I : T v B ‘ E I Ñ E v is surjective. Here B is the corresponding Banach manifold and E Ñ B is the Banachvector bundle. Therefore we can construct a tubular neighborhood as follows. For any p J near ˜ V N JI which is represented by a J -thickened solution ` C , v , t y α u , t p φ α u , t e α u ˘ , the J -thickened solution implies thatˆ F I p v , e I q “ error . Here e “ t e α u α P J is decomposed as p e I , e JI q where e I P E I and e JI P E JI and the error term can be controlled by the size of e JI . When the neighborhood of ˜ V N JI is sufficientlysmall, the error term is sufficiently small. Then by the implicit function theorem, thereis a unique pair p v , e I q lying in the same Banach manifold such thatˆ F I p v , e I q “ , p v , e I q ´ p v , e I q P Im ˆ Q I . This constructs a Γ J -invariant neighborhood ˜ N N JI of ˜ V N JI , a Γ J -equivariant projection˜ ν JI : ˜ N N JI Ñ ˜ V N JI . The implicit function theorem also implies that ˜ S JI induces anequivalence of microbundles. Hence ˜ S JI is transverse along ˜ V N JI and φ N JI is an orbifoldembedding.It then follows that p φ N JI is a bundle embedding covering φ N JI . It is obvious that thepair p φ N JI , p φ N JI q satisfies the (TANGENT BUNDLE CONDITION) of Definition 7.20. It remainsto shrink the charts and coordinate changes so that Item (a) and Item (b) of Definition7.22 are satisfied. We only show it for a pair I ď J , from which one can obtain a wayof shrinking all the charts and coordinate changes so that all T ‚ JI becomes a coordinatechanges. First, one can find open subsets U ‚ I Ă U N I such that ψ N I ` U ‚ I X p S N I q ´ p q ˘ “ F ‚ I , ψ N I ` U ‚ I X p S N I q ´ p q ˘ “ F ‚ I . Moreover, we may take U ‚ I such that for all J P I , U ‚ J Ă ` č I ď J ˜ N N JI ˘ { Γ J (11.10)where ˜ N N JI is the one we chose by Lemma 11.3. Then take U ‚ JI “ U ‚ I X p φ N JI q ´ p U ‚ J q Ă U N JI . That is, U ‚ JI is the “induced” domain from the shrinkings C ‚ I and C ‚ J . Define T ‚ JI “p U ‚ JI , φ ‚ JI , p φ ‚ JI q where p φ ‚ JI , p φ ‚ JI q is the restriction of p φ N JI , p φ N JI q onto U ‚ JI . T ‚ JI satisfies Item (a) of Definition 7.22 automatically, because the footprint of U ‚ I , U ‚ J and U ‚ JI are precisely F ‚ I , F ‚ J and F ‚ J . To verify Item (b), suppose x n is a sequenceof points in U ‚ JI which converges to x P U ‚ I and y n “ φ ‚ JI p x n q converges to y P AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 119 U ‚ J . Notice that each ˜ ψ ´ J p y n q is a sequence of Γ J -orbits of J -thickened solutions with˜ S JI p π ´ J p y n qq “
0. Then ˜ S JI p π ´ J p y qq “ ψ ´ J p y q Ă ˜ N N JI X ˜ S ´ JI p q “ ˜ V N JI ùñ y P φ N JI p U N JI q . Since φ N JI is injective, continuous, and x n Ñ x , we must have x P U ‚ JI and φ ‚ JI p x q “ φ N JI p x q “ y . This establishes Item (b) and finishes the proof. (cid:3) Therefore, we have constructed all objects needed for establishing Proposition 11.1.Two more items need to be verified. Firstly, the (COCYCLE CONDITION) for all the co-ordinate changes T ‚ JI follows immediately from the construction. Second, all the chartsand coordinate changes have canonical orientations, as the linearized operator is of thetype of a real Cauchy–Riemann operator over a Riemann surface with cylindrical endswithout boundary, and the asymptotic constrains at infinities of the cylindrical ends aregiven by complex submanifolds or orbifolds. Therefore Proposition 11.1 is proved.11.3. Constructing a good coordinate system.
To obtain a good coordinate systemfrom the objects constructed by Proposition 11.1, one first needs to make them satisfythe (OVERLAPPING CONDITION) property of 7.24). This can be done by shrinking thecharts. Our method is modified from the one used in the proof of [MW17, Lemma 5.3.1].Indeed, for all i P I , choose precompact shrinkings F ˝ p i Ă F ‚ p i such that there holds M “ N ď i “ F ˝ p i . We also choose an ordering of I as I , . . . , I M such that I k ď I l ùñ k ď l. Choose intermediate precompact shrinkings between F ˝ p i and F ‚ p i as F ˝ p i “ : G p i Ă F p i Ă ¨ ¨ ¨ Ă G Mp i Ă F Mp i : “ F ‚ p i . (11.11)Then for all I k P I , define F ˝ I k : “ ´ č i P I k F kp i ¯ r ´ ď i R I k G kp i ¯ . These are open subsets of the moduli space M . Lemma 11.5.
The collection t F ˝ I | I P I u satisfies the (OVERLAPPING CONDITION) prop-erty, namely, F ˝ I X F ˝ J ‰ H ùñ I ď J or J ď I. Proof.
Choose any pair I k , I l P I with F ˝ I k X F ˝ I l ‰ H . Without loss of generality, assume that k ă l . We claim that I k ď I l . Suppose it is notthe case, then there exists i P I k r I l . Take x in the intersection. Then by the definitionof F ˝ I and (11.11), x P F ˝ I k Ă F kp i Ă G lp i . On the other hand, x P F ˝ I l Ă M r G lp i Ă M r G lp i . This is a contradiction. Hence I k ď I l . (cid:3) AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 120
Therefore, we can shrink the charts C ‚ I provided by Proposition 11.1 to subcharts C ˝ I whose footprints are F ˝ I . The shrinkings induce shrunk coordinate changes T ˝ JI from T ‚ JI .If for any I P I , the shrunk F ˝ I “ H , then we just delete C ‚ I from the collection of chartsand redefine the set I . Then by Definition 7.24, the collection A ˝ : “ ´ t C ˝ I | I P I u , t T ˝ JI | I ď J P I u ¯ form a virtual orbifold atlas on M in the sense of Definition 7.24.Lastly, from A ˝ one can obtain a good coordinate system. More precisely, chooseprecompact open subsets F I Ă F ˝ I , @ I P I which still cover M . Then by Theorem 7.28, there exists a precompact shrinking A of A ˝ which is a good coordinate system (see Definition 7.27), denoted by A “ ´ t C I | I P I u , t T JI | I ď J P I u ¯ . The virtual cycle.
It is obvious that the good coordinate system A has a thick-ening (see Definition 7.35). Then by Theorem 7.37, there exist transverse multi-valuedperturbations t on A whose perturbed zero locus | ˜ s ´ p q| Ă | A | is an oriented compactweighted branched topological manifold. Using the strongly continuous map ev : A Ñ ¯ X Γ W ˆ M Γ we can pushforward the fundamental class of | s ´ p q| to a rational class ev ˚ r M Γ s vir P H ˚ p ¯ X Γ W ; Q q b H ˚ p M Γ ; Q q . The correlation function of the form (8.2) can then be defined. This formally ends ourconstruction of the GLSM correlation functions, except that we need to verify the desiredproperties in order to give a cohomological field theory on the Chen–Ruan cohomologyof ¯ X W .Lastly we would like to prove that the virtual cycle is independent of the various CHOICES we made during the construction. The main reason for this independence isthat the good coordinate systems coming from any two different systems of
CHOICES canbe put together to a good coordinate system over the space M Γ ˆ r , s .First, from the general argument, if we take a shrinking of the good coordinate system(i.e., a “reduction” in the sense of McDuff–Wehrheim), then the resulting virtual cycleremains in the same homology class.Then we compare two different systems of CHOICES made during the inductive con-struction of this section. One can stratify M Γ ˆ r , s by M Π ˆ t u , M Π ˆ t u , M Π ˆ p , q (11.12)Suppose we have two systems of structure as stated in Proposition 11.1, say basic charts C r s p i , i “ , . . . , N , C r s q j , j “ , . . . , N , sum charts C r s I , I P I : “ t p ,...,p N u r tHu , C r s I , I P I : “ t q ,...,q N u r tHu and weak coordinate changes. Then by multiplying them with r , q (resp. p , s ), oneobtains a system of basic charts indexed by elements in t p , . . . , p N u \ t q , . . . , q N u .They can be viewed as a system of basic charts that cover the first two types of strataof (11.12). Then by repeating the previous inductive construction with M Γ replaced by M Γ ˆ r , s . One can complete the induction process to construct a weak virtual orbifold AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 121 atlas ˜ A over the product M Γ ˆ r , s , or more precisely, prove an analogue of Proposition11.1 for M ˆ r , s .Then from the weak atlas ˜ A one can shrink it to make a good coordinate system, suchthat when restrict to the two boundary strata, gives shrunk good coordinate systemsof M Γ . Then a transverse perturbation provides a cobordism between the weightedbranched manifolds for the two initial systems of CHOICES .There were other
CHOICES that we made for the construction, for example, the
CHOICE on a Riemannian metric in order to define the gauge fixing condition. Different such
CHOICES can all be compared by using the cobordism argument. We leave such compar-ison to the reader to check.12.
Properties of the Virtual Cycles
In the last section we verify the properties of the virtual fundamental cycles of themoduli spaces of gauged Witten equation listed in Theorem 8.1. As we have shown inSection 8, these properties imply the axioms of the CohFT.12.1.
The dimension property.
The (DIMENSION) property of the virtual cycles fol-lows directly from the construction and the index calculation given in Section 9.12.2.
Disconnected graphs.
It is easy to see that the (DISCONNECTED GRAPH) propertycan be proved by induction on the numbers of connected components. Hence it sufficesto consider the case that Γ “ Γ \ Γ . On the other hand, this property is not as easy as itseems, as there is no obvious notion of “products” of good coordinate systems or virtualorbifold atlases. Hence one needs to work more carefully. Here we prove the followingproposition about the product of two moduli spaces. Abbreviate X “ M Γ , Y “ M Γ , Z “ X ˆ Y. Proposition 12.1.
There exist the following objects.(a) Good coordinate systems A X , A Y , and A Z on X , Y , and Z respectively. Let the charts be indexed by I , J and K respectively.(b) A strongly continuous map ev Z : A Z Ñ ¯ X Γ W ˆ ¯ X Γ W ˆ M r Γ ˆ M r Γ . (c) An injective map ι : K Ñ I ˆ J . (d) For each K P K , a precompact open embedding η K : C Z,K ã Ñ C X,I ˆ C Y,J , where ι p K q “ I ˆ J P I ˆ J . They satisfy the following properties.(a) If ι p K q “ I ˆ J , ι p K q “ I ˆ J with K ď K , then I ď I and J ď J .(b) For each K with ι p K q “ I ˆ J , if we view C Z,K as a precompact shrinking of C X,I ˆ C Y,J via the open embedding η K then for all pairs K ď K , the coordinatechanges T K K are the induced coordinate changes.(c) The evaluation maps are compatible. Namely, if ι p K q “ I ˆ J , thenev Z,K “ p ev X,I ˆ ev Y,J q ˝ η K . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 122
Then we can construct a perturbation on (a shrinking of) A Z which is of “producttype.” Indeed, suppose t X and t Y be transverse perturbations of A X and A Y respectivelysuch that the perturbed zero loci are compact oriented weighted branched manifoldscontained in the virtual neighborhoods | A X | and | A Y | . The perturbations consist ofchartwise multi-valued sections t X,I : U X,I m Ñ E X,I , t
Y,J : U Y,J m Ñ E Y,J . Via the embeddings η K , one can restrict the product of t X and t Y to a perturbation t Z defined by t Z,K : U Z,K m Ñ E Z,K , t
Z,K “ t X,I b t Y,J ˝ η K , where ι p K q “ I ˆ J. The collection obvious satisfies the compatibility condition with respect to coordinatechanges, hence a valid perturbation of t Z on A Z . We call it the induced perturbation, orthe product perturbation. Moreover, if both t X and t Y are transverse, so is t Z . However,it is not obvious whether the perturbed zero locus is compact or not. Proposition 12.2.
There exist transverse perturbations t X , t Y of A X and A Y respec-tively satisfying the following conditions.(a) The induced perturbation t Z is transverse.(b) Via the inclusion | A Z | ã Ñ | A X | ˆ | A Y | , one has (as topological spaces equippedwith the quotient topology) | ˜ s ´ Z p q| “ | ˜ s ´ X p q| ˆ | ˜ s ´ Y p q| . (12.1) Proof.
The first property about transversality can always be achieved by generic smallperturbations. Hence we only prove Item (b). Take precompact shrinkings F X,I Ă F X,I , @ I P I ; F Y,J Ă F Y,J , @ J P J such that X “ ď I P I F X,I , Y “ ď J P J F Y,J . Then we can take two sequences of precompact shrinkings F X,I Ă ¨ ¨ ¨ Ă U k ` X,I Ă U kX,I Ă ¨ ¨ ¨ Ă U X,I , I P I and F Y,J Ă ¨ ¨ ¨ Ă U k ` Y,J Ă U kY,J Ă ¨ ¨ ¨ Ă U Y,J , J P J such that č k ě U kX,I “ F X,I , č k ě U kY,J “ F Y,J . (12.2)Define | U kX | : “ ğ I P I U kX,I {O , | U kY | “ ğ J P J U kY,J { O . Also choose thickenings N X “ t N X,I I | I ď I u , N Y “ t N Y,J J | J ď J u of A X and A Y (see Definition 7.35, since the bundles E X,I , E Y,J naturally split as direct sums, athickening only contains a collection of tubular neighborhoods). By Theorem 7.37, thereexist two sequence of perturbations ˜ s X,k and ˜ s Y,k satisfying | ˜ s ´ X,k p q| Ă | U kX | , | ˜ s ´ Y,k p q| Ă | U kY | . Let the induced perturbation on A Z be ˜ s Z,k . We claim that for k sufficiently large, (12.1)holds and ˜ s ´ Z,k p q is sequentially compact. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 123
First we show (12.1) in the set-theoretic sense. Suppose it is not the case. Then thereexist a subsequence (still indexed by k ) and two sequences of points x k P | ˜ s ´ X,k p q| , y k P | ˜ s ´ Y,k p q| such that p x k , y k q R | ˜ s ´ Z,k p q| . By the precompactness of | U kX | , | U kY | and (12.2), there existsubsequences (still indexed by k ) such thatlim k Ñ8 x k “ x P X, lim k Ñ8 y k “ y P Y. Then there exists some K P K with ι p K q “ I ˆ J , and u Z, P U Z,K , u X, P U X,I , u Y, P U Y,J such that p x , y q “ ψ Z,K p u q “ p ψ X,I p u X, q , ψ Y,J p u Y, qq , η K p u Z, q “ p u X, , u Y, q . Claim.
For k sufficiently large, there exists u X,k P U X,I (resp. u Y,k P U Y,J ) which repre-sents x k (resp. y k ) in the virtual neighborhood, namely π X p u X,k q “ x k , π Y p u Y,k q “ y k . Proof of the claim.
Since I and J are finite sets, by taking a subsequence, we may assumethat there is I P I (resp. J P J ) such that sequence x k (resp. y k ) is represented by asequence of points u X,k P U X,I (resp. u Y,k P U Y,J ), i.e. π X p u X,k q “ x k , π Y p u Y,k q “ y k . Since U X,I and U Y,J are precompact, by taking a further subsequence, we may assumethat u X,k Ñ u X, P F X,I Ă U X,I , u Y,k Ñ u Y, P F Y,J Ă U Y,J . Then ψ X,I p u X, q “ x , ψ Y,J p u Y, q “ y . Then by the (OVERLAPPING CONDITION) ofthe atlases A and A , we know I ď I or I ď I (resp. J ď J or J ď J ).If I ď I , then u X, P U X,II . Since U X,II Ă U X,I is an open subset, the convergence u X,k Ñ u X, implies that for k sufficiently large, u X,k P U X,II . Hence x k “ π X p u X,k q “ π X p u X,k q , where u X,k : “ φ X,II p u X,k q . On the other hand, if I ď I and dim U X,I “ dim U X,I , the embedding φ X,I I is anopen embedding hence is invertible. Then u X, P U X,I I and u X, “ φ X,I I p u X, q . Since φ ´ X,I I p U X,I I q is an open subset of U X,I and contains the element u X, , for k sufficientlylarge, there exists u X,k P U X,I I with u X,k “ φ X,I I p u X,k q . Hence x k “ π X p u X,k q “ π X p u X,k q . Lastly, assume that I ď I but dim U X,I ă dim U X,I . Then the embedding φ X,I I has positive codimension. Let N X,I I Ă U X,I be the tubular neighborhood given by N X . If u X,k R N X,I I , then the limit u X, R N X,I I . This contradicts the fact that x P F X,I X F X,I . Hence u X,k P N X,I I . We also know that the perturbations ˜ s X,k are N X -normal. Hence u X,k P N X,I I X ˜ s ´ X,I,k p q implies that u X,k P Im φ X,I I . Hence thereexists u X,k P U X,I I with u X,k “ φ X,I I p u X,k q and hence x k “ π X p u X,k q . The situation for u Y,k is completely the same. (cid:3)
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 124
Therefore, in the topology of the chart U X,I and U Y,J , one has p u X,k , u
Y,k q Ñ p u X, , u Y, q P η K p s ´ Z,K,k p qq Ă s ´ X,I,k p q ˆ s ´ Y,I,k p q . Since η K p U Z,K q is an open subset of U X,I ˆ U Y,J , for k sufficiently large, there existunique u Z,k P U Z,K such that η K p u Z,k q “ p u X,k , u
Y,k q , and u Z,k converges to u Z, . Hence˜ s Z,K p u Z,k q “ ùñ p x k , y k q P | ˜ s ´ Z,k p q| . This is a contradiction. Hence for k sufficiently large, as sets, | ˜ s ´ Z,k p q| “ | ˜ s ´ X,k p q| ˆ | ˜ s ´ Y,k p q| . Fix a large k and abbreviate ˜ s X,k “ ˜ s X , ˜ s Y,k “ ˜ s Y , ˜ s Z,k “ ˜ s Z . We prove that | ˜ s ´ Z p q| issequentially compact in the quotient topology. Given any sequence x l P | ˜ s ´ ˆ p q| identifiedwith a pair p x ,l , x ,l q P | ˜ s ´ p q| ˆ | ˜ s ´ p q| , by the compactness of | ˜ s ´ p q| ˆ | ˜ s ´ p q| , wemay assume a subsequence (still indexed by l ) converges to p x , , x , q P | ˜ s ´ p q| ˆ| ˜ s ´ p q| “ | ˜ s ´ ˆ p q| . Then there exists J P I ˆ with ι p J q “ I ˆ I and u P U J , u , P U ,I , u , P U ,I such that x , “ π ,I p u , q , x , “ π ,I p u , q , η J p u q “ p u , , u , q . Similar to the proof of the previous claim, for l sufficiently large, there exists u ,l P U ,I and u ,l P U ,I such that x ,l “ π ,I p u ,l q , x ,l “ π ,I p u ,l q ; u ,l Ñ u , , u ,l Ñ u , . Since η J p U ˆ ,J q Ă U ,I ˆ U ,I is an open subset, for l sufficiently large, p u ,l , u ,l q P η J p U J q .Denote u l “ η ´ J p u ,l , u ,l q . Then u l Ñ u . Since the map s ´ ˆ ,J p q Ñ | ˜ s ´ ˆ p q| is continuous we see x l “ π ˆ ,J p u l q Ñ π ˆ ,J p u q “ x . Hence | ˜ s ´ ˆ p q| is sequentially compact.Lastly, when ˜ s X and ˜ s Y are transverse, so is ˜ s Z . Then | ˜ s ´ Z p q| is second countable.Hence sequential compactness is equivalent to compactness. Since the identity map | ˜ s ´ Z p q| Ñ } ˜ s ´ Z p q} is continuous and bijective, and the subspace topology is Hausdorff, this map is actually ahomeomorphism. Hence | ˜ s ´ Z p q| is compact Hausdorff and second countable. Moreover, | ˜ s ´ X p q| and | ˜ s ´ Y p q| are also compact, Hausdorff and second countable. It is easy tosee that the map | ˜ s ´ Z p q| Ñ | ˜ s ´ X p q| ˆ | ˜ s ´ Y p q| sends converging sequences to convergingsequences, hence is continuous. Since the domain of the map is compact and the targetis Hausdorff, this map is a homeomorphism. (cid:3) Moreover, the strongly continuous evaluation maps clearly agrees. Namely, one has | A ˆ | | ev ˆ | / / | η | (cid:15) (cid:15) ¯ X Γ W ˆ M Γ \ Γ (cid:15) (cid:15) | A | ˆ | A | | ev |ˆ| ev | / / ` ¯ X Γ W ˆ M Γ ˘ ˆ ` ¯ X Γ W ˆ M Γ ˘ . By restricting to the zero locus, one obtains the (DISCONNECTED GRAPH) property for thetwo graphs Γ and Γ . Namely p ev Γ \ Γ q ˚ “ M Γ \ Γ ‰ vir “ p ev Γ q ˚ “ M Γ ‰ vir b p ev q ˚ “ M Γ ‰ vir . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 125
Proof of Proposition 12.1.
Recall the construction virtual orbifold atlases (seeProposition 11.1) and good coordinate systems on moduli spaces. Suppose we can con-struct a collection of charts ! C X,i “ p U X,i , E
X,i , S
X,i , ψ
X,i , F
X,i q | i “ , . . . , M ) , with footprints F X,i cover X and a collection of charts ! C Y,j “ p U Y,i , E
Y,j , S
Y,j , ψ
Y,j , F
Y,j q | j “ , . . . , N ) , with footprints F Y,j cover Y . By the method of Section 11, we have a collection of charts C ˝ X,I “ p U ˝ X,I , E ˝ X,I , S ˝ X,I , ψ ˝ X,I , F ˝ X,I q , I P I : “ t ,...,M u r H and C ˝ Y,I “ p U ˝ Y,I , E ˝ Y,I , S ˝ Y,I , ψ ˝ Y,I , F ˝ Y,I q , I P I : “ t ,...,N u r H . Their footprints (which could be empty) are F ˝ X,I “ č i P I F X,i , F ˝ Y,I “ č j P I F Y,j . For each pair I ď J (resp. I ď J ), one has coordinate changes T ˝ X,JI (resp. T ˝ Y,J I )from C ˝ X,I to C ˝ X,J (resp. from C ˝ Y,I to C ˝ Y,J ), whose footprints are F ˝ X,J (resp. F ˝ Y,J ).We denote the two collections of data A ˝ X and A ˝ Y , but keep in mind that, althoughthe coordinate changes satisfy the (COCYCLE CONDITION) , they are not virtual orbifoldatlases since they may not satisfy the (OVERLAPPING CONDITION) .Order elements of I and I as I , . . . , I m ; I , . . . , I n such that I k ď I l ùñ k ď l ; I k ď I l ùñ k ď l. Choose precompact shrinkings of all F X,i as G X,i, Ă F X,i, Ă ¨ ¨ ¨ Ă G X,i,m Ă F X,i,m “ F X,i ;and G Y,j Ă F Y,j Ă ¨ ¨ ¨ Ă G mY,j Ă F mY,j “ F Y,j . such that X “ M ď i “ G X,i, , Y “ N ď j “ G Y,j . Furthermore, for all k “ , . . . , m , choose precompact shrinkings as G kY,j “ : G kY,j, Ă F kY,j, Ă ¨ ¨ ¨ Ă G kY,j,n Ă F kY,j,n : “ F kY,j . Then for I “ I k P I , define F ‚ X,I k : “ ´ č i P I k F X,i,k ¯ r ´ ď i R I k G X,i,k ¯ ;for 1 ď k ď m and I “ I l P I , define F k, ‚ Y,I l : “ ´ č j P I l F kY,j,l ¯ r ´ ď j R I l G kY,j,l ¯ . (12.3)One has the following covering properties of the above open sets. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 126
Lemma 12.3.
One has X “ ď I P I F ‚ X,I ; and for all k P t , . . . , m u , Y “ ď I P I F k, ‚ Y,I . Proof.
The same as the proof of Lemma 11.5. (cid:3)
Define the partial order on K “ I ˆ I by inclusion, i.e. I ˆ I ď J ˆ J ðñ I ď J, I ď J . (12.4)For K “ I ˆ I “ I k ˆ I l , define F ‚ K “ F ‚ I k ˆ I l : “ F ‚ X,I k ˆ F kY,I l Ă X ˆ Y. Lemma 12.3 implies that the collection of F ‚ K for all K P K is an open cover of X ˆ Y .Furthermore, one has the (OVERLAPPING CONDITION) with respect to the partial order(12.4). Lemma 12.4.
The collection F ‚ K satisfy the (OVERLAPPING CONDITION) property ofDefinition 7.24. Namely, for each pair K , K P K , one has F ‚ K X F ‚ K ‰ H ùñ K ď K or K ď K . Proof.
Suppose K “ I k ˆ I l , K “ I k ˆ I l . Suppose p x, y q P F ‚ K X F ‚ K . Then x P F ‚ X,I k X F ‚ X,I k . Then by the (OVERLAPPINGCONDITION) of the collection F ‚ X,I , either I k ď I k or I k ď I k . Without loss of general-ity, assume the former is true. Then k ď k . We then need to prove that either I l ď I l ,or k “ k , and in the latter case, either I l ď I l or I l ď I l .If k “ k , then one has y P F k , ‚ Y,I l X F k , ‚ Y,I l . Then it is similar to the proof of Lemma 11.5, that either I l ď I l or I l ď I l . It remainsto consider the case that k ă k . Suppose in this case I l ď I l does not hold. Thenthere exists j ˚ P I l r I l . One one hand, one has y P č j P I l F k Y,j,l Ă F k Y,j ˚ ,l Ă G k Y,j ˚ ,l . The last inclusion follows from the property of G kY,j,l and F kY,j,l and the fact that k ă k .On the other hand, one has y P Y r ď j R I l G k Y,j,l Ă Y r G k Y,j ˚ ,l . This is a contradiction. Hence I l ď I l . (cid:3) Define J : “ ! p k, I q | k P t , . . . , m u , I P I ) . Define the partial order on J by p k, I q ď p l, J q ðñ k ď l, I ď J . AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 127
The above construction also provides a covering of Y by F ‚ Y,J : “ F k, ‚ Y,I , J “ p k, I q P J . Recall the latter is defined by (12.3). Clearly these open sets cover Y . Moreover, thisopen cover satisfies the (OVERLAPPING CONDITION) . Lemma 12.5.
The open cover t F ‚ Y,J | J P J u of Y satisfies (OVERLAPPING CONDITION) of Definition 7.24.Proof. It is similar to the proof of Lemma 12.4. The details are left to the reader. (cid:3)
Now we start to construct good coordinate systems A X , A Y and A Z . The collection t F ‚ X,I u satisfies the (OVERLAPPING CONDITION) property of Definition 7.24. Therefore,by the same method as in Section 11, one can construct a good coordinate system A X “ ´ t C X,I | I P I u , t T II | I ď I P I u ¯ of X whose collection of footprints are arbitrary precompact shrinkings F X,I Ă F ‚ X,I theunion of which still equals X . We take the shrinkings F X,I so close to F ‚ X,I so that westill have Z “ X ˆ Y “ ď I k P I , I P I F X,I k ˆ F k, ‚ Y,I . On the other hand, for an arbitrary collection of precompact shrinkings F Y,J Ă F ‚ Y,J , @ J P J , one can construct a good coordinate system on Y A Y : “ ´ C Y,J “ p U Y,J , E
Y,J , S
Y,J , ψ
Y,J , F
Y,J q | J P J ( , T Y,J J | J ď J P J (¯ Notice that one has the inclusion U Y,I Ă U ˝ Y,J . Moreover, one can take the shrinkings F Y,J so close to F ‚ Y,J such that the collection F _ Z,K : “ F X,I k ˆ F Y,J , where K “ I k ˆ I P K “ I ˆ I and J “ p k, I q still cover the product Z “ X ˆ Y . The (OVERLAPPING CONDITION) remains true for F _ Z,K . Then choose precompact shrinkings F Z,K Ă F _ Z,K for all K P K that still cover Z “ X ˆ Y , one can construct a good coordinate system A Z on Z “ X ˆ Y by shrinkingthe product charts C ˝ Z,K “ C ˝ X,I ˆ C ˝ Y,J , where K “ I ˆ J P K to a chart C Z,K “ p U Z,K , E
Z,K , S
Z,K , ψ
Z,K , F
Z,K q . Moreover, we can make the shrinking so small that for all K , U Z,K Ă U X,I ˆ U Y,J Ă U ˝ X,I ˆ U ˝ Y,I , where K “ I ˆ I , I “ I k , J “ p k, I q . Then for the three good coordinate systems A X , A Y and A Z , we see the map ι : K Ñ I ˆ J and the precompact open embeddings η K : C Z,K Ñ C X,I ˆ C Y,J , where ι p K q “ I ˆ J are obviously defined. The properties of ι , η K and the evaluation maps are very easy tocheck. This finishes the proof of Proposition 12.1. AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 128
Cutting edge.
Let Γ be a stable decorated dual graph and let Π be the decorateddual graph obtained by shrinking a loop in Γ which is still stable. Let the underlyingstable dual graphs be Γ and Π respectively. Then one has the commutative diagram M Π / / st (cid:15) (cid:15) M Γ st (cid:15) (cid:15) M Π ι γ / / M Γ . (12.5)When construct the virtual orbifold atlas on M Γ , one always build from lower strata tohigher strata, and in each chart C I the orbifold U I always contains the gluing parameters.Hence the vertical map st : M Γ Ñ M Γ induces a strongly continuous map st : A Γ Ñ M Γ which is transverse to the orbifold divisor M Π . By the construction of Subsection 7.8, itgives a virtual orbifold atlas on M Π . On the other hand, by looking at the procedure ofconstructing the virtual orbifold atlas on M Π , one can see this induced atlas is the sameas one that we can construct, basically by turning off a corresponding gluing parameter.Then by Proposition 7.42, one has p ι Π q ˚ ev ˚ “ M Π ‰ vir “ ev ˚ “ M Γ ‰ vir X “ M Π ‰ . This finishes the proof of the (CUTTING EDGE) property.12.4.
Composition.
Let Π be a decorated dual graph with a chosen edge labelled by apair of twisted sectors r ¯ τ s , r ¯ τ s ´ . Let ˜ Π be the decorated dual graph obtained by normal-ization, namely, cutting the edge. Then one has the following commutative diagram M Π / / ev Π (cid:15) (cid:15) M ˜ Π ev ˜ Π (cid:15) (cid:15) ∆ r ¯ τ s / / p ¯ X r ¯ τ s W q . (12.6)Here the vertical arrows are evaluations at the tails obtained by cutting off the chosenedge. However, in general the square is not Cartesian. This is because automorphisms ofelements in M ˜ Π may not glue to an automorphism of the corresponding element in M Π .First suppose it is a Cartesian square. Then when constructing the virtual orbifold atlason M ˜Π , by thickening the obstruction spaces, one can make the strongly continuous map ev ˜ Π transverse to the diagonal. Then by the construction of Subsection 7.8, one obtainsan induced virtual orbifold atlas on M Π . Moreover, by Proposition 7.42, one has p ev Π q ˚ “ M Π ‰ vir “ p ev ˜ Π q ˚ “ M ˜ Π ‰ vir r PD r ∆ r ¯ τ s s . In general when (12.6) is not Cartesian, M Π differs from the actual fibre product by aninteger factor which is the degree of the mapq : M ˜Π ˆ M Π M Π Ñ M ˜ Π . The above property of the virtual cycle still holds in this general case. This finishes theproof of the (COMPOSITION) property.
AUGED LINEAR SIGMA MODEL IN GEOMETRIC PHASES 129
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