Gromov-Witten theory of toroidal orbifolds and GIT wall-crossing
GGROMOV-WITTEN THEORY OF TOROIDAL ORBIFOLDS AND GITWALL-CROSSING
ROB SILVERSMITH
Abstract.
Toroidal 3-orbifolds ( S ) /G , for G a finite group, were some of the earliest examples ofCalabi-Yau 3-orbifolds to be studied in string theory. While much mathematical progress towards thepredictions of string theory has been made in the meantime, most of it has dealt with hypersurfacesin toric varieties. As a result, very little is known about curve-counting theories on toroidal orbifolds.In this paper, we initiate a program to study mirror symmetry and the Landau-Ginzburg/Calabi-Yau(LG/CY) correspondence for toroidal orbifolds. We focus on the simplest example [ E /µ ] , where E ⊆ P is the elliptic curve V ( x + x + x ) . We study this orbifold from the point of GIT wall-crossingusing the gauged linear sigma model, a collection of moduli spaces generalizing spaces of stablemaps. Our main result is a mirror symmetry theorem that applies simultaneously to the differentGIT chambers. Using this, we analyze wall-crossing behavior to obtain an LG/CY correspondencerelating the genus-zero Gromov-Witten invariants of [ E /µ ] to generalized Fan-Jarvis-Ruan-Witteninvariants. Introduction
The gauged linear sigma model.
Landau-Ginzburg/Calabi-Yau (LG/CY) correspondencesare conjectural relations between invariants of certain moduli spaces. On one hand, Gromov-Wittentheory provides a collection of “virtual curve counts” on a Calabi-Yau orbifold Z . These are integralsover the moduli stacks M g,n ( Z, β ) of twisted stable maps ([1]). On the other hand, Fan-Jarvis-Ruan ([20], based on ideas of Witten) constructed moduli stacks W Zg,n parametrizing roots of linebundles on orbifold curves. These are “combinatorial” in nature, whereas the spaces M g,n ( Z, β )are “geometric”. One may generate
Fan-Jarvis-Ruan-Witten (FJRW) invariants by integratingcohomology classes over W Zg,n . Using motivation from string theory, Witten [35] predicted thateither of these sets of invariants — Gromov-Witten or FJRW — could be computed from the other.A far-reaching conjecture was precisely formulated by Ruan ([30]), and was recently proven byChiodo-Iritani-Ruan ([8]) in the case where Z is a Calabi-Yau hypersurface in weighted projectivespace . The form of the conjecture is described in more detail below.Toroidal 3-orbifolds [( S ) /G ] are some of the earliest examples studied in string theory ([18]).They form a rich class of very explicit Calabi-Yau orbifolds (see the classification [22]). Nevertheless,in many ways we know very little about them; for example, the program of mirror symmetry forCalabi-Yau orbifolds has been worked out only for complete intersections in toric stacks, whereasmost toroidal orbifolds are not of this type. Similarly LG/CY correspondences have not been studiedin this context.Our goal in this paper is to initiate a program towards filling both of these gaps. Our strategy isbased on a common generalization of the moduli stacks M g,n ( Z, β ) and W Zg,n above, collectivelycalled the gauged linear sigma model , or GLSM. It was proposed by Witten and formulatedmathematically by Fan-Jarvis-Ruan ([21]). The main feature of these new stacks is that they takeas input a GIT presentation [
V // θ G ] . Dolgachev-Hu ([19]) and Thaddeus ([32]) studied how suchGIT quotients change if θ crosses a wall of a certain finite chamber decomposition, and similarlythe GLSM stacks depend on a chamber of this decomposition. There is a geometric chamber of Date : September 24, 2018. a r X i v : . [ m a t h . AG ] M a r his decomposition whose GLSM moduli stack is M g,n ( Z, β ) , and a so-called pure Landau-Ginzburg(LG) chamber whose GLSM moduli stack is W Zg,n . The LG/CY correspondence is thus recast as a“GIT wall-crossing” phenomenon.We fully work out the genus-zero LG/CY correspondence in the simplest example of a toroidalorbifold, [ E /µ ] for E an elliptic curve, and will apply our technique more generally in a subsequentarticle. [ E /µ ] is a complete intersection in a GIT quotient [ C // θ ( C ∗ ) ] . This quotient has achamber decomposition with 16 chambers, namely the 16 hyperoctants ( ± , ± , ± , ± ) in R . Wefind the geometric chamber to be the hyperoctant (+ , + , + , +), and the pure LG-chamber to be( − , − , − , +) . We choose a sequence of wall-crossings(+ , + , + , +) → (+ , + , − , +) → (+ , − , − , +) → ( − , − , − , +)connecting these two chambers, and in each of the four chambers we describe the correspondingGLSM moduli stacks as parametrizing sections of certain line bundles on orbifold curves. To eachof these chambers is then associated a collection of numerical invariants.GLSM moduli stacks depend upon an additional parameter (cid:15) ∈ Q > . The special cases M g,n ( Z, β )and W Zg,n above correspond to (cid:15) → ∞ . We use techniques based on those of Ciocan-Fontanine andKim ([13]) to prove an all-chamber mirror theorem : Theorem 8.1.
Let { J (cid:15),θ } be the generating functions of GLSM invariants defined in Section 7.3.Then there is an explicit invertible transformation identifying J (cid:15),θ with J ∞ ,θ . Theorem 8.1 is the core of the paper, as well as its most notable aspect. Finding an appropriatestatement of mirror symmetry for each GIT chamber is the most difficult part of the LG/CYcorrespondence. Previous examples of LG/CY correspondences have all relied on proving mirrortheorems in each chamber separately, whereas our proof is uniform in θ , i.e. applies simultaneouslyin all chambers.Setting (cid:15) → θ = (+ , + , + , +) recovers an instance of the mirror theorem for toric stacks ([16, 7]), and setting (cid:15) → θ = ( − , − , − , +) gives a Landau-Ginzburg mirror theorem similar tothe one in [10]. The purpose of Theorem 8.1 is that one may compute (a restriction of) J ,θ foreach θ : Corollary (Section 9) . There are explicit hypergeometric functions I θ that encode the GLSMinvariants of [ C // θ ( C ∗ ) ] . For example (using notation defined throughout the paper), the series I (+ , + , − , +) ( q, (cid:126) ) = (cid:88) β x ∈ Z ≥ β y ∈ Z ≥ β z ∈ ( − / Z > (cid:114)Z β a ∈ (1 / Z ≥ q β x x q β y y q β z +1 / z q β a a (cid:81) ρ ∈ R β ρ < − (cid:81) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− ( D ρ + ( β ρ − ν ) (cid:126) ) (cid:81) ρ ∈ R β ρ ≥ (cid:81) ≤ ν ≤(cid:100) β ρ (cid:101)− ( D ρ + ( β ρ − ν ) (cid:126) ) A x A y (cid:104)− β (cid:105) encodes the GLSM invariants of the chamber (+ , + , − , +).Finally, we relate the functions I θ across the four GIT chambers above. We think of I θ as aholomorphic map from a certain space N to a subquotient H ( θ ) of the Chen-Ruan cohomology H ∗ CR ([ C // θ ( C ∗ ) ]). There are two problems with comparing the functions I θ . First, they do nothave the same codomain. Second, they are not defined on all of N ; in fact, each is defined on adifferent small open set. Thus we relate them in two steps:(1) We find a natural sequence of graded isomorphisms (Theorem 5.4) H (+ , + , + , +) ∼ = H (+ , + , − , +) ∼ = H (+ , − , − , +) ∼ = H ( − , − , − , +) . In fact, as (cid:15) moves within a chamber decomposition of Q > , this theorem is also a type of wall-crossing . To avoidconfusion, we use the term only to refer to walls of GIT chambers.
2) We find a sequence of analytic continuations of I θ on N (Section 10).The method of analytic continuation is based on [10]. Together, these prove: Theorem 10.7.
After analytic continuation and identification of GLSM state spaces, the functions I θ differ by (explicit) linear transformations.The first genus-zero LG/CY correspondence was proved for the quintic threefold by Chiodo andRuan ([10]). It has since been proven for several other classes of targets, including Calabi-Yauhypersurfaces in weighted projective spaces ([8]), many classes of Calabi-Yau complete intersectionsin weighted projective spaces ([14, 15]), and some other examples ([28, 31]). (Acosta ([3]) alsodeveloped a similar correspondence for non-Calabi-Yau hypersurfaces in weighted projective spaces.)All of these used techniques quite different from those presented here; [8], [14], [28], and [31] used adirect computational method, and [15] (following previous work for hypersurfaces in [26]) used areduction to the crepant transformation conjecture, previously established in the relevant cases in[17].While toroidal 3-orbifolds are our primary motivation, we expect Theorem 8.1 to apply muchmore broadly, to large classes of complete intersections in GIT quotients carrying certain torusactions. Because of this, we view it as a general conceptual approach to LG/CY correspondences.In the future we hope to explore the generality in which this technique applies. Plan of the paper.
Section 2 contains background facts about GIT quotients and orbifold curves.In Sections 3 and 4, we introduce the “target” stacks Z ( θ ) ⊆ [ C // θ ( C ∗ ) ] and their associatedmoduli stacks LGQ (cid:15) ,m ( Z ( θ ) , β ). In Section 5 we define the GLSM state spaces H ( θ ), and theGLSM invariants, which are integrals over the moduli stacks LGQ (cid:15) ,m ( Z ( θ ) , β ). In Section 6 wedefine natural group actions on our moduli stacks, to be used for fixed-point localization. Section7 contains the definitions of various generating functions, which we use to prove our all-chambermirror theorem in Section 8. Finally, in Sections 9 and 10 we compute the series I θ ( q, (cid:126) ) for general θ , and relate the resulting formulas to each other. Section 11 is a table of notation. Acknowledgements.
I would like to thank Yongbin Ruan for introducing me to the field, suggestingthe problem, and providing guidance along the way. I am grateful to Rohini Ramadas and DustinRoss for helpful conversations. 2.
Background
We work over C . We denote by µ d the group of d th roots of unity in C . Geometric invariant theory and stack quotients.Definition 2.1.
Let V be a smooth affine variety, let G be a reductive algebraic group acting on V , and let θ : G → C ∗ be a character of G . This defines a G -action on the trivial bundle V × C by g · ( v, z ) = ( g · v, θ ( g ) z ). The G -invariant sections of V × C are called θ -equivariant functions .The θ -unstable locus V uns ( θ ) in V is the subvariety defined by the vanishing of all N θ -equivariantfunctions, for N ≥
1. The θ -semistable locus V ss ( θ ) is the complement of V uns ( θ ) . The θ -stablelocus V s ( θ ) is the set of semistable points with finite G -stabilizer whose G -orbit is closed in V ss ( θ ).In this paper we will always have V s ( θ ) = V ss ( θ ) . In this case the
GIT stack quotient [ V // θ G ] :=[ V ss ( θ ) /G ] is a smooth separated Deligne-Mumford stack. Proposition 2.2.
Let G be a group acting (on the left) on a variety V , and let S be any scheme.There is a natural correspondence between(1) Maps f : S → [ V /G ] ,(2) Principal G -bundles P on S together with a G -equivariant map φ : P → V , and
3) Principal G -bundles P on S together with a section σ φ of the associated fiber bundle P × G V → S. Proof.
The equivalence of (1) and (2) is by definition of a map S → [ V /G ]. We show that (2) and(3) are equivalent. A section σ : S → P × G V gives a map S → [ V /G ] by composition with theprojection
P × G V → [ V /G ] . Conversely, given a G -equivariant map P → V , we define a section S → P × G V by mapping s (cid:55)→ ( p, φ ( p )) . It is straightforward to check these are inverse to eachother. (cid:3)
With V , G , and S as in Proposition 2.2, let ρ : G → C ∗ be a character of G . This induces a G -equivariant structure on the trivial bundle V × C by g · ( v, z ) := ( g · v, ρ ( g ) z ) . (We write V × C ρ to keep track of the G -action.) There is an associated line bundle L ρ = [( V × C ρ ) /G ]on [ V /G ] . Abusing notation, we also write L ρ for restrictions of L ρ to substacks [ V // θ G ]. Proposition 2.3.
Let S be a scheme, and let f : S → [ V /G ] be a map, with corresponding principal G -bundle P and map φ : P → V. Then for any character ρ of G , f ∗ L ρ ∼ = P × G C ρ ∼ = σ ∗ φ ( P × G ( V × C ρ )) . An m -marked prestable orbifold curve ( C, b , . . . , b m ) is a balanced twisted nodal m -pointed curve in the sense of [2]. That is, ´etale locally at each point P it is either:(1) isomorphic to [ C /µ d P ] for some d P , where µ d P acts by multiplication, and P is identifiedwith 0, or(2) isomorphic to [ V ( xy ) /µ d P ] , where V ( xy ) ⊆ C is the union of the coordinate axes, µ d P actsby multiplication by opposite roots of unity on x and y , and P is identified with (0 , m distinct marked points b , . . . , b m of type (1), including all of those with d P > d P as the order of P and µ d P as the isotropy group of P . We often write C instead of( C, b , . . . , b m ) . Remark . For points P of type (1), there is a canonical identification of the isotropy group of P with µ d P ; the canonical generator is that which acts by multiplication by e πi/d P on T P C. However,this is not true for points of type (2), since each element acts by opposite roots of unity on the twobranches. Instead, there is a canonical identification after choosing a branch of the node.Note that d P = 1 for all but finitely many points P of C . An m -marked prestable orbifold curveadmits a coarse moduli space map to an ordinary m -marked prestable curve C . Olsson ([27]) provedthat families of m -marked orbifold curves whose coarse moduli spaces have arithmetic genus g forman algebraic stack M tw g . We will only be interested in the case g = 0 . For the purposes of this paperwe restrict to an open substack of M tw0 , as follows. Definition 2.6 ([29]) . A genus zero m -marked orbifold curve is stable if each irreducible componenthas at least three marked points or nodes. An m -marked is a stable genus zero m -marked orbifold curve such that all marked points and nodes are orbifold points of order 3.Next we review some facts about line bundles on orbifold curves. Definition 2.7.
Let C be an m -marked prestable orbifold curve. A line bundle on C is a stack L with a map to C, such that L is ´etale locally isomorphic on C it is isomorphic to one of thefollowing, corresponding to the cases in Definition 2.4:(1) [ C × C /µ d P ] , where µ d P acts by multiplication on the first copy of C and linearly on thesecond copy, or
2) [ V ( xy ) × C /µ d P ] , where µ d P acts on V ( xy ) as in item (2) of Definition 2.4, and linearly on C . Definition 2.8.
In case (1), e πi/d P ∈ µ d P acts on the second copy of C by multiplication by e πik/d P for some 0 ≤ k < d P . We call the rational number mult P ( L ) := k/d P the multiplicity orthe monodromy of L at P . If mult P ( L ) = 0, we say L has trivial monodromy at P . Remark . We also refer to the multiplicity of L at a node of C . As in Remark 2.5, this iswell-defined only after choosing a branch of the node. In this case we will refer to the multiplicity of L “on one side of the node.”One can similarly define vector bundles and their duals, sections, tensor products, and direct sumson orbifold curves. These behave largely the same as on nonstacky curves, with a few differences.For example, local sections of L at an orbifold point P of C are µ d P -invariant sections as in (1)above, so in particular, if L has nontrivial monodromy at P then every local section of L vanishes at P . More specifically, if we define the order of vanishing of a section via pulling back along an ´etalecover by a scheme, then the order of vanishing of a section at an orbifold point P is an element ofmult P ( L ) + Z . Also, the monodromy of a tensor product of line bundles is given bymult P ( L ⊗ L (cid:48) ) = mult P ( L ) + mult P ( L (cid:48) ) mod 1 . Isomorphism classes of line bundles on orbifold curves may be easily understood via the divisor-linebundle correspondence for smooth orbifold curves . We state it only for genus zero curves, as theseare all we consider.
Definition 2.10. A Weil divisor on a smooth orbifold curve is a (finite) formal sum D = (cid:80) P ∈ C a P P of points. The degree deg( D ) of D is (cid:80) P ∈ C a P d P , where d P is the order of P . The degree is clearlyadditive under addition of Weil divisors.The notion of rational equivalence of Weil divisors on orbifold curves is identical to that forschemes. For genus zero orbifold curves, it reduces to the following. Definition 2.11.
Two Weil divisors D = (cid:80) P ∈ C a P P and D (cid:48) = (cid:80) P ∈ C a (cid:48) P P on a genus zero smoothorbifold curve are rationally equivalent if for each P ∈ C we have a P ≡ a (cid:48) P mod d P for all P , anddeg( D ) = deg( D (cid:48) ) . We write [ D ] of the rational equivalence class of D. In particular, the divisors d P P are rationally equivalent for all points P . We have the followingcorrespondence: Proposition 2.12 (See [34]) . The additive group of Weil divisors on a smooth orbifold curve C upto rational equivalence is naturally isomorphic to the group of line bundles on C up to isomorphism,with the operation of tensor product. As for schemes, the zeroes and poles of a rational section of a line bundle L define a divisor,and this determines one direction of the correspondence. From this we see that the multiplicitymult P ( L ) is identified with the rational number a P d P mod 1. The correspondence also allows us todefine the degree deg( L ) of a line bundle, additive under tensor product.We will often refer to the log-canonical bundle ω C, log on a 3-stable curve C . This is a line bundlewhose sections are (correctly defined) holomorphic 1-forms on C , twisted by the divisor (cid:80) i b i ofmarked points. As with ordinary nodal curves, these holomorphic 1-forms may have simple polesat nodes. It will be important that ω C, log has trivial monodromy at each orbifold point, and hasdegree 2 g − m = − m. The curve P , := [( C (cid:114) { } ) / C ∗ ], where C ∗ acts with weights 3 and 1 on the coordinatesrespectively, will be particularly useful to us. This curve is smooth, and has a single orbifold markedpoint of order 3 at [1 : 0] , which we refer to as ∞ . (In particular, it is not 3-stable.) The group f Weil divisor classes (and hence the group of isomorphism classes of line bundles) is generatedby a single element [ ∞ ]. Following convention we refer to the corresponding isomorphism class ofline bundles as O P , (1) , and its tensor powers by O P , ( n ) . Note that deg( O P , ( n )) = n/ . The logcanonical bundle ω P , , log (viewing P , as a 1-marked orbifold curve) has degree − − , so itis isomorphic to O P , ( − . The inertia stack and Chen-Ruan cohomology.
Let X be a smooth complex orbifold,i.e. a smooth connected Deligne-Mumford stack of finite type over C . Definition 2.13.
Suppose X = [ M/G ] , where M is a smooth scheme and G is an abelian group.Then the inertia stack of X is the quotient IX := [ ˜ M /G ] , where ˜ M is the scheme parametrizingpairs ( m, g ) where m ∈ M and g ∈ G m , where G m ⊆ G is the stabilizer of m .For fixed g ∈ G , let ˜ M ( g ) be the open closed subscheme of ˜ M of elements of the form ( m, g ) . The rigidified inertia stack is the union IX := (cid:91) g ∈ G [ ˜ M ( g ) / ( G/ (cid:104) g (cid:105) )] . (Note: In the cases we consider, ˜ M ( g ) is empty for all but finitely many g . More generally, therigidified inertia stack is slightly more difficult to define.) The inertia stack and rigidified inertiastack are defined in much more generality (see [1]), but we will only need the cases above. Remark . The rigidified inertia stack has the same coarse moduli space as the inertia stack.Indeed, the only difference between the two is that the inertia stack has “extra” stack structure.For example, Bµ := [Spec C /µ ] has inertia stack IBµ ∼ = Bµ (cid:116) Bµ (cid:116) Bµ , and rigidified inertiastack IBµ ∼ = Bµ (cid:116) Spec C (cid:116) Spec C . Terminology . Connected components of IX and IX are called sectors . Both IX and IX contain X as a connected component, namely the quotient [ ˜ M ( e ) /G ] for e ∈ G the identity. Thiscomponent is referred to as the untwisted sector of IX or IX , and other components are called twisted sectors . A twisted sector X (cid:48) has a corresponding element g ∈ G , so we refer to X (cid:48) as a g -twisted sector . Remark . Note that there is a forgetful map IX → X that realizes each component of IX asa closed substack of X . There are also inversion automorphisms υ on IX and IX, which send( m, g ) (cid:55)→ ( m, g − ) . Definition 2.17.
The
Chen-Ruan cohomology of X is defined, as a C -vector space, to be H ∗ CR ( X ) := H ∗ ( IX, C ) , where the right side denotes the singular cohomology of the coarse moduli space.The grading of H ∗ CR ( X ) is different from the usual one. To describe it we use the following: Definition 2.18. If X (cid:48) is a g -twisted sector, then for generic ( m, g ) ∈ X (cid:48) we diagonalize theautomorphism of T m M induced by g . The eigenvalues are roots of unity e πiα j with 0 ≤ α j < . The age of X (cid:48) , denoted age( X (cid:48) ) , is defined to be (cid:80) j α j . We may now define the grading: H kCR ( X ) := (cid:77) X (cid:48) H k − X (cid:48) ) ( X (cid:48) , C ) . Remark . There is a natural graded embedding H ∗ ( X, C ) (cid:44) → H ∗ CR ( X ), induced by the inclusion X (cid:44) → IX of the untwisted sector. otation . For each sector X (cid:48) , there is a class 1 X (cid:48) ∈ H ∗ CR ( X ) that is the unit in H ∗ ( X (cid:48) , C ) . (Its degree may be nonzero under the grading above.) If ˜ M ( g ) is connected we will write 1 X (cid:48) = 1 g . We will also write 1 X (cid:48) or 1 g for the corresponding class on the nonrigidified inertia stack.Here are two important properties of Chen-Ruan cohomology:(1) There is a natural notion of cup product on H ∗ CR ( X ), compatible with the grading. (Notethat the cup product on H ∗ ( IX, C ) is not compatible with the grading as defined.)(2) If X is proper, there is a (perfect) Poincar´e pairing on H ∗ CR ( X ) , defined by (cid:104) α, β (cid:105) X = (cid:82) IX α ∪ υ ∗ β. (The υ ∗ β in the integrand makes the pairing compatible with the grading.)2.4. Cohomology of P /µ . One of the basic objects of this paper is the stack quotient [ P /µ ] , where µ acts by multiplication on the first coordinate. In this section we consider only singularcohomology of the coarse moduli space P /µ , not Chen-Ruan cohomology. Write [ x : x : x ]for points of P , and η : P → [ P /µ ] for the quotient map. We use the following notation: TheSymbol Locus in P Symbol Locus in [ P /µ ]˜ L Line V ( x ) ⊆ P L η ( ˜ L )˜ P Point [1 : 0 : 0] ∈ P P η ( ˜ P )˜ L (cid:48) Any line through ˜ P L (cid:48) η ( ˜ L (cid:48) )˜ P (cid:48) Any point on ˜ L P (cid:48) η ( ˜ P (cid:48) ) µ -fixed locus in P is ˜ L ∪ ˜ P . The lines ˜ L (cid:48) are µ -invariant, and these are the only µ -invariantlines (other than ˜ L ). H ∗ ( P /µ , Z ) is generated by { , [ L ] , [ L (cid:48) ] , [ P ] , [ P (cid:48) ] } . Note that 3[ L ] = 3[ L (cid:48) ] and 3[ P ] = 3[ P (cid:48) ] . Therefore we define the generators H := [ L ] = [ L (cid:48) ] and P := [ P ] = [ P (cid:48) ] of the complex cohomology H ∗ ( P /µ , C ) . Remark . We usually consider not [ P /µ ] but the line bundle [ O P ( − /µ ]. Here O P ( −
3) isthe quotient (( C (cid:114) { (0 , , } ) × C ) / C ∗ , where C ∗ acts with weights (1 , , , − . The group µ actson the (quasi-)homogeneous coordinates of O P ( −
3) by ζ · [ x : x : x : p x ] = [ ζx : x : x : p x ].The pullback map H ∗ ( P /µ , C ) → H ∗ ( O P ( − /µ , C ) is an isomorphism, and we will also use thesymbols H and P to denote the corresponding classes in the latter.3. The targets Z ( θ )3.1. Notation.
We begin by fixing notation that we will use throughout the paper. It is essentiallyin agreement with the notation of [21]. We let V = C with coordinates( x , x , x , y , y , y , z , z , z , a, p x , p y , p z ) . Let G = ( C ∗ ) , with action on V by( t x , t y , t z , t a ) · ( x , x , x , y , y , y , z , z , z , a, p x , p y , p z )= ( t x t − a x , t x x , t x x , t y t − a y , t y y , t y y , t z t − a z , t z z , t z z , t a a, t − x p x , t − x p y , t − x p z ) . Define another group C ∗ R = C ∗ (denoted thus to avoid confusion) acting on V by t R · ( x , x , x , y , y , y , z , z , z , a, p x , p y , p z )= ( x , x , x , y , y , y , z , z , z , a, t R p x , t R p y , t R p z ) . Remark . The groups G and C ∗ R are “independent” in that (cid:104) G, C ∗ R (cid:105) ⊆ GL( V ) is isomorphic to G × C ∗ R . et W : V → C be the G -invariant function W := p x ( ax + x + x ) + p y ( ay + y + y ) + p z ( az + z + z ) . It is C ∗ R -homogeneous of degree 1. Terminology . In the literature, W is referred to as a superpotential on V , and C ∗ R is known asan R -charge .As G × C ∗ R acts diagonally, V is a direct sum of 1-dimensional G × C ∗ R -representations, corre-sponding to the list of characters R = { (cid:98) t x − (cid:98) t a , (cid:98) t x , (cid:98) t x , (cid:98) t y − (cid:98) t a , (cid:98) t y , (cid:98) t y , (cid:98) t z − (cid:98) t a , (cid:98) t z , (cid:98) t z , (cid:98) t a , − (cid:98) t x + (cid:98) t R , − (cid:98) t y + (cid:98) t R , − (cid:98) t z + (cid:98) t R } , where (cid:98) t x is the character ( t x , t y , t z , t a , t R ) (cid:55)→ t x , and similarly for the others.The critical locus Crit( W ) of W is G -invariant. Let X : = [ V /G ] Z : = [Crit( W ) /G ] , with ι : Z (cid:44) → X the natural embedding. X and Z are nonseparated Artin stacks, but we considercertain open GIT quotient substacks, as follows.3.2. The characters θ . We study GIT quotients [
V // θ G ] = [ V ss ( θ ) /G ] where θ : G → C ∗ varies.The Euclidean space parametrizing θ : ( t x , t y , t z , t a ) (cid:55)→ t e x x t e y y t e z z t e a a is isomorphic to Z ⊗ R = R .The 16 GIT chambers are those on which the signs of e x , e y , e z , and e a are constant. We definecharacters Θ = { θ xyza , θ xyaz , θ xayz , θ axyz } representing four of the chambers: θ xyza ( t x , t y , t z , t a ) = t x t y t z t a θ xyaz ( t x , t y , t z , t a ) = t x t y t − z t a θ xayz ( t x , t y , t z , t a ) = t x t − y t − z t a θ axyz ( t x , t y , t z , t a ) = t − x t − y t − z t a . (The multiples of 3 will simplify notation later.) We then define for θ ∈ Θ : X ( θ ) : = [ V // θ G ] ⊆ XZ ( θ ) = [Crit( W ) // θ G ] : = [(Crit( W ) ∩ V ss ( θ )) /G ] ⊆ Z. Again, we use ι to denote the embedding Z ( θ ) (cid:44) → X ( θ ). Terminology . If x (resp. y , z ) is in the subcript of θ, we will say “ x (resp. y , z ) is a subscriptvariable.” Similarly we refer to “superscript variables,” and to modifying a character by “moving x from the superscript to the subscript.” (In every case, a is a superscript variable.) Remark . As mentioned in the introduction, the characters θ xyza and θ axyz are of primary interest,as from them we will construct moduli spaces previously studied in Gromov-Witten theory andFan-Jarvis-Ruan-Witten theory, respectively. The characters θ xyaz and θ xayz will provide a means ofinterpolating between these moduli spaces. By symmetry of x , y , and z , everything that followsregarding the characters in Θ works equally well for the characters θ xzay , θ yzax , θ yaxz , and θ zaxy . Characters on walls of the chamber decomposition (such as θ xay ( t x , t y , t z , t a ) := t x t − y t a ) are notconsidered. The corresponding GIT quotients are not well-behaved, and the moduli spaces of Section4 are not defined in this situation.The other missing characters are those where a is a subscript variable, e.g. θ xyza ( t x , t y , t z , t a ) := t x t y t − z t − a . These are not needed to carry out the interpolation mentioned. However, this case maybe of independent interest and we hope to return to it in the future. erminology . The GIT chambers are called phases in the physics literature, and variousmanifestations of GIT wall-crossing (such as the LG/CY correspondence) are known as phasetransitions .The following will help us state the definitions of moduli spaces in Section 4. Define characters ϑ xyza ( t x , t y , t z , t a ) = t x t y t z t a ϑ xyaz ( t x , t y , t z , t a ) = t x t y t − z t a t R ϑ xayz ( t x , t y , t z , t a ) = t x t − y t − z t a t R ϑ axyz ( t x , t y , t z , t a ) = t − x t − y t − z t a t R . of G × C ∗ R . These lift the characters in Θ to G × C ∗ R ⊇ G. The GIT quotients X ( θ ) and Z ( θ ) . A routine calculation of the equations defining V uns ( θ ) andCrit( W ) yields the following characterization:(1) If x is a superscript variable of θ , then V ( x , x , x ) ⊆ V uns ( θ ). If x is a subscript variableof θ, then V ( p x ) ⊆ V uns ( θ ). For every θ we have V ( a ) ⊆ V uns ( θ ) . These three conditionsentirely cut out V uns ( θ ) in V .(2) X ( θ xyza ) is isomorphic to the total space of the rank 3 vector bundle [ O P ( − /µ ] over[( P ) /µ ] . Here µ acts on each copy of P and O P ( −
3) as in Section 2.4. Z ( θ xyza ) isisomorphic to the complete intersection [ E /µ ] inside the zero section of this vector bundle,where E is the µ -invariant elliptic curve V ( x + x + x ) ⊆ P . (3) X ( θ xyaz ) ∼ = (cid:104) ( O P ( − × [ C /µ ]) (cid:46) µ (cid:105) , where µ acts on C by scaling. Z ( θ xyaz ) ∼ =[( E × Bµ ) /µ ], where Bµ ⊆ [ C /µ ] is the origin.(4) X ( θ xayz ) ∼ = (cid:104) ( O P ( − × [ C /µ ] ) (cid:46) µ (cid:105) , and Z ( θ xayz ) ∼ = [( E × ( Bµ ) ) /µ ].(5) X ( θ axyz ) ∼ = (cid:2) [ C /µ ] /µ (cid:3) ∼ = [ C / ( µ ) ], and Z ( θ axyz ) ∼ = B (( µ ) ) is the origin. Remark . Using e.g. the j -invariant, we may check that E is isomorphic to a quotient of C bythe lattice generated by { , e πi/ } , and the µ -action lifts to the multiplication action of µ on C .In this picture, we may identify H , ( E ) with C dτ, where τ is the coordinate on C . The µ -actionon H , ( E ) is by multiplication.Since we have H , ( E ) ∼ = H , ( E ) ⊗ , the diagonal µ -action on H , ( E ) is trivial. In otherwords, the nonvanishing holomorphic 3-form on E (unique up to scaling) is invariant under the µ -action, so it descends to Z ( θ xyza ). That is, Z xyza is Calabi-Yau. Remark . In every case, C ∗ R acts trivially on Z ( θ ). For example, C ∗ R acts on X ( θ xyza by scalingon the fibers of the vector bundle [( O P ( − /µ ] , so acts trivially on Z ( θ xyza ) since Z ( θ xyza ) liesinside the zero section. Similarly, C ∗ R acts on X ( θ axyz ) by scaling the coordinates of [ C /µ ], so actstrivially on the origin. Remark . We may check that for θ ∈ Θ , V ss ( θ ) is equal to V ss ( ϑ ) , for ϑ the lift of θ definedabove.3.3. Toric divisors.
We will often refer to the toric divisors D ρ ∈ H ( X ( θ ) , C ) and their pullbacks ι ∗ D ρ ∈ H ( Z ( θ ) , C ) . A character ρ : G → C ∗ defines a line bundle L ρ as in Section 2.1. For ρ ∈ R , the corresponding coordinate s ρ is a section of L ρ . As usual, abusing notation we also write L ρ and s ρ for the restriction to each quotient X ( θ ) ⊆ X. We define D ρ := c ( L ρ ).For θ = θ xyaz , we compute D ρ and ι ∗ D ρ explicitly. Observe that X ( θ ) admits projection mapspr x , pr y : X ( θ ) → [ O P ( − /µ ] and pr z : X ( θ ) → [ C / ( µ ) ] . We consider the vanishing loci in ( θ ) of the sections s ρ . The sections s ρ x , s ρ x , s ρ x are pulled back along pr x . They cut out thefibers in [ O P ( − /µ ] over the coordinate lines in [ P /µ ], and similarly for s ρ y , s ρ y , s ρ y . UsingSection 2.4, these substacks give classes [ L ] , [ L (cid:48) ] , [ L (cid:48) ] , respectively, and all of these are equal to H. Thus we have D ρ x = D ρ x = D ρ x = pr ∗ x ( H ) =: H x D ρ y = D ρ y = D ρ y = pr ∗ y ( H ) =: H y . The sections s ρ z , s ρ z , s ρ z are pulled back along pr z . They cut out the coordinate planes in [ C /µ ].These are trivial in H ( X ( θ ) , C ), i.e. D ρ z = D ρ z = D ρ z = 0 . The section s ρ a is nonvanishing, so D ρ a = 0. By the same argument, D ρ pz = 0. Finally, s ρ px isagain pulled back along pr x , and vanishes along the zero section of [ O P ( − /µ ] . Since x p x isa well-defined function on [ O P ( − /µ ] (with coordinates as above), which vanishes to order 3along L (cid:48) = { x = 0 } and to order 1 along the zero section. Hence the class of the zero section is − L (cid:48) ] = − H, so D ρ px = − H x D ρ py = − H y . Moduli spaces of sections of line bundles
LG-quasimaps.
For each θ ∈ Θ, we define moduli spaces of
LG-quasimaps . These wereintroduced in [21], though in our examples the definitions simplify significantly.
Definition 4.1. A prestable genus zero m -marked LG-quasimap to X ( θ ) is a tuple ( C, u, κ ) , where(i) ( C, b , . . . , b m ) is an m -marked 3-stable curve,(ii) u : C → [ V / ( G × C ∗ R )] = [ X/ C ∗ R ] is a morphism of stacks, and(iii) κ : u ∗ L (cid:99) t R → ω C, log is an isomorphism of line bundles on C ,such that all marked points, nodes, and generic points of components map to [ X ( θ ) / C ∗ R ] =[ V // ϑ ( G × C ∗ R )]. A prestable genus zero m -marked LG-quasimap to Z ( θ ) is a prestable genus zero m -marked LG-quasimap to X ( θ ) that factors through [ Z/ C ∗ R ] (cid:44) → [ X/ C ∗ R ]. Definition 4.2.
Let (
C, u, κ ) be a prestable genus zero m -marked LG-quasimap to X ( θ ). A point P of C for which u ( P ) ∈ [ V un ( θ ) / ( G × C ∗ R )] ⊆ [ V / ( G × C ∗ R )] is called a basepoint of u .We will now reinterpret these definitions more algebraically. From Section 2.1, Definition 4.1(ii) isthe same as a principal ( C ∗ ) -bundle P on C — we will denote the five corresponding line bundlesby L x , L y , L z , L a , L R , and their degrees by β x , β y , β z , β a , β R — and a section σ of E := P × ( C ∗ ) V = ( L x ⊗ L ∗ a ) ⊕ L x ⊕ L x ⊕ ( L y ⊗ L ∗ a ) ⊕ L y ⊕ L y ⊕ ( L z ⊗ L ∗ a ) ⊕ L z ⊕ L z ⊕⊕ L ⊗ a ⊕ ( L ⊗− x ⊗ L R ) ⊕ ( L ⊗− y ⊗ L R ) ⊕ ( L ⊗− z ⊗ L R ) . Using (iii) we may forget about L R altogether and replace the data (ii) and (iii) with the data ofthe line bundles L x , L y , L z , L a and a section σ of E = ( L x ⊗ L ∗ a ) ⊕ L x ⊕ L x ⊕ ( L y ⊗ L ∗ a ) ⊕ L y ⊕ L y ⊕ ( L z ⊗ L ∗ a ) ⊕ L z ⊕ L z ⊕⊕ L ⊗ a ⊕ ( L ⊗− x ⊗ ω C, log ) ⊕ ( L ⊗− y ⊗ ω C, log ) ⊕ ( L ⊗− z ⊗ ω C, log ) . We will write L ρ := u ∗ L ρ for the summands of E , i.e. E = (cid:77) ρ ∈ R L ρ , nd σ = ( σ x , σ x , σ x , σ y , σ y , σ y , σ z , σ z , σ z , σ a , σ p x , σ p y , σ p z ) . Thus an LG-quasimap (
C, u, κ ) to X ( θ ) is the same data as a tuple ( C, L , σ ) , where L is shorthandfor ( L x , L y , L z , L a ) . LG-quasimaps to Z ( θ ) are similarly reinterpreted, and we use the notations( C, u, κ ) and ( C, L , σ ) interchangeably. Definition 4.3.
Let ( C, L , σ ) be an LG-quasimap to X ( θ ) or Z ( θ ). The degree of ( C, L , σ ) is thetuple of rational numbers β := ( β x , β y , β z , β a ) . (We do not need to include β R as it is necessarilyequal to − m .) Notation . For an arbitrary character ρ of G × C ∗ R , we define L ρ := u ∗ L ρ = P × G × C ∗ R C ρ anddenote by β ρ the degree of L ρ . Definition 4.5.
For (cid:15) ∈ Q > , we say an m -marked, genus-zero LG-quasimap to Z ( θ ) or X ( θ ) is (cid:15) -stable if(1) The length (cid:96) σ ( P ) of σ at each point P is at most 1, and(2) ω C, log ⊗ L (cid:15)ϑ is ample.For the general definition of length, see [21]. We describe it in the case θ = θ xyaz , from whichthe other cases are clear. The length is a sum over components of the unstable locus V uns ( θ ). If( σ x , σ x , σ x ) = (0 , ,
0) at a point P ∈ C, then the minimum order of vanishing of these sections at P is the contribution to the (cid:96) σ ( P ) from the component { ( x , x , x ) = (0 , , } of V uns ( θ ) . We denotethis contribution by (cid:96) σx ( P ). The contribution (cid:96) σy ( P ) is defined similarly. The contribution (cid:96) σz ( P ) fromthe component { p z = 0 } is even simpler — it is just the order of vanishing of σ p z at P . Similarly (cid:96) σa ( P ) is the order of vanishing of σ a at P . Finally, we define (cid:96) σ ( P ) = (cid:96) σx ( P ) + (cid:96) σy ( P ) + (cid:96) σz ( P ) + (cid:96) σa ( P ) . Definition 4.6.
Let (
C, u, κ ) be an LG-quasimap to X ( θ ) or Z ( θ ) of degree β , and let P be abasepoint of u . The degree β ( P ) = ( β x ( P ) , β y ( P ) , β z ( P ) , β a ( P )) of the basepoint P is defined, for θ = θ xyaz , by ( β x ( P ) , β y ( P ) , β z ( P ) , β a ( P )) := ( (cid:96) σx ( P ) , (cid:96) σy ( P ) , − (cid:96) σz ( P ) , (cid:96) σa ( P )) . The reason for this definition is as follows. Restricting (
C, u, κ ) to C (cid:114) P gives a section σ | C (cid:114) P of E| C (cid:114) P . There is a way (unique up to isomorphism) to extend ( E| C (cid:114) P , σ | C (cid:114) P ) to ( E (cid:48) , σ (cid:48) ) , where E (cid:48) is a vector bundle on C and σ (cid:48) ∈ H ( C, E (cid:48) ), such that (cid:96) σ (cid:48) ( P ) = 0 . This bundle E (cid:48) is associated toa space of LG-quasimaps of degree β − β ( P ) . Therefore the degree of (
C, u, κ ) is equal to the degree“over the generic points of C ”, plus the sum of the degrees of all basepoints. Remark . As in [13], we may also define (cid:15) -stability for (cid:15) equal to either of the symbols 0+ and ∞ . A quasimap is (0+)-stable if it is (cid:15) -stable for all (cid:15) sufficiently small, and ∞ -stable if it is (cid:15) -stablefor all (cid:15) sufficiently large. Theorem 4.8 ([21], Theorem 1.1.1) . For each θ ∈ Θ , (cid:15) ∈ [0+ , ∞ ] , m ∈ Z ≥ and β ∈ ( Z ) , thereis a finite type, separated Deligne-Mumford stack LGQ (cid:15) ,m ( X ( θ ) , β ) of families of (cid:15) -stable genus zero m -marked LG-quasimaps to X ( θ ) of degree β .There is also a finite type, separated, proper Deligne-Mumford stack
LGQ (cid:15) ,m ( Z ( θ ) , β ) of familiesof (cid:15) -stable genus zero m -marked LG-quasimaps to Z ( θ ) of degree β . Definition 4.9 (Graph spaces) . We will often use the slightly modified moduli spaces LGQG (cid:15) ,m ( X ( θ ) , β )and LGQG (cid:15) ,m ( Z ( θ ) , β ), called graph spaces or spaces of LG-graph quasimaps . They parametrize (cid:15) -stable m -marked genus zero LG-quasimaps with a parametrized component , i.e. a map τ : C → P of degree 1. The stability condition is then imposed only on (the closure of) C (cid:114) (cid:98) C , where (cid:98) C is theparametrized component. .2. Effective and extremal degrees.Definition 4.10.
We say ( β, m ) is θ -effective if LGQ (cid:15) ,m ( Z ( θ ) , β ) is nonempty. Proposition 4.11. If ( β, m ) is effective, we have: • β a ≥ , • β x ≥ if x is a superscript variable, and • β x ≤ m − if x is a subscript variable.(By symmetry these statements hold with x replaced by y or z .)Proof. First, for all θ the condition in Definition 4.1, together with the fact that { a = 0 } ⊆ V uns ( θ ) , implies that L a has a global section that is nonvanishing at the generic point of each component of C . This implies β a ≥ . If x is a superscript variable, the fact that { ( x , x , x ) = (0 , , } ⊆ V uns ( θ ) shows that at leastone of L x ⊗ L ∗ a and L x has nonnegative degree. Since β a ≥ β x ≥ . If x is a subscript variable, since { p x = 0 } ⊆ V uns , we have that L ⊗− x ⊗ ω C, log has nonnegativedegree, i.e. β x ≤ m − . (cid:3) Corollary 4.12.
Let β ϑ be as in Notation 4.4. Whenever ( β, m ) is effective, we have β ϑ ∈ Z ≥ . Definition 4.13.
Even if the conditions of Proposition 4.11 are satisfied, we may have − m + (cid:15)β ϑ < , in which case Condition 2 of Definition 4.5 is never satisfied. We call such tuples ( β, m ) unstable .Explicitly, ( β, m ) is unstable when(1) m = 2 and β = β ( θ, , or(2) m = 1 and β ϑ > /(cid:15), or(3) m = 0 and β ϑ > /(cid:15) . Remark . It is the existence of unstable tuples that allows explicit calculation in Section 9.
Extremal Degrees.
Let θ = θ xyaz , as this is the example we work out in later sections. As weobserved in Section 4.2, if β x , β y , or β a is negative, the moduli space is empty. Also, if β z > m − , thenthe line bundle L − z ⊗ ω C, log has negative degree, so the moduli space is empty for the same reason.Therefore we say that the pair ((0 , , m − , , m ) is extremal , and write β ( θ, m ) := (0 , , m − , Observation 4.15.
We will often use the fact that β ( θ, m + m ) = β ( θ, m + 1) + β ( θ, m + 1) . Remark . It follows from Remark 4.12 that for ( β, m ) effective, we have β ϑ = 0 if and only if( β, m ) = ( β ( θ, m ) , m ) is extremal. Definition 4.17. If C is irreducible and the “degree over the generic point” β − (cid:80) P β ( P ) fromDefinition 4.6 is equal to β ( θ, m ) , then we say C is contracted by u . Similarly we can say anirreducible component C (cid:48) of C is contracted .The extremal degree β ( θ, m ) will play essentially the same role for us as β = 0 does in Gromov-Witten theory, with matters complicated slightly by the fact that for θ (cid:54) = θ xyza , β ( θ, m ) is a functionof m .4.3. Connections to quasimaps and spin structures.
Given an (cid:15) -stable LG-quasimap to Z ( θ ) , we may extract a quasimap to Z ( θ ) (in the sense of [7]) as follows. Definition 4.18.
By Remark 3.7, C ∗ R acts trivially on Crit( W ) . This implies that[ Z/ C ∗ R ] ∼ = Z × B C ∗ R has a projection map pr to Z. For an LG-quasimap (
C, u, κ ) , the pair ( C, pr ◦ u ) is a prestablequasimap to Z ( θ ), called the quasimap associated to ( C, u, κ ). emark . As C ∗ R acts nontrivially on X ( θ ) , there is no way to extract a quasimap from aLG-quasimap ( C, u, κ ) to X ( θ ), unless u maps C into the locus X R ( θ ) ⊆ [ X ( θ ) / C ∗ R ] of points whoseisotropy group contains C ∗ R . In this case we obtain a quasimap to X rig R ( θ ) , the rigidification of X R ( θ )by C ∗ R (see [1]). From the C ∗ R -action on X ( θ ) we see that X rig R ( θ ) is isomorphic to[(( P ) i × Bµ − i ) /µ ] ⊆ [( O P ( − i × [ C /µ ] − i ) /µ ]for some i depending on θ, where P denotes the zero section of O P ( −
3) and Bµ denotes the originin [ C /µ ] . For instance, for θ = θ xyaz there is a quasimap to X ( θ ) associated to ( C, L , σ ) exactlywhen σ z = σ z = σ z = σ p x = σ p y = 0 . The space X rig R ( θ ) will later allow us to reduce statementsabout LG-quasimaps to known facts about quasimaps.Notice that the quasimap associated to ( C, u, κ ) captures information about the superscriptvariables. We may also extract complementary data related to the subscript variables, as follows:
Proposition 4.20. If x is in the subscript of θ , then forgetting everything except for C, L x , and σ p x gives maps from LGQ (cid:15) ,m ( Z ( θ ) , β ) and LGQ (cid:15) ,m ( X ( θ ) , β ) to a space R m,(cid:15) ( − β x + m − of (prestable) spin structures , see [11] .If there are several subscript variables, this gives maps to a product of spaces of spin structures,fibered over M ,m . This follows from the following lemma, adapted from [29].
Lemma 4.21. If x is in the subscript of θ, then if L x and L x ⊗ L ∗ a have nontrivial monodromy ateach marked point, they have no global sections.Proof. The proof of Lemma 1.5 of [29] immediately generalizes to any line bundle on a nodal genuszero twisted curve such that a tensor power is a twist down of ω C, log by a effective divisor. (cid:3) Remark . The assumption of nontrivial monodromy is very important, and we will later definenumerical invariants to vanish when this assumption is not satisfied. (See Sections 5.1 and 5.2.)
Remark . In the case θ = θ axyz one can straightforwardly mimic the entire argument of [29], whichproves the analog of Theorem 8.1 for a different class of moduli spaces coming from hypersurfaces inweighted projective spaces. Indeed for θ axyz , Sections 6 and 8 are extremely simple, since the torusaction of Section 6.2 is trivial.Consider the space LGQ (cid:15) ,m ( Z ( θ xyza ) , β ) . Let ( C, L , σ ) be an (cid:15) -stable genus-zero m -marked LG-quasimap to Z ( θ ). As σ p x = σ p y = σ p z = 0 by the condition that σ land in Z ( θ ), we may rephrasethe data of ( C, L , σ ) once again, and we arrive at exactly the data of an (cid:15) -stable quasimap (in thesense of [13]) to Z ( θ ). That is, we haveLGQ (cid:15) ,m ( Z ( θ ) , β ) = Q (cid:15) ,m ( Z ( θ ) , β ) , where the latter space is a moduli stack of quasimaps. (The β on the right must be reinterpretedslightly as a character of G .) This isomorphism is, of course, the motivation for Definition 4.1. Note,however, that LGQ (cid:15) ,m ( X ( θ ) , β ) is not isomorphic to Q (cid:15) ,m ( X ( θ ) , β ) . One may wonder why, in this setup, we express the orbifold (cid:2) ( O P ( − /µ (cid:3) as the complicatedtoric variety X ( θ xyza ) = [ C // ( C ∗ ) ], rather than the more natural-seeming (and isomorphic) toricvariety [ C // (( C ∗ ) × µ )] . The reason is a slightly complicated one. In fact, we could have used either presentation. However,we will later calculate an important generating function I θ ( q, z ), which is defined using the modulispace LGQG , ( Z ( θ ) , β ) . This space parametrizes LG-quasimaps (
C, u, κ ) to Z ( θ ) , where C ∼ = P , . ne can easily see from the definitions in [21] that there is no change if P , is replaced with P ; inother words, the stack structure plays no role! The reason is that part of the data of u is a principal µ -bundle on C , and a principal µ -bundle on P , is trivial, since the orbifold fundamental groupof P , is trivial.This issue disappears in our setup, essentially because the line bundle L a may still be nontrivialfor ( C, L , σ ) ∈ LGQG , ( Z ( θ ) , β ) . As a result, I θ ( q, z ) contains much less information when usingthe “natural” presentation of X ( θ xyza ) than it does when using our presentation. This method offinding more informative presentations of orbifolds is alluded to in [12], and is related to the notionof S -extended I -functions from [16].5. Evaluation maps, compact type state space, and invariants
Evaluation maps.
The universal curve UQ (cid:15) ,m ( Z ( θ ) , β ) over LGQ (cid:15) ,m ( Z ( θ ) , β ) has a universal map u rig to Z , by Definition 4.18. By the condition in Definition 4.1, all marked points and (relative)nodes of UQ (cid:15) ,m ( Z ( θ ) , β ) map to Z ( θ ) ⊆ Z. Thus by Section 4.4 of [1], there are evaluation maps ev i : LGQ (cid:15) ,m ( Z ( θ ) , β ) → IZ ( θ ) , the rigidified inertia stack of Z ( θ ), recording the image of the i thmarked point. In [21] there is a more general (and more subtle) notion of evaluation map definedon LGQ (cid:15) ,m ( X ( θ ) , β ), and taking values in IX ( θ ). There are also evaluation maps on graph spaces: ev i : LGQG (cid:15) ,m ( X ( θ ) , β ) → IX ( θ ) × P . Compact type state space.
Part of the setup of the gauged linear sigma model in [21] is aspecial graded vector space H ( θ ), with a pairing, called the state space . It is defined via relative Chen-Ruan cohomology groups. In our case, we may work with a particularly simple subspace H ( θ ). Definition 5.1.
A sector of IX ( θ ) is called narrow if for each subscript variable, the correspondingsector of [ C /µ ] is compact, i.e. is isomorphic to Bµ . We denote the (open and closed) substackof narrow sectors by IX ( θ ) nar ⊆ IX ( θ ). The cohomology classes of narrow sectors are a directsummand H ∗ CR ( X ( θ )) nar ⊆ H ∗ CR ( X ( θ )) . Definition 5.2.
The ambient narrow state space H ( θ ) is the image ι ∗ ( H ∗ CR ( X ( θ )) nar ) ⊆ H ∗ CR ( Z ( θ )). Remark . The Poincar´e pairing on H ∗ CR ( Z ( θ )) descends to H ( θ ) . H ( θ ) inherits a grading from H ∗ CR ( Z ( θ )), but it is not the correct one for our purposes. Forexample, for θ = θ axyz this would result in a vector space concentrated in degree zero, due to the factthat Z ( θ ) ∼ = [Spec C / ( µ ) ] . In fact, this issue also arises when trying to define a graded pullbackmap on Chen-Ruan cohomology. Instead, we define the grading as follows.First, the ages of elements of H ∗ CR ( Z ( θ )) are calculated from the normal bundle to the embeddingof a g -twisted sector not into Z ( θ ) , but into the ambient space X ( θ ) . (These are equivalent in thecase where Z ( θ ) intersects the g -fixed locus in X ( θ ) transversely, which is the case for θ = θ xyza . However, it is not true for θ = θ axyz , where Z ( θ ) is contained in every g -fixed locus.)Second, we add to each degree the somewhat mysterious shift dim( Z ( θ )) − . We do this in orderto obtain the following:
Theorem 5.4.
There is a graded isomorphism between H ( θ ) and H ( θ (cid:48) ) for any θ, θ (cid:48) ∈ Θ . Inparticular, the graded subspaces have the dimensions: H ( θ ) H ( θ ) H ( θ ) H ( θ )1 4 4 1(1) Note that our definition of a narrow sector is different from that in [21]; however, it is the correct one for this setting. emark . Proving this is just a calculation; we call it a theorem because it is one of the partsof Ruan’s original LG/CY conjecture ([30]). There is a method ([9]) for proving more generalstatements of this form, via careful use of an orbifold Thom isomorphism theorem, but it does notyet apply to this case.
Proof.
We record the enlightening parts of the proof here.
Compact type state space of Z ( θ xyza ) . Let θ = θ xyza . The inclusion ι : Z ( θ ) (cid:44) → X ( θ ) factorsas ι (cid:48) ◦ ι (cid:48)(cid:48) , where ι (cid:48) is the inclusion [( P ) /µ ] (cid:44) → X ( θ ) . Since ι (cid:48) is a homotopy equivalence, we haveIm( ι ∗ ) ∼ = Im (( ι (cid:48)(cid:48) ) ∗ ).The points of ( P ) with nontrivial stabilizer are those points ( p , p , p ) , where p , p , p ∈ P areall fixed by multiplication of the first coordinate by ζ . In other words, using the notation of Section2.4, p i ∈ ˜ L or p i = ˜ P . From this we see that the orbifold locus in [( P ) /µ ] is a union of eightcomponents. The seven components isomorphic to L × L × P , L × P × P , and P × P × P do not intersect the critical locus, since ˜ P (cid:54)∈ E. The component L × L × L intersects [ E /µ ] in3 = 27 points (each isomorphic to Bµ ). That is, Proposition 5.6.
The rigidified inertia stack IZ ( θ ) of Z ( θ ) is isomorphic to the disjoint union of Z ( θ ) and · points. It is easy to check that a C -basis of Im(( ι (cid:48)(cid:48) ) ∗ ) consists of the pullbacks of the classes { , H x , H y , H z , H x H y , H x H z , H y H z , H x H y H z } . The images of the classes 1 ζ , ζ ∈ H ∗ CR ( X ( θ )) are the sums of the 27 ζ -twisted and ζ -twistedclasses, respectively, in H ∗ CR ( Z ( θ )). The corresponding ages are 1 and 2, respectively. We thusobtain the table (1) for θ xyza for H ( θ xyza ) . . Notation . We refer to e.g. ( ι (cid:48)(cid:48) ) ∗ ( H y H z ) by the more-cumbersome notation (1 ⊗ H y ⊗ H z ) , anddenote ( ι (cid:48)(cid:48) ) ∗ (1 ζ ) and ( ι (cid:48)(cid:48) ) ∗ (1 ζ ) by (1 ⊗ ⊗ ζ and (1 ⊗ ⊗ ζ , respectively, since it will simplifyour notation in the remaining cases. Compact type state space of Z ( θ axyz ) . Let θ = θ axyz . Recall from Section 3.2 that X ( θ ) ∼ =[ C / ( µ ) ] , with Z ( θ ) = [ pt/ ( µ ) ] the origin. As all fixed loci are connected, the components of IX ( θ ) and IZ ( θ ) are both in bijection with ( µ ) . The pullback map ι ∗ is surjective, and the narrowsectors correspond to elements of ( µ ) that act trivially on C . We can easily write down theseelements, and calculating their ages gives Element Age( ζ, ζ, ζ,
1) 3( ζ , ζ, ζ,
1) 4( ζ, ζ , ζ,
1) 4( ζ, ζ, ζ ,
1) 4( ζ , ζ , ζ,
1) 5( ζ , ζ, ζ ,
1) 5( ζ, ζ , ζ ,
1) 5( ζ , ζ , ζ ,
1) 6( ζ , ζ , ζ , ζ ) 5( ζ, ζ, ζ, ζ ) 4We denote the class associated to ( ζ, ζ, ζ,
1) by (1 ζ ⊗ ζ ⊗ ζ ) , and similarly for the first eight rowsof this list. The last two we denote (1 ζ ⊗ ζ ⊗ ζ ) ζ and (1 ζ ⊗ ζ ⊗ ζ ) ζ , respectively. The shifteddegree of a class γ is 2 age( γ ) − Z ( θ ) − γ ) −
6. Thus again we obtain (1).
Compact type state space of Z ( θ xyaz ) . Finally, we include the computation for H ∗ ( θ ) where θ = θ xyaz , because this case will be worked out in detail throughout the paper. ecall that X ( θ ) ∼ = [(( O P ( − × [ C /µ ]) /µ ] Z ( θ ) ∼ = [( E × Bµ ) /µ ] = [ E / ( µ ) ] . The elements (1 , , (1 , ζ ) , (1 , ζ ) , ( ζ, ζ ) , ( ζ , ζ ) ∈ ( µ ) do not give narrow sectors. We write thenarrow sectors associated to the other elementsof ( µ ) :Group element Sectors( ζ,
1) [ E /µ ]( ζ ,
1) [ E /µ ]( ζ , ζ ) 9 points( ζ, ζ ) 9 pointsThe image of the pullback is calculated in the same way as it was for θ xyza . In particular, wehave a basis for H ( θ ): { (1 ⊗ ⊗ ζ ) , ( H x ⊗ ⊗ ζ ) , (1 ⊗ H y ⊗ ζ ) , ( H x ⊗ H y ⊗ ζ ) , (1 ⊗ ⊗ ζ ) , ( H x ⊗ ⊗ ζ ) , (1 ⊗ H y ⊗ ζ ) , ( H x ⊗ H y ⊗ ζ ) , (1 ⊗ ⊗ ζ ) ζ , (1 ⊗ ⊗ ζ ) ζ } . Here (1 ⊗ ⊗ ζ ) ζ is the sum of the first set of 9 points above, and ( H x ⊗ ⊗ ζ ) is the pullbackof the class H x on the ( ζ, X ( θ ) , isomorphic to [ O P ( − /µ ] . Calculating the(properly shifted) degrees gives (1) again.It is straightforward to carry out the calculation for θ xayz , with the same result. This proves thetheorem. (cid:3) Remark . This graded isomorphism holds for the larger state spaces H ( θ ) defined in [21] as well. Explicit isomorphisms.
In Section 10, use an explicit identification of H ( θ ) with H ( θ (cid:48) ), whichwe describe here. Let θ be such that x is a superscript variable, and let θ (cid:48) be the character obtainedby moving x to the subscript. Elements of H ( θ ) are of the form (1 ⊗ α y ⊗ γ z ) g or( H x ⊗ α y ⊗ γ z ) g with g ∈ µ . We send: (1 ⊗ α y ⊗ γ z ) g (cid:55)→ (1 ζ ⊗ α y ⊗ γ z ) g ∈ H ( θ (cid:48) )( H x ⊗ α y ⊗ γ z ) g (cid:55)→ (1 ζ ⊗ α y ⊗ γ z ) g ∈ H ( θ (cid:48) ) . Repeating this process and its inverse gives explicit graded isomorphisms between H ( θ ) and H ( θ (cid:48) )for any θ, θ (cid:48) ∈ Θ . Remark . For each θ , there is a special generator with degree zero. This is the element where g = 1 ∈ µ and all entries of the tensor are 1 or 1 ζ , for x a superscript or subscript variablerespectively. We will abbreviate it by 1 θ .5.2. Virtual class and invariants.
We define open and closed substacks:LGQ (cid:15) ,m ( X ( θ ) , β ) nar : = m (cid:92) i =1 ev − i ( IX ( θ ) nar ) ⊆ LGQ (cid:15) ,m ( X ( θ ) , β )LGQ (cid:15) ,m ( Z ( θ ) , β ) nar : = m (cid:92) i =1 ev − i ( IZ ( θ ) nar ) ⊆ LGQ (cid:15) ,m ( Z ( θ ) , β ) . heorem 5.10 ([21]) . The complex R • π ∗ E is a perfect obstruction theory on LGQ (cid:15) ,m ( X ( θ ) , β ) nar . It induces (via cosection localization , see [25, 21] ) a virtual fundamental class [LGQ (cid:15) ,m ( Z ( θ ) , β ) nar ] vir ∈ H ∗ (LGQ (cid:15) ,m ( Z ( θ ) , β ) nar , C ) . There is similarly a virtual fundamental class on each graph space
LGQG (cid:15) ,m ( Z ( θ ) , β ) nar . By a general fact about cosection localization, ι ∗ [LGQ (cid:15) ,m ( Z ( θ ) , β ) nar ] vir = [LGQ (cid:15) ,m ( X ( θ ) , β ) nar ] vir , where the latter is the virtual fundamental induced by the perfect obstruction theory R • π ∗ E . Using this, we may define
LG-quasimap invariants : Definition 5.11.
Let α , . . . , α m ∈ H ( θ ) . Then we define (cid:104) α ψ a , . . . , α m ψ a m (cid:105) (cid:15),θ ,m,β : = (cid:90) [LGQ (cid:15) ,m ( Z ( θ ) ,β ) nar ] vir m (cid:89) i =1 ( ψ a i i ev ∗ i α i ) (cid:104) α ψ a , . . . , α m ψ a m (cid:105) (cid:15),θ,Gr ,m,β : = (cid:90) [LGQG (cid:15) ,m ( Z ( θ ) ,β ) nar ] vir m (cid:89) i =1 ( ψ a i i ev ∗ i α i ) . Remark . Theorem 5.10 also states that the unshifted virtual dimension of LGQ (cid:15) ,m ( Z ( θ ) , β ) nar is m . This implies that if α , . . . , α m have degrees k , . . . , k m , then (cid:104) α ψ a , . . . , α m ψ a m (cid:105) (cid:15),θ ,m,β vanishesunless (cid:80) i ( k i + a i ) = m. Similarly LGQG (cid:15) ,m ( Z ( θ ) , β ) nar has unshifted virtual dimension m + 3 . Remark . We may define these invariants for arbitrary α ∈ H ∗ CR ( Z ( θ )) , with the conventionthat they vanish if α i is supported on IX ( θ ) (cid:114) IX ( θ ) nar for some i . (We refer to such α i as broad ,as we used narrow to refer to elements of H ( θ ) . )6. Equivariant localization on
LGQG (cid:15) ,m ( X ( θ ) , β ) and LGQ (cid:15) ,m ( X ( θ ) , β )In this section we define two natural group actions on the moduli spaces; a C ∗ -action onLGQG (cid:15) ,m ( X ( θ ) , β ) induced by the C ∗ -action on P , and a torus action on LGQ (cid:15) ,m ( X ( θ ) , β ) inducedby the torus action on the toric variety X ( θ ) . The C ∗ -action on LGQG (cid:15) ,m ( X ( θ ) , β ) . The C ∗ -action we define, as well as the graph spacesthemselves, are essentially combinatorial tools for analyzing the generating functions defined inSection 7. Recall that an LG-graph quasimap to X ( θ ) is a tuple ( C, L , σ, τ ), where τ : C → P is adegree-one (nonrepresentable) map. For λ ∈ C ∗ , let λ · [ s : t ] = [ λs : t ] denote the standard (left)action on P . Then ( C, L , σ, τ ) (cid:55)→ ( C, L , σ, τ ◦ λ )is a (right) C ∗ -action on LGQG (cid:15) ,m ( X ( θ ) , β ). C ∗ -fixed locus and normal bundles. An LG-graph quasimap ( C, L , σ, τ ) to X ( θ ) is C ∗ -fixed iffor each λ ∈ C ∗ there exists an automorphism φ of C commuting with σ such that φ ◦ τ = τ ◦ λ. Alternatively, let (cid:98) C ◦ := C (cid:114) { τ − (0) , τ − ( ∞ ) } and (cid:98) C := (cid:98) C ◦ ⊆ C . Then ( C, L , σ, τ ) is C ∗ -fixed if(1) (cid:98) C ◦ contains no marked points, nodes, or basepoints of σ , and(2) (cid:98) C is contracted by u . The term unshifted virtual dimension is nonstandard, and we define it by this property. In the statement in [21], thecondition instead would read (cid:80) i ( k i + a i − age i ) = m − (cid:80) age i , where age i is the age of the twisted sector in whichev i lands. otation . We denote by (cid:98) C the closure of (cid:98) C ◦ , and we write C and C ∞ for τ − (0)and τ − ( ∞ ), respectively. We write • := C ∩ (cid:98) C and ˇ • := C ∞ ∩ (cid:98) C . The point • may be a smooth(possibly orbifold) point (in which case C is a single point), or it may be a node (in which case C is a nodal curve). Proposition 6.2. A C ∗ -fixed m -marked LG-quasimap ( C, L , σ, τ ) to X ( θ ) of degree β defines apartition B (cid:116) B ∞ of { , . . . , m } and a partition of tuples β + β ∞ = β , such that ( β , | B | + 1) and ( β ∞ , | B ∞ | + 1) are θ -effective or unstable.Proof. The fact that τ ( b i ) is either 0 or ∞ for each i defines a partition { , . . . , m } = B (cid:116) B ∞ .If • is a node, then ( C , L| C , σ | C ) is an (cid:15) -stable LG-quasimap to X ( θ ) . (Here C has the extramarking • .) We denote by β its degree, and similarly β ∞ . If there is no node at • (resp. ˇ • ), thenwe define β to be the degree of the basepoint at • (resp ˇ • , see Definition 4.6).If there are nodes at • and ˇ • , we can check that (cid:98) β = 0 , so β + β ∞ = β. If one or both of • and ˇ • is not a node, β + β ∞ by the definition of the degree of a basepoint. (cid:3) Remark . The tuple ( B , β ) is unstable exactly when • is a smooth point.Proposition 6.2 allows us to define open and closed substacks F B ∞ ,β ∞ B ,β of LGQG (cid:15) ,m ( X ( θ ) , β ) C ∗ ,consisting of those LG-quasimaps that induce the partition B (cid:116) B ∞ of { , . . . , m } and the partition β + β ∞ of β. (We refer to these as “components” of LGQG (cid:15) ,m ( X ( θ ) , β ) C ∗ , though they are almostnever connected.) If ( β , | B | + 1) and ( β ∞ , | B ∞ | + 1) are effective rather than unstable, then F B ∞ ,β ∞ B ,β ∼ = LGQ (cid:15) , | B | + • ( X ( θ ) , β ) × IZ ( θ ) LGQ (cid:15) , | B ∞ | +ˇ • ( X ( θ ) , β ∞ ) , (2)fibered over the evaluation maps ev • and ev ˇ • . When • and ˇ • are both nodes, we may calculate the C ∗ -equivariant Euler class of the virtualnormal bundle to F B ∞ ,β ∞ B ,β . One may check that the C ∗ -moving infinitesimal deformations comefrom smoothing the nodes and deforming the map τ (equivalently, moving the points τ ( • ) and τ (ˇ • )). By a classical computation, smoothing the nodes contributes factors (cid:126) − ψ • and − (cid:126) − ψ ˇ • , pulled back to the fiber product (2). (These are the weights of the deformation spaces T • C ⊗ T • (cid:98) C and T ˇ • C ∞ ⊗ T ˇ • (cid:98) C , respectively. Here we use the natural identification H ∗ C ∗ (Spec C , C ) ∼ = C [ (cid:126) ].)Deforming τ gives factors are (cid:126) and − (cid:126) , the weights of the tangent spaces T P and T ∞ P . Thusthe C ∗ -equivariant Euler class of the virtual normal bundle is ( − (cid:126) )( (cid:126) − ψ • )( − (cid:126) − ψ ˇ • ) . Definition 6.4.
We define here a special component F (cid:48) β := F m,β(cid:63),β (2) of LGQG (cid:15) ,m + (cid:63) ( X ( θ ) , β ) C ∗ ,which we will use in Sections 7.2 and 9. Definition 6.5.
We may restrict all constructions in this section to the space LGQG (cid:15),θ ,m ( Z ( θ ) , β ).Denote by F β the analog of the F (cid:48) β ; then there is a fibered square F β F (cid:48) β LGQG (cid:15) ,m +1 ( Z ( θ ) , β ) LGQG (cid:15) ,m +1 ( X ( θ ) , β )Finally, we define special classes in H C ∗ ( P , C ) . Let p and p ∞ denote the pushforwards of1 ∈ H ∗ C ∗ (Spec C , C ) ∼ = C [[ (cid:126) ]] along the equivariant inclusions 0 (cid:44) → P and ∞ (cid:44) → P , respectively. As in [13], one may define the factors by convention so that this remains true for ( β , | B | + 1) and ( β ∞ , | B ∞ | + 1)unstable. hen (choosing a C ∗ -action on O P (1)) we have p | = (cid:126) p ∞ | ∞ = − (cid:126) p | ∞ = p ∞ | = 0 . The torus action on
LGQ (cid:15) ,m ( X ( θ ) , β ) . Torus actions on spaces of stable maps were used byKontsevich to carry out explicit computations of Gromov-Witten invariants of toric varieties. Theyreduce the complicated geometry of curves in toric varieties to combinatorics of fixed point sets,which are finite and explicit. We will use the torus actions on spaces of LG-quasimaps to obtain arecursive structure, leading to the proof of Theorem 8.1. In fact, to our knowledge all of the manysuch “mirror” theorems in Gromov-Witten theory use torus-fixed-point localization.For clarity, in this section we take θ = θ xyaz unless stated otherwise. For everything we do, theappropriate changes to make for the other characters will be clear.There is a natural T = ( C ∗ ) action on V by scaling the coordinates. As all group actions on V that we have discussed are by scaling coordinates, they all commute. Thus we obtain T -actions on X ( θ ) and [ X ( θ ) / C ∗ R )] for each θ ∈ Θ . The latter induces a T -action on LGQ (cid:15) ,m ( X ( θ ) , β ) , and thevarious bundles and maps we consider have natural T -equivariant lifts. For example, ψ i and ev i have natural equivariant lifts since since they are defined via the geometry of maps to [ X ( θ ) / C ∗ R ].Similarly E = P × G × C ∗ R V has a natural lift induced by the action on V . T -fixed locus. By a classical argument of Gromov-Witten theory, T -fixed LG-quasimaps to X ( θ )are those that send C into the closure of 1-dimensional T -orbits in [ X/ C ∗ R ], and send all nodes,markings, and ramification points of ( C, u, κ ) to the T -fixed locus of [ X/ C ∗ R ].We check that the T -fixed locus of [ X/ C ∗ R ] is where: • p x = p y = 0, • z = z = z = 0, • at most one of x , x , and x is nonzero, and • at most one of y , y , and y is nonzero.These are exactly the coordinate points of X R ( θ ) ∼ = [(( P ) × Bµ ) /µ ] × B C ∗ R . Similarly, the 1-dimensional T -orbits of [ X/ C ∗ R ] (with proper closure) are the coordinate lines in X R ( θ ). Corollary 6.6. A T -fixed LG-quasimap ( C, u, κ ) to X ( θ ) has an associated T -fixed quasimap u rig to X ( θ ) . Corollary 6.7.
The T -fixed locus in LGQ (cid:15) ,m ( X ( θ ) , β ) is proper. As a result of the last fact, we can very closely mimic the T -localization arguments for quasimapsin [13, 7]. Definition 6.8.
Write K ∼ = C ( λ , . . . , λ ) for the localized T -equivariant cohomology of a point. Definition 6.9.
Consider a 1-dimensional T -orbit X µ,ν in X ( θ ) between T -fixed points µ and ν. (If such an X µ,ν exists we say µ and ν are T -adjacent .) We define the tangent weight w ( µ, ν ) to be c ( T µ X µ,ν ) ∈ H T ( µ, C ) ∼ = K. Remark . Everything in this section also applies to the graph space LGQG (cid:15) ,m ( X ( θ ) , β ) . Generating functions for genus zero LG-quasimap invariants
Sections 7, 8 and 9 are based on Sections 5 and 7 of [13] and Section 5 of [7], respectively,with minor but necessary modifications at each step. (The techniques in [13] and [7] follow thoseof Givental ([23]).) We define and compare generating functions J (cid:15),θ ( t, q, (cid:126) ), S (cid:15),θ ( t, q, (cid:126) ), and P (cid:15),θ ( t, q, (cid:126) ), encoding LG-quasimap invariants and LG-graph quasimap invariants of Z ( θ ). (Notethat the space LGQ (cid:15) ,m ( X ( θ ) , β ) and the T -action defined in the last section do not appear in thissection.) We continue to work with θ = θ xyaz . .1. Double brackets.
From now on, we fix a basis { γ j } for H ( θ ). Let { γ j } be a dual basis withrespect to the Poincar´e pairing on the nonrigidified inertia stack IZ ( θ ). Let t = (cid:80) j t j γ j ∈ H ( θ ).For α , . . . , α k ∈ H ( θ ) , and a , . . . , a k ∈ Z ≥ , we define the double bracket (compare with [13, 29]): (cid:104)(cid:104) α ψ a , . . . , α k ψ a k k (cid:105)(cid:105) (cid:15),θ ,k : = (cid:88) β,m q β q − ( k + m )3 z m ! (cid:104) α ψ a , . . . , α k ψ a k k , t, . . . , t (cid:105) (cid:15),θ ,k + m,β (3) = (cid:88) β,m q β − β ( θ,k + m ) m ! (cid:104) α ψ a , . . . , α k ψ a k k , t, . . . , t (cid:105) (cid:15),θ ,k + m,β . Here m ≥ β runs over degrees with ( β, m ) θ -effective. The shifting factor q − ( k + m )3 z , whichdoes not appear in [13], makes the double bracket an element of C [[ q x , q y , q − z , q a ]] rather than C [[ q x , q y , q a ]](( q − z )). We also define graph space double brackets by replacing (cid:104)·(cid:105) (cid:15),θ ,k + m,β in (3) with (cid:104)·(cid:105) (cid:15),θ,Gr ,k + m,β . Notation . We write C [[ q ]] as shorthand for C [[ q x , q y , q − z , q a ]]. (Analogously for θ (cid:54) = θ xyaz .)7.2. Conventions for unstable tuples.
For small k , some terms of (3) correspond to unstabletuples ( β, k + m ) (recall Definition 4.13). In the following sections, setting those terms to zerowould not give the correct relations between generating functions. To fix this, we now define certaininvariants corresponding to unstable tuples.First, we motivate these conventions. We apply C ∗ -localization to the graph space invariant (cid:104) α ψ a , . . . , α m ψ a m m , α (cid:63) ⊗ p ∞ (cid:105) (cid:15),θ,Gr ,m + (cid:63),β = (cid:90) [LGQG (cid:15) ,m ( Z ( θ ) ,β )] vir (cid:89) i ev ∗ i ( α i ) ψ a i i ∪ ev ∗ (cid:63) ( α (cid:63) ⊗ p ∞ ) . (4)The result is a sum over the fixed loci F m ,β m ∞ ,β ∞ . Consider the term corresponding to the locus F β = F m,β(cid:63),β ( θ, . The tuple ( β ( θ, ,
2) is unstable, which implies that ˇ • is a smooth point with themarking (cid:63) . Thus by the computation in Section 6.1, if the tuple ( m, β ) is stable, the normal bundleto F β is ( − (cid:126) )( (cid:126) − ψ • ) , under the identification of F β with LGQ (cid:15) ,m + {•} ( Z ( θ ) , β ) . Also, ev ∗ (cid:63) ( p ∞ )restricts on this locus to − (cid:126) and ev (cid:63) is identified with ev • . It follows that (4) can be written as (cid:90) [LGQ (cid:15) ,m + {•} ( Z ( θ ) ,β )] vir (cid:89) i ev ∗ i ( α i ) ψ a i i ∪ ev ∗ (cid:63) ( α (cid:63) ) (cid:126) ( (cid:126) − ψ • ) = (cid:104) α ψ a , . . . , α m ψ a m m , α (cid:63) (cid:126) ( (cid:126) − ψ • ) (cid:105) (cid:15),θ ,m + • ,β . This relation allows us to define invariants for ( β, m ) unstable, in the case where one entry ofthe bracket is of the form α (cid:126) ( (cid:126) − ψ i ) . That is, we set (cid:104) α ψ a , . . . , α m ψ a m m , α (cid:126) ( (cid:126) − ψ m +1 ) (cid:105) (cid:15),θ ,m +1 ,β to be thecontribution of F β to the equivariant integral (cid:104) α ψ a , . . . , α m ψ a m m , α ⊗ p ∞ (cid:105) (cid:15),θ,Gr ,m + (cid:63),β . Remark . When used in LG-quasimap invariants (rather than LG-graph quasimap invariants),we may treat (cid:126) as a formal variable, rather than a C ∗ -equivariant class on P . The function J (cid:15),θ ( t, q, (cid:126) ) . Using the conventions in the last section, we define J (cid:15),θ ( t, q, (cid:126) ) := (cid:88) j γ j (cid:104)(cid:104) γ j (cid:126) ( (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , ∈ H ( θ )[[ q ]](( (cid:126) − )) . Using IZ ( θ ) instead of IZ ( θ ) will make our notation much simpler. This is discussed in Section 3.1 of [7]. This should be thought of formally as a function H ( θ ) → H ( θ )[[ q ]](( (cid:126) − )) , without worrying aboutconvergence.) The unstable tuples contributing to J (cid:15),θ ( t, q, (cid:126) ) are:( β, m + 1) = ( β ( θ, ,
1) ( β, m + 1) = ( β ( θ, , β, m + 1) = ( β,
1) with β ϑ < /(cid:15) Calculating the terms coming from the tuples ( β,
1) with β ϑ < /(cid:15) (in the case (cid:15) = 0+) is thesubject of Section 9. We compute the other two terms here. The term ( β ( θ, , . This term is defined, according to Section 7.2, as the F β -contribution tothe sum (cid:88) j γ j (cid:104) γ j ⊗ p ∞ (cid:105) (cid:15),θ,Gr ,(cid:63),β ( θ, . (6) Claim.
LGQG (cid:15) ,(cid:63) ( Z ( θ ) , β ( θ, • A parametrized curve C τ −→ P , with a marked orbifold point (cid:63) , and • A constant map C → [ E /µ ] without basepoints, and with trivial monodromy at (cid:63) . Proof.
The line bundles L x , L y , L a , and L ρ pz have degree zero, and thus are trivial. (We may seefrom Proposition 2.12 that line bundles on P , have trivial monodromy at (cid:63) .) Since u lands in[ Z ( θ ) / C ∗ R ] , the sections σ z , σ z , σ z , σ p x , σ p y are all zero. Thus up to isomorphism, ( C, L , σ ) carriesonly the data of the parametrized marked curve C , the sections σ x i and σ y i , and the line bundle L z . As L ρ pz is trivial, we have L ⊗ z ∼ = ω C, log . However, there is a unique such bundle up to isomorphism,with monodromy 2/3 at (cid:63) . It has automorphism group µ , acting by multiplication on fibers, whichcommutes with κ : L ⊗ z → ω C, log . The sections σ x i and σ y i define a map C → [ E /µ ] . It has trivial monodromy as L a is trivial,and has no basepoints since L x and L y are trivial. (cid:3) F β is the locus where τ ( (cid:63) ) = ∞ , so it is isomorphic to [ E /µ ] × Bµ . (The Bµ comes from theautomorphisms of L z .) We see that F β is a twisted sector of IZ ( θ ).The virtual fundamental class is [ F β ] vir = [ F β ], and ev (cid:63) is the µ -rigidification map to asector [ E /µ ] ⊆ I ( Z ( θ )). The class p ∞ restricts to − (cid:126) on F β , and the normal bundle to F β (cid:44) → LGQG (cid:15) ,(cid:63) ( Z ( θ ) , β ( θ, (cid:63) on P , and has Euler class − (cid:126) . Thus (6) is equal to: (cid:88) j γ j (cid:90) [ E /µ ] × Bµ ev ∗ (cid:63) ( γ j ) = (1 ⊗ ⊗ ζ ) (cid:90) [ E /µ ] × Bµ ev ∗ (cid:63) (( H x ⊗ H y ⊗ ζ ) ) = 1 θ , (7)the twisted sector of H ( θ ) from Remark 5.9. The term ( β ( θ, , . This term is the F β -contribution to the sum (cid:88) j γ j (cid:104) γ j ⊗ p ∞ , t (cid:105) (cid:15),θ,Gr ,(cid:63),β ( θ, . (8)LGQG (cid:15) , (cid:63) ( Z ( θ ) , β ( θ, C with two order3 orbifold points to [ E /µ ] , together with a 3rd root of ω C, log ∼ = O C . L a may be nontrivial, andthere are three choices (one trivial) for the 3rd root L z . The resulting stack is isomorphic to IZ ( θ ) , and in particular the isomorphism is ev (cid:63) (after composition with the rigidification IZ ( θ ) → IZ ( θ )).The virtual fundamental class was defined to vanish on the components where L z has trivialmonodromy (Remark 5.13), and on the other components it restricts to the fundamental class. Theunion of these components is IZ ( θ ) nar . Since L x , L y , L z , L a have degree zero, they have opposite onodromies at b and (cid:63), so ev = υ ◦ ev . The normal bundle has contributions (cid:126) and − (cid:126) frommoving the images of (cid:63) and b , respectively, on P . Again, p ∞ | F β = − (cid:126) . Thus (8) is equal to (cid:88) j γ j (cid:90) IZ ( θ ) nar (cid:126) υ ∗ γ j ∪ t = 1 (cid:126) (cid:88) j γ j (cid:104) t, γ j (cid:105) Z ( θ ) = t/ (cid:126) . (9) Remark . In both of these calculations, the moduli spaces described parametrized maps LG-quasimaps without basepoints . More generally, m -marked LG-quasimaps of degree β ( θ, m ) neverhave basepoints. Thus we observe that integrals over these moduli spaces are independent of (cid:15) . Inparticular, the coefficient of q (0 , , , in J (cid:15),θ ( t, q, (cid:126) ) is independent of (cid:15). Proposition 7.4. J (cid:15),θ (0 , q, (cid:126) ) is homogeneous of degree zero, when deg( q ) := 0 and deg( (cid:126) ) = 2 .Proof. By Section 7.2, any term of J (cid:15),θ (0 , q, (cid:126) ) may be expressed as the F β contribution to (cid:88) j γ j (cid:104) γ j ⊗ p ∞ (cid:105) (cid:15),θ,Gr , ,β = (ev ) ∗ (ev ∗ p ∞ ) ∈ H ( θ ) . (10)From Theorem 4.8, LGQG , ( Z ( θ ) , β ) has unshifted virtual dimension 4, so this pushforward hasrelative dimension 1. (This statement depends on correctly shifting degrees as in Section 5.1.) Thus(10) has degree zero. (cid:3) Corollary 7.5. J (cid:15),θ ( t, q, (cid:126) ) ∈ H ( θ )[[ q, (cid:126) − ]] . Proof.
A priori, J (cid:15),θ ( t, q, (cid:126) ) may have positive powers of (cid:126) coming from the unstable terms. However, J (cid:15),θ (0 , q, (cid:126) ) includes all unstable terms, and is homogeneous of degree zero, so this does not occur. (cid:3) Remark . By Remark 4.16, only the tuples on the first line of (5) appear in the case (cid:15) = ∞ . Inparticular, J ∞ ,θ ( t, q, (cid:126) ) = 1 θ + t (cid:126) + O (1 / (cid:126) ) . Combining this with Remark 7.3 implies J (cid:15),θ ( t, q, (cid:126) ) = 1 θ + t (cid:126) + O ( q )+ O (1 / (cid:126) ) , where O ( q ) ∈ H ( θ )[[ q ]]has no q (0 , , , -coefficient.7.4. The S (cid:15),θ -operator and its inverse. We define operators on H ( θ )[[ q ]](( (cid:126) − )) by: S (cid:15),θ ( t, q, (cid:126) )( γ ) = (cid:88) j γ j (cid:104)(cid:104) γ j (cid:126) − ψ , γ (cid:105)(cid:105) (cid:15),θ , ( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) )( γ ) = (cid:88) j γ j (cid:104)(cid:104) γ j , γ − (cid:126) − ψ (cid:105)(cid:105) (cid:15),θ , . Remark . Under some conditions, we may make sense of applying these operators to power seriesin (cid:126) . The details are in Section 5.1 of [13].
Proposition 7.8. S (cid:15),θ ( t, q, (cid:126) ) = Id + O (1 / (cid:126) ) and ( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) ) = Id + O (1 / (cid:126) ) .Proof. As in Corollary 7.5, the only terms of S (cid:15),θ ( t, q, (cid:126) )( γ ) with nonnegative powers of (cid:126) comefrom unstable tuples ( β, m + 2) . The only such tuple is ( β ( θ, , . The corresponding term is thecontribution of F β = F , (0 , , , , (0 , , , to (cid:88) j γ j (cid:104) (cid:126) γ j ⊗ p ∞ , γ (cid:105) (cid:15),θ , ,β ( θ, This contribution was calculated (Section 7.3, Equation (8)), and it is equal to γ . The argument for( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) ) is the same. (cid:3) roposition 7.9. ( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) ) (cid:16) S (cid:15),θ ( t, q, (cid:126) )( γ ) (cid:17) = γ. Proof.
The series (cid:104)(cid:104) γ ⊗ [0] , δ ⊗ [ ∞ ] (cid:105)(cid:105) (cid:15),θ,Gr is a sum of equivariant integrals, hence a power series in (cid:126) . Applying localization gives a sum overfixed components where τ ( b ) = 0 ∈ P and τ ( b ) = ∞ ∈ P . These fixed components were describedin Section 6.1 as fibered products, and their normal bundles were calculated. These yield: (cid:104)(cid:104) γ ⊗ [0] , δ ⊗ [ ∞ ] (cid:105)(cid:105) (cid:15),θ,Gr = (cid:88) j (cid:104)(cid:104) γ j (cid:126) − ψ , γ (cid:105)(cid:105) (cid:15),θ , (cid:104)(cid:104) δ, γ j − (cid:126) − ψ (cid:105)(cid:105) (cid:15),θ , . (11)The constant term in (cid:126) of the right side is the contribution from the fixed component F ,β ( θ, ,β ( θ, . Again, this calculation is essentially the one from Section 7.3 (Equation (8)), and the answer is (cid:82) IZ ( θ ) υ ∗ γ ∪ δ .As this is the only term of the right side of (11) with a nonnegative power of (cid:126) , and the left sideof (11) is a power series in (cid:126) , we conclude: (cid:88) j (cid:104)(cid:104) γ j (cid:126) − ψ , γ (cid:105)(cid:105) (cid:15),θ , (cid:104)(cid:104) δ, γ j − (cid:126) − ψ (cid:105)(cid:105) (cid:15),θ , = (cid:90) IZ ( θ ) υ ∗ γ ∪ δ. Using this, we have:( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) ) (cid:16) S (cid:15),θ ( t, q, (cid:126) )( γ ) (cid:17) = (cid:88) j γ j (cid:104)(cid:104) γ j , (cid:80) j (cid:48) γ j (cid:48) (cid:104)(cid:104) γ j (cid:48) (cid:126) − ψ , γ (cid:105)(cid:105) (cid:15),θ , − (cid:126) − ψ (cid:105)(cid:105) (cid:15),θ , = (cid:88) j,j (cid:48) γ j (cid:104)(cid:104) γ j (cid:48) (cid:126) − ψ , γ (cid:105)(cid:105) (cid:15),θ , (cid:104)(cid:104) γ j , γ j (cid:48) − (cid:126) − ψ (cid:105)(cid:105) (cid:15),θ , = (cid:88) j γ j (cid:90) IZ ( θ ) υ ∗ γ ∪ γ j = γ. (cid:3) The P -series. Finally, we define: P (cid:15),θ ( t, q, (cid:126) ) := (cid:88) j γ j (cid:104)(cid:104) γ j ⊗ p ∞ (cid:105)(cid:105) (cid:15),θ,Gr ∈ H ( θ )[[ q, (cid:126) ]] . Proposition 7.10. P (cid:15),θ ( t, q, (cid:126) ) = 1 θ + O ( q ) .Proof. This coefficient of q (0 , , , in P (cid:15),θ ( t, q, (cid:126) ) is (cid:88) m,j γ j m ! (cid:104) γ j ⊗ p ∞ , t, . . . , t (cid:105) (cid:15),θ,Gr m,β ( θ,m ) , (12)The moduli spaces LGQG (cid:15),θ , m ( Z ( θ ) , (0 , , − m , L a is trivial, and(2) Components corresponding to LG-quasimaps where L a has nontrivial monodromy at somemarked point.Since L a has degree zero, these are the only possibilities.First we consider components of type (1). As L x and L y are trivial the union of these componentsis isomorphic to [ E /µ ] × W , m ( P ), where W , m ( P ) is a moduli space of spin curves (see[11]). It has dimension 1 + m and parametrizes A parametrized 1 + m -marked curve C , and • A 3rd root of ω C, log . Under this identification the evaluation maps ev i are given by the product(id , mult b i ( L z )) : [ E /µ ] × W , m ( P ) → [ E /µ ] × IBµ . Write γ j = γ j, ⊗ γ j, , where γ j, ∈ H ∗ ( E /µ , C ) and γ j, ∈ H ∗ CR ( Bµ ) . Similarly write t = t ⊗ t ∈ H ∗ ( E /µ , C ) ⊗ H ∗ CR ( Bµ ). Then (12) is equal to the product1 m ! (cid:32)(cid:90) [ E /µ ] γ j, ∪ t m (cid:33) (cid:32)(cid:90) [ W , m ( P )] vir ev ∗ ( γ j, ⊗ p ∞ ) ∪ m (cid:89) i =1 ev ∗ i ( t ) (cid:33) , (13)Using the projection formula, we rewrite the second integral as (cid:82) P p ∞ ∪ α, where α ∈ H m ( P , C )is a nonequivariant class pushed forward from W , m ( P ). Thus the second integral vanishes unless m = 0. The case m = 0 has been computed (Section 7.3, Equation (8)), and this term is equal to 1 θ .For components of type (2), the moduli space is only a fibered product, not a product, but wesimilarly find that terms with m > m = 0, since L a is a 3rd root of the trivial bundle, and when m = 0 there are no nontrivial3rd roots of the trivial bundle. Thus components of type (2) do not contribute to the coefficient of q (0 , , , . (cid:3) Factorization of J (cid:15),θ ( t, q, (cid:126) ) . Next we apply C ∗ -localization to P (cid:15),θ . As in the proof of Proposition7.9, we have P (cid:15),θ ( t, q, (cid:126) ) = (cid:88) j,j (cid:48) γ j (cid:104)(cid:104) γ j (cid:48) (cid:126) ( (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , (cid:104)(cid:104) ( − (cid:126) ) γ j , γ j (cid:48) − (cid:126) ( − (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , . Now factoring gives P (cid:15),θ ( t, q, (cid:126) ) = (cid:88) j γ j (cid:104)(cid:104) ( − (cid:126) ) γ j , (cid:80) j (cid:48) γ j (cid:48) (cid:104)(cid:104) γ j (cid:48) (cid:126) ( (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , − (cid:126) ( − (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , = (cid:88) j γ j (cid:104)(cid:104) ( − (cid:126) ) γ j , J (cid:15),θ ( t, q, (cid:126) ) − (cid:126) ( − (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , = (cid:88) j γ j (cid:104)(cid:104) γ j , J (cid:15),θ ( t, q, (cid:126) )( − (cid:126) − ψ ) (cid:105)(cid:105) (cid:15),θ , = ( S (cid:15),θ ) (cid:63) ( t, q, − (cid:126) )( J (cid:15),θ ( t, q, (cid:126) )) . The last expression contains no positive powers of (cid:126) , but P (cid:15),θ ( t, q, (cid:126) ) contains no negative powers of (cid:126) . Thus P (cid:15),θ ( t, q, (cid:126) ) = P (cid:15),θ ( t, q ) ∈ H ( θ )[[ q ]] . Applying Proposition 7.9, we have:
Corollary 7.11. J (cid:15),θ ( t, q, (cid:126) ) = S (cid:15),θ ( t, q, − (cid:126) )( P (cid:15),θ ( t, q )) . By Proposition 7.8 and Section 7.3, we have P (cid:15),θ ( t, q ) = P (cid:15),θ ( q ) · θ ∈ C [[ q ]] · θ is independent of t and J (cid:15),θ ( t, q, (cid:126) ) = P (cid:15),θ ( q ) · θ + O (1 / (cid:126) ) . In particular, Remark 7.6 shows that for (cid:15) = ∞ , J ∞ ,θ ( t, q, (cid:126) ) = 1 θ + O (1 / (cid:126) ) , hence P ∞ ,θ ( q ) = 1 θ . This implies J ∞ ,θ ( t, q, (cid:126) ) = S ∞ ,θ ( t, q, (cid:126) )(1 θ ) , hich also follows from the string equation, correctly adapted as a combination of that for stablemaps ([1], Theorem 8.3.1) and that appearing in FJRW theory ([20], Theorem 4.2.9).8. Mirror theorems
Setup.
In this section we will relate J ∞ ,θ ( t, q, (cid:126) ) to J (cid:15),θ ( t, q, (cid:126) ) , and we describe here preciselyhow they are related.In Section 7, we could have taken t ∈ H ( θ )[[ q ]] rather than t ∈ H ( θ ) with no changes. Doing so,we may formally view J (cid:15),θ ( t, q, (cid:126) ) as a map H ( θ )[[ q ]] → H ( θ )[[ q, (cid:126) − ]] . It is well-known (see [33])that the image of J ∞ ,θ ( t, q, (cid:126) ) lies on and determines a (germ of a) cone L θ ⊆ H ( θ )[[ q, (cid:126) − ]] . Wewill show that J (cid:15),θ ( t, q, (cid:126) ) also lies on this cone also for all (cid:15) .From Section 7.5, J (cid:15),θ ( t, q, (cid:126) ) differs by an element of C [[ q ]] from S (cid:15),θ ( t, q, (cid:126) )(1 θ ) . Thus the claimthat J (cid:15),θ ( t, q, (cid:126) ) lies on the cone L θ follows from the claim that S (cid:15),θ ( t, q, (cid:126) )(1 θ ) lies on L θ . We will prove:
Theorem 8.1 (All-chamber mirror theorem) . There is an automorphism T (cid:15),θ of H ( θ )[[ q ]] suchthat S (cid:15),θ ( t, q, (cid:126) )(1 θ ) = S ∞ ,θ ( T (cid:15),θ ( t ) , q, (cid:126) )(1 θ ) = J ∞ ,θ ( T (cid:15),θ ( t ) , q, (cid:126) ) . In particular, the image of S (cid:15),θ ( t, q, (cid:126) )(1 θ ) is the same as the image of J ∞ ,θ ( t, q, (cid:126) ) . To find T (cid:15),θ ( t ), we expand: S (cid:15),θ ( t, q, (cid:126) )(1 θ ) = 1 θ + 1 (cid:126) (cid:88) β,m,j ( β,m ) (cid:54) =( β ( θ, , γ j q β − β ( θ,m ) m ! (cid:104) γ j , θ , t, . . . , t (cid:105) (cid:15),θ , m,β + O (1 / (cid:126) )= 1 θ + 1 (cid:126) (cid:88) j γ j (cid:104)(cid:104) γ j , θ (cid:105)(cid:105) (cid:15),θ , − θ + O (1 / (cid:126) ) . (For consistency, the unstable part (cid:80) j γ j (cid:104) γ j , θ (cid:105) (cid:15),θ , ,β ( θ, is defined to be 1 θ .) Set: T (cid:15),θ ( t ) := (cid:88) j γ j (cid:104)(cid:104) γ j , θ (cid:105)(cid:105) (cid:15),θ , − θ . Observation 8.2. J ∞ ,θ ( T (cid:15),θ ( t ) , q, (cid:126) ) = S (cid:15),θ ( t, q, (cid:126) )(1 θ ) + O (1 / (cid:126) ) . To see that T (cid:15),θ ( t ) is an automorphism of H ( θ )[[ q ]] , we equate coefficients of q (0 , , , (cid:126) − inCorollary 7.11. Using Remark 7.6, we see that T (cid:15),θ ( t ) = t + O ( q ) . Our strategy for establishing the equality of S (cid:15),θ ( t, q, (cid:126) )(1 θ ) and J ∞ ,θ ( T (cid:15),θ ( t ) , q, (cid:126) ) is to calculateboth using T -localization. Since both involve integrals over LGQ (cid:15) ,m ( Z ( θ ) , β ) , not LGQ (cid:15) ,m ( X ( θ ) , β ) , we need to rewrite them. In particular, the coefficients of S (cid:15),θ ( t, q, (cid:126) )( γ ) are integrals of the form (cid:90) [LGQ (cid:15) ,m ( Z ( θ ) ,β )] vir m (cid:89) i =1 ψ a i i ev ∗ i ( α i ) . (14)From Section 5.1, we only consider classes α i = ι ∗ a i pulled back from IX ( θ ) . Thus if e :LGQ (cid:15) ,m ( Z ( θ ) , β ) (cid:44) → LGQ (cid:15) ,m ( X ( θ ) , β ) is the natural embedding, we may rewrite the above as (cid:90) e ∗ [LGQ (cid:15) ,m ( Z ( θ ) ,β )] vir m (cid:89) i =1 ψ a i i ev ∗ i ( a i ) . Here the word cone refers to a subset that is preserved under multiplication by elements from the “base ring” C [[ q ]] . It is also a fact (which we will not need) that H ( θ )[[ q ]](( (cid:126) − )) has a symplectic structure and that the cone is aLagrangian submanifold. ere we use the fact that evaluation maps and ψ classes are compatible with ι. Now we use thegeneral fact about cosection localization that ι ∗ [LGQ (cid:15) ,m ( Z ( θ ) , β )] vir = [LGQ (cid:15) ,m ( X ( θ ) , β )] vir (15)to we rewrite the S -operator (see [25]). For α ∈ H ( θ ) , write ˜ α for the corresponding element of H ∗ CR ( X ( θ )) nar / (ker ι ∗ ) . Then we have S (cid:15),θ ( t, q, (cid:126) )( γ ) = (cid:88) β,m,j q β − β ( θ, m ) m ! γ j (cid:90) [LGQ (cid:15) , m ( X ( θ ) ,β )] vir ev ∗ ˜ γ j (cid:126) − ψ ∪ ev ∗ ˜ γ ∪ ev ∗ ˜ t · · · ev ∗ m +2 ˜ t. (Note LGQ (cid:15) ,m ( X ( θ ) , β ) is not proper; however, by (15) the virtual fundamental class is a homologyclass, in particular compactly supported, so the integral makes sense.) We also see that S (cid:15),θ ( t, q, (cid:126) )( γ )is pulled back from IX ( θ ):˜ S (cid:15),θ ( t, q, (cid:126) )( γ ) := (cid:88) β,m,j q β − β ( θ, m ) m ! ˜ γ j (cid:90) [LGQ (cid:15) , m ( X ( θ ) ,β )] vir ev ∗ ˜ γ j (cid:126) − ψ ∪ ev ∗ ˜ γ ∪ ev ∗ ˜ t · · · ev ∗ m +2 ˜ t. There is no Poincar´e pairing on IX ( θ ), as it is not proper; however, we may still view the elements˜ γ j ∈ H ∗ CR ( X ( θ )) nar / (ker ι ∗ ) as a dual basis to { ˜ γ j } in the following sense. Since IX ( θ ) deformationretracts to IX rig R ( θ ) , which is proper and contains IZ ( θ ) , there is a cohomology class Z such that Z ∩ X rig R ( θ ) = ι ∗ [ IZ ( θ )] ∈ H ∗ ( IX ( θ )). Then the Poincar´e pairing on Z ( θ ) induces the perfect pairingon H ∗ CR ( X ( θ )) nar / (ker ι ∗ ): (cid:104) α, β (cid:48) (cid:105) Z := (cid:90) [ X rig R ( θ )] α ∪ β ∪ Z, and under this pairing { ˜ γ j } and { ˜ γ j } are again dual bases. Alternatively, { Z ∪ ˜ γ j } and { ˜ γ j } aredual bases with respect to a perfect pairing Z ∪ H ∗ CR ( X ( θ )) nar / (ker ι ∗ ) ⊗ H ∗ CR ( X ( θ )) nar / (ker ι ∗ ) → C . For this reason, we define Z (cid:15),θ ( t, q, (cid:126) )( γ ) := Z ∪ ˜ S (cid:15),θ ( t, q, (cid:126) )( γ ) . (16)It is then sufficient to show: Z ∞ ,θ ( T (cid:15),θ ( t ) , q, (cid:126) ) = Z (cid:15),θ ( t, q, (cid:126) )(1 θ ) . Remark . It will simplify things to choose lifts of classes in H ( θ ), rather than working withelements of H ∗ CR ( X ( θ )) nar / (ker ι ∗ ) . Therefore in our notation we view ˜ γ j , ˜ γ j , ˜ γ, ˜ t , and Z (cid:15),θ ( t, q, (cid:126) )( γ )as elements of H ∗ CR ( X ( θ )) nar [[ q, (cid:126) − ]] . Of course, they are not well-defined, but their pullbacks to Z ( θ ) are.We are now in a situation to follow the arguments of [13]. For each fixed point µ of IX ( θ ),consider the T -equivariant integral Z (cid:15),θµ ( t, q, (cid:126) )( γ ) := i ∗ µ Z (cid:15),θ ( t, q, (cid:126) )( γ ) = (cid:90) [ X rig R ( θ )] δ µ ∪ Z (cid:15),θ ( t, q, (cid:126) )( γ ) , where δ µ ∈ H ∗ CR,T, loc ( X ( θ )) is the equivariant fundamental class of µ and i µ : µ (cid:44) → IX ( θ ) is theinclusion. Precisely, as IX ( θ ) has isolated fixed points, the Atiyah-Bott localization formula statesthat its localized equivariant cohomology groups H ∗ CR,T, loc ( X ( θ )) are generated by pushforwards f fundamental classes of the fixed points from the equivariant cohomology of the fixed locus. Werewrite Z (cid:15),θµ ( t, q, (cid:126) )( γ ) = (cid:88) m,β q β − β ( θ, m ) m ! (cid:90) [LGQ (cid:15) , m ( X ( θ ) ,β )] vir ev ∗ δ µ (cid:126) − ψ ∪ ev ∗ ˜ γ ∪ ev ∗ ˜ t ∪ · · · ∪ ev ∗ m +2 ˜ t. (17)These are the objects of interest in the next section.8.2. Localization and recursion.
From Section 6.2, (17) has a natural equivariant lift, so weapply the fixed-point localization formula to write: Z (cid:15),θµ ( t, q, (cid:126) )( γ ) = (cid:88) m,β,F q β − β ( θ, m ) m ! (cid:90) [ F ] vir i ∗ F (cid:16) ev ∗ δ µ (cid:126) − ψ ∪ ev ∗ ˜ γ ∪ · · · ∪ ev ∗ m +2 ˜ t (cid:17) e ( N vir F ) , (18)where i F : F (cid:44) → LGQ (cid:15) , m ( X ( θ ) , β ) is the inclusion of a component of the fixed locus, [ F ] vir is thevirtual fundamental class from the T -fixed part of R • π ∗ E , and N vir F is the T -equivariant virtualnormal bundle, defined to by the T -moving part of R • π ∗ E .We recall terminology from [13]. Definition 8.4.
For each fixed point µ of IX ( θ ) , we partition the components of the T -fixed locusof LGQ (cid:15) , m ( X ( θ ) , β ) into three subsets: • V ( µ, β, m ) consists of components for which the first marking does not map to µ , • In( µ, β, m ) consists of components for which the first marking maps to µ and is on acontracted component of C (see Definition 4.17), and • Rec( µ, β, m ) consists of components for which the first marking maps to µ and is not ona contracted component of C . In this case, u rig sends this component to a fixed curve in IX ( θ ) connecting µ to a unique other fixed point, which we denote by ν .This is Lemma 7.5.1 of [13]: Lemma 8.5.
For each ( β, ( k j ) j ) with (cid:80) j k j = m the coefficient of q β − β ( θ, m ) (cid:81) j t k j j in Z (cid:15),θµ ( γ ) isa rational function of (cid:126) with coefficients in K (see Section 6.2). This rational function decomposesas a finite sum of rational functions with denominators either powers of (cid:126) , of powers of linear factors (cid:126) − α , where − nα is one of the weights of the T -representation T µ ( IX ( θ )) for some n ∈ Z > .Proof. The proof in [13] requires essentially no modification, and we summarize it here.From (18), the coefficient of q β − β ( θ, m ) (cid:81) j t k j j in Z (cid:15),θµ ( t, q, (cid:126) )( γ ) is: (cid:88) F m !) (cid:90) [ F ] vir i ∗ F (cid:16) ev ∗ δ u (cid:126) − ψ ∪ A (cid:17) e ( N vir F ) = (cid:88) F,a m ! (cid:126) a +1 ) (cid:90) [ F ] vir i ∗ F ( ψ a ev ∗ δ u ∪ A ) e ( N vir F ) , (19)where A is the product of factors from the evaluation maps 2 , . . . , m , and depends on β and( k j ) j . • On components in V ( µ, β, m ), the factor ev ∗ δ µ restricts to zero. • On components in In( µ, β, m ) , ψ is nonequivariant, hence nilpotent, so the denominatorsare (bounded) powers of (cid:126) . • On components in Rec( µ, β, m ), the ψ is an equivariant class. However, if d is thedegree of u rig on the component containing b , then the fibers of the ( T ∗ b C ) ⊗ d is naturallyisomorphic to T ∗ µ X µ,ν from Section 6.2. Thus the left side of (19) has a simple pole at ψ = − w ( µ,ν ) d . (cid:3) This is Lemma 7.5.2 of [13], and is essentially unchanged from Proposition 4.4 of [23]. emma 8.6. Z (cid:15),θµ ( t, q, (cid:126) ) satisfies the recursion Z (cid:15),θµ ( t, q, (cid:126) ) = R (cid:15),θµ ( t, q, (cid:126) ) + (cid:88) ν T -adjacentto µ ∞ (cid:88) d =1 q dβ ( µ,ν ) C µ,ν,d (cid:126) + w ( µ,ν ) d Z (cid:15),θν (cid:18) t, q, w ( µ, ν ) d (cid:19) , (20) such that • R (cid:15),θµ ( t, q, (cid:126) ) is a power series in / (cid:126) such that for each ( β, ( k j ) j ) with (cid:80) k j = m, thecoefficient of q β − β ( θ, m ) (cid:81) j t k j j is a polynomial in / (cid:126) , • β ( µ, ν ) is a degree dependent only on µ and ν , • w ( µ, ν ) is the tangent weight defined in Section 6.2, and • C µ,ν,d is independent of (cid:15). The proof is similar to that in [13]. Note also that in the case θ = θ axyz the second term is zero,as every component is contracted. Proof. Z (cid:15),θµ ( t, q, (cid:126) ) has a single unstable term, from the unstable tuple ( β ( θ, , . Using Section7.4, the contribution is (cid:104) [ Z ( θ )] ∪ ˜ γ, δ µ (cid:105) . We analyze the contributions from V ( µ, β, m ) , In( µ, β, m ), and Rec( µ, β, m ) to (18).As in Lemma, 8.5, the contribution of components in V ( µ, β, m ) , is zero.Consider a fixed component in In( µ, β, m ) . This parametrizes LG-quasimaps such that theassociated quasimap C → [ E /µ ] sends b to µ and contracts the component containing b . As in theproof of Lemma 8.5, the contribution is a power series in 1 / (cid:126) , whose q β − β ( θ, m ) (cid:81) j t k j j -coefficientis an element of K [1 / (cid:126) ] . We define R (cid:15),θµ ( t, q, (cid:126) ) to be the sum of contributions from components inIn( µ, β, m ).We now consider a fixed component M ∈ Rec( µ, β, m ) , corresponding to the term of (18): q β − β ( θ, m ) m ! (cid:90) [ M ] vir ev ∗ δ µ (cid:126) − ψ ∪ ev ∗ ˜ γ ∪ · · · ∪ ev ∗ m +2 ˜ te ( N vir M ) . (21)Let ν be as in Definition 8.4. LG-quasimaps in (As with µ , ν naturally lives in IX ( θ ) T rather than X ( θ ) T . ) By gluing LG-quasimaps, we can write M as a fibered product M (cid:48) × IX ( θ ) M (cid:48)(cid:48) , where M (cid:48) is a T -fixed component of LGQ (cid:15) , • ( X ( θ ) , β (cid:48) ) and M (cid:48)(cid:48) is a T -fixed component of LGQ (cid:15) ,m +1+ˇ • ( X ( θ ) , β − β (cid:48) ).(Note that the meaning of ˇ • differs very slightly from that in Section 6.1.) Here β (cid:48) is the degree of( C (cid:48) , u (cid:48) , κ (cid:48) ). The maps to IX ( θ ) are, in the first case, the evaluation map at • , and in the secondcase, the evaluation map at ˇ • , composed with the inversion map on IZ ( θ ) . (See Sections 2.2 and2.3.)As C (cid:48) has a single marked point, a single node, and no basepoints, we have β (cid:48) z = 0 . Similarly β (cid:48) a = 0 . Also, by the characterization of 1-dimensional T -orbits in Section 6.2, either β (cid:48) x = 0 or β (cid:48) y = 0 , and by the noncontractedness of C (cid:48) , the other is a positive integer. Thus it is of the form dβ ( µ, ν ) , where β ( µ, ν ) is either (1 , , ,
0) or (0 , , , β (cid:48) − β ( θ, (cid:54) = (0 , , , . We wish to write (21) as a product of integrals over M (cid:48) and M (cid:48)(cid:48) . To do this, we need to computethe virtual class [ M ] vir in terms of [ M (cid:48) ] vir and [ M (cid:48)(cid:48) ] vir . Smoothing the node o : C (cid:48) ∩ C (cid:48)(cid:48) gives thedistinguished triangle of relative perfect obstruction theories: R • π ∗ E → R • π ∗ E| C (cid:48) ⊕ R • π ∗ E| C (cid:48) ⊕ → R • π ∗ E| o → R • π ∗ E [1] . The term R • π ∗ E| o is isomorphic as a G -bundle over M to the trivial bundle with fiber V , concentratedin degree zero.We instead need a perfect obstruction theory relative to the stack M tw0 ,m (Section 2.2). Thedifference comes from the relative tangent complex T A / M tw0 ,m . This is equal to P × G × C ∗ R g , concentrated n degree -1, where g is the Lie algebra of G. Thus if we denote by F • ( C ) the perfect obstruction theoryof LGQ (cid:15) ,m ( X ( θ ) , β ) relative to M tw0 ,m , and by F • ( C (cid:48) ), etc., the corresponding perfect obstructiontheories on M (cid:48) , etc., the triangle above becomes F • ( C ) → F • ( C (cid:48) ) ⊕ F • ( C (cid:48)(cid:48) ) → F • ( o ) → F • [1] , where every fiber of F ( o ) can be canonically identified with T ν IX ( θ ) . As the T -fixed points of IX ( θ )are isolated, T ν IX ( θ ) has no T -fixed part.Now, to be able to make statements about the absolute obstruction theory of LGQ (cid:15) ,m ( X ( θ ) , β ) , we need to analyze the tangent complex of M tw0 ,m . Again we have a triangle T M tw0 ,m ( C ) → T M tw0 ,m ( C (cid:48) ) ⊕ T M tw0 ,m ( C (cid:48)(cid:48) ) → T M tw0 ,m ( o ) → T M tw0 ,m [1] , and here T M tw0 ,m ( o ) is the deformation space of the node o , with each fiber canonically isomorphicto T o C (cid:48) ⊗ T o C (cid:48)(cid:48) . The factor T o C (cid:48) gives a topologically trivial bundle over M up to torsion, with T -weight w ( ν, µ ) /d . The factor T o C (cid:48)(cid:48) may be topologically nontrivial (depending on M (cid:48)(cid:48) ), but inany case the T -action on T M tw0 ,m ( o ) is nontrivial. Thus we have, exactly as in [13]:[ M ] vir = [ M (cid:48) ] vir × [ M (cid:48)(cid:48) ] vir e T ( N vir M ) = e T ( T ν IX ( θ )) e T ( N vir M (cid:48) ) e T ( N vir M (cid:48)(cid:48) )( w ( ν,µ ) d − ψ M (cid:48)(cid:48) ) . Since ev | M (cid:48)(cid:48) is a constant map to ν ∈ IX ( θ ), and w ( ν, µ ) = − w ( µ, ν ) , (21) is equal to the product: (cid:32) q β (cid:48) − β ( θ, (cid:90) [ M (cid:48) ] vir ev ∗ δ µ ( (cid:126) + w ( µ,ν ) d ) e T ( N vir M (cid:48) ) (cid:33) (cid:32) q ( β − β (cid:48) ) − β ( θ,m +2) m ! (cid:90) [ M (cid:48)(cid:48) ] vir ev ∗ ( δ ν ) ∪ ev ∗ (˜ γ ) ∪ (cid:81) i ev ∗ i ( t )( − w ( µ,ν ) d − ψ ) e T ( N vir M (cid:48)(cid:48) ) (cid:33) The factor (cid:126) + w ( µ,ν ) d is pulled back from H ∗ T × C ∗ (Spec C , C ) , so may be factored out. The resultingintegral C µ,ν,d is over a moduli space of sections with no basepoints , hence it is independent of (cid:15). Summing over M , β , and m , we get (20). (cid:3) Lemma 8.7.
Define ( qe L ρ ) β = q β e β ρ . Then for any γ ∈ H ( θ )[[ q ]] , the expression D ( Z (cid:15),θµ ) := (cid:16) Z (cid:15),θµ ( t, qe Y zL ϑ , (cid:126) )( γ ) (cid:17) (cid:16) Z (cid:15),θυ ( µ ) ( t, q, − (cid:126) )( γ ) (cid:17) is an element of K [[ q, Y, (cid:126) ]] . Proof.
There is an important line bundle U ( L ) β ,β θ on each quasimap graph space, defined in [13].We define a modified version for LG-quasimap graph spaces.The line bundle L ϑ on [ V / ( G × C ∗ R )] induces an embedding [ V / ( G × C ∗ R )] (cid:44) → [ C N +1 / C ∗ ] . Given anLG-graph quasimap (
C, u, κ, τ ) of degree β , composing gives a prestable graph quasimap C → P N of degree β ϑ . (The fact that this quasimap is prestable in the sense of [13] comes from the fact that V ss ( ϑ ) = V ss ( θ ) . ) Note that L ϑ = u ∗ L ϑ has trivial monodromy at every marked point of C , andindeed the map C → P N factors through the coarse moduli space C of C by the definition of C .Done in families, this construction yields a map LGQG (cid:15) ,m ( X ( θ ) , β ) Q ( β ϑ ), where Q ( β ϑ ) is a stackof prestable nonorbifold graph quasimaps to P N of degree β ϑ .The stack Q ( β ϑ ) has a forget-and-contract map to a stack Q (cid:48) ( β ϑ ) as in Section 3 of [13],remembering only the restriction of a quasimap to the parametrized component; all marked pointsare forgotten (possible since the orbifold structure has been removed), and nodes are replaced withbasepoints of degree equal to the total degree of the line bundle “on the other side” of the node.The stack Q (cid:48) ( β ϑ ) parametrizes sections of line bundles on P , with no stability conditions — in fact,it is a projective space. Denote by U ( L ) ϑ the pullback to LGQG (cid:15) ,m ( X ( θ ) , β ) of O Q (cid:48) ( β ϑ ) (1) . nstead of forgetting the marked points, one may replace them with basepoints. Fix degrees β (1)and β (2) . Write β ϑ (1) and β ϑ (1) for the corresponding integers as in Definition 4.5. Then there is amap Φ : LGQG (cid:15) ,m ( X ( θ ) , β ) → Q (cid:48) ( β ϑ + β (1) ϑ + β (2) ϑ ) , which as above sends ( C, u, κ ) to a quasimap P → P N , with “artificial” basepoints added at τ ( b )and τ ( b ) . We define U ( L ) β (1) ,β (2) ϑ := Φ ∗ O Q (cid:48) ( β ϑ + β (1) ϑ + β (2) ϑ ) (1) . Let F µ (2 + m, β ) ⊆ LGQG (cid:15) , m ( X ( θ ) , β ) T be the open and closed substack of T -fixed (but notnecessarily C ∗ -fixed) LG-quasimaps for which the parametrized component is contracted to µ . It is C ∗ -invariant but not C ∗ -fixed, as there may be basepoints, nodes, and marked points mapped by τ to P (cid:114) { , ∞} . Write γ = (cid:88) β q β γ β ∈ H ( θ )[[ q ]]and consider the series of T -equivariant integrals: (cid:88) m,β q β − β ( θ, m ) m ! (cid:88) β (1) ,β (2) q β (1) q β (2) (22) (cid:90) [ F µ (2+ m,β )] vir e c ( U ( L ) β (1) ,β (2) ϑ ) Y ev ∗ (˜ γ β (1) ⊗ p ) ev ∗ (˜ γ β (2) ⊗ p ∞ ) (cid:81) mi =3 ev ∗ i ( t ) e T ( N vir F µ (2+ m,β ) ) ∈ K [[ (cid:126) ]] . Since the denominator is a class in the T -equivariant cohomology of F µ (2 + m, β ), it does not contain (cid:126) . We apply C ∗ -localization to compute the integral. The contribution from a fixed component F B ∞ ,β ∞ B ,β ,µ := F B ∞ ,β ∞ B ,β ∩ F µ (2 + m, β ) is zero unless b ∈ B and b ∈ B ∞ . In this case, since we haveseen that e ( N vir F B ∞ ,β ∞ B ,β ,µ | F µ (2+ m,β ) ) = ( − (cid:126) )( (cid:126) − ψ • )( − (cid:126) − ψ ˇ • ), we get the integral: (cid:90) [ F B ∞ ,β ∞ B ,β ∩ F µ (2+ m,β )] vir e c ( U ( L ) β (1) ,β (2) ϑ ) Y ev ∗ (˜ γ β (1) ) ev ∗ (˜ γ β (2) ) (cid:81) mi =3 ev ∗ i ( t ) e T (cid:16) N vir F µ (2+ m,β ) (cid:17) ( (cid:126) − ψ • )( − (cid:126) − ψ ˇ • ) . (23)As before we may write F B ∞ ,β ∞ B ,β ,µ as a (not fibered) product M × (cid:99) M × M ∞ , where M = (LGQ (cid:15) , | B | + • ( X ( θ ) , β )) Tµ (cid:99) M = (LGQG (cid:15) , ( X ( θ ) , β ( θ, T × C ∗ µ M ∞ = (LGQ (cid:15) , | B ∞ | +ˇ • ( X ( θ ) , β ∞ )) Tµ . Here the superscript µ refers only to components where the extra marked point (or, for (cid:99) M , the entirecurve C ) is mapped to µ . (cid:99) M is a union of points, each corresponding to choices of monodromies.As in Lemma 8.6, we write (23) as a product of integrals over M and M ∞ . Smoothing the nodes • and ˇ • shows N vir F µ (2+ m,β ) = N vir M ⊕ N vir M ∞ , where the normal bundles on the right are taken relative to the ambient spaces LGQ (cid:15) , | B | + • ( X ( θ ) , β )and LGQ (cid:15) , | B ∞ | +ˇ • ( X ( θ ) , β ∞ ). The line bundle U ( L ) β (1) ,β (2) ϑ can be expressed on the product M × (cid:99) M × M ∞ as follows. As the construction above involves restricting to the parametrized component, the ap M × (cid:99) M × M ∞ → Q (cid:48) ( β ϑ + β (1) ϑ + β (2) ϑ ) is constant, so the restriction of U ( L ) β (1) ,β (2) ϑ | M × (cid:99) M × M ∞ is topologically trivial.We compute the ( T × C ∗ )-weight as follows. Let ( C, u, κ ) ∈ M × (cid:99) M × M ∞ , with C ∼ = C ∪ (cid:98) C ∪ C ∞ . Then Φ(
C, u, κ ) is a quasimap P → P N , given in coordinates by[ s : t ] (cid:55)→ [ a s β ϑ + β (1) ϑ t β ∞ ϑ + β (2) ϑ : · · · : a N s β ϑ + β (1) ϑ t β ∞ ϑ + β (2) ϑ ] . Here the a i s are determined by µ . The weight of U ( L ) β (1) ,β (2) ϑ at ( C, u, κ ) is equal to the weight of O Q (cid:48) ( β ϑ + β (1) ϑ + β (2) ϑ (1) at Φ( C, u, κ ). By the definition of the map Φ, this is the T -weight of L ϑ at µ ,denoted w µ,ϑ . From the choice of coordinates in Section 6.1, the C ∗ -weight is ( β ϑ + β (1) ϑ ) (cid:126) .Now we may factor the integral (23) as: e ( w µ,ϑ ) Y (cid:90) [(LGQ (cid:15) , | B | + • ( X ( θ ) ,β )) Tµ ] vir e ( β ϑ + β ϑ (1)) (cid:126) Y ev • ( δ µ ) ev (˜ γ β (1) ) (cid:81) | B | i =2 ev ∗ i ( t ) e T ( N vir(LGQ (cid:15) , | B | + • ( X ( θ ) ,β )) Tµ )( (cid:126) − ψ • ) · (cid:90) [(LGQ (cid:15) , | B ∞| +ˇ • ( X ( θ ) ,β ∞ )) Tυ ( µ ) ] vir ev ˇ • ( υ ∗ δ µ ) ev (˜ γ β (2) ) (cid:81) | B ∞ | i =2 ev ∗ i ( t ) e T ( N vir(LGQ (cid:15) , | B ∞| +ˇ • ( X ( θ ) ,β ∞ )) Tυ ( µ ) )( − (cid:126) − ψ ˇ • ) . For compactness, we write this as e ( w µ,ϑ ) Y S ( | B | ) S ( | B ∞ | ) . Summing gives: e ( w µ,ϑ ) Y (cid:88) B ,B ∞ ,β ,β ∞ ,β (1) ,β (2) q β + β ∞ + β (1)+ β (2) − β ( θ, | B | + | B ∞ | ) ( | B | + | B ∞ | )! S ( | B | ) S ( | B ∞ | )= e ( w µ,ϑ ) Y (cid:88) m ,m ∞ ,β ,β ∞ ,β (1) ,β (2) q β + β ∞ + β (1)+ β (2) − β ( θ,m + m ∞ ) m ! m ∞ ! S ( m ) S ( m ∞ )= e ( w µ,ϑ ) Y (cid:88) m ,β ,β (1) q β + β (1) − β ( θ,m +1) m ! S ( m ) (cid:88) m ∞ ,β ∞ ,β (2) q β ∞ + β (2) − β ( θ,m ∞ +1) m ∞ ! S ( m ∞ ) = e ( w µ,ϑ ) Y (cid:88) β (1) ( qe Y (cid:126) L ϑ ) β (1) Z (cid:15),θµ ( t, qe Y (cid:126) L ϑ , (cid:126) )( γ β (1) ) (cid:88) β (2) q β (2) Z (cid:15),θυ ( µ ) ( t, q, − (cid:126) )( γ β (2) ) = e ( w µ,ϑ ) Y (cid:16) Z (cid:15),θµ ( t, qe Y (cid:126) L ϑ , (cid:126) )( γ ) (cid:17) (cid:16) Z (cid:15),θυ ( µ ) ( t, q, − (cid:126) )( γ ) (cid:17) . (cid:3) We have now assembled all of the necessary pieces to prove our mirror theorem.
Proof of Theorem 8.1.
The theorem now follows from Uniqueness Lemma 7.7.1 of [13], applied tothe systems: { Z (cid:15),θµ ( t, q, (cid:126) )(1 θ ) , µ ∈ IX ( θ ) T }{ Z ∞ ,θµ ( T (cid:15),θ ( t ) , q, (cid:126) )(1 θ ) , µ ∈ IX ( θ ) T } . (As in Section 3.7.3, Item (3) of [7], we modify Condition (5) of the Uniqueness Lemma slightly.) Inparticular, the Uniqueness Lemma requires five properties to hold, and they are verified in:(1) Lemma 8.5,(2) Lemma 8.6,(3) Lemma 8.7,
4) Observation 8.2, and(5) Remark 7.3. (cid:3) Calculating the I -functions In this section, we compute I θ ( q, (cid:126) ) := J ,θ (0 , q, (cid:126) ) for any θ ∈ Θ. The following threeobservations allow explicit computations.
Observation 9.1.
LGQG , ( X ( θ ) , β ) is proper. To see this, consider ( C, L , σ ) = ( C, u, κ ) ∈ LGQG , ( X ( θ ) , β ) . If x is a superscript variable, then β x ≥ . Hence the bundle L p x ∼ = L − x ⊗ ω C, log has negative degree, since deg ω C, log = − − < . If x is a subscript variable, we saw inProposition 4.21 that L x and L x ⊗ L ∗ a have no global sections. From these, properness follows by astandard argument.Alternatively, one may show that LGQG , ( X ( θ ) , β ) is isomorphic to LGQG , ( Z (cid:48) ( θ ) , β ) , where Z (cid:48) ( θ (cid:48) ) ∼ = [(( P ) × Bµ ) /µ ] is the critical locus of a polynomial W (cid:48) inside a quotient X (cid:48) ( θ (cid:48) ) . ThenTheorem 4.8 asserts that
LGQG , ( Z (cid:48) ( θ ) , β ) is proper. Observation 9.2.
The universal curve U (cid:48) β over the distinguished fixed part F (cid:48) β (see Definition6.4) is trivial, with fibers canonically isomorphic to P , , as follows. Recall that F (cid:48) β parametrizesLG-quasimaps ( C, u, κ ) where C has a single marking b with τ ( b ) = ∞ ∈ P . Further, the degree β is concentrated at τ − (0) . The (cid:15) -stability condition implies that τ − (0) is a single point, and u has a basepoint of degree β there. All fibers are canonically identified with P , , so U (cid:48) β → F (cid:48) β is atrivial family. We write (cid:36) for the projection U (cid:48) β → P , . Observation 9.3.
For each summand L ρ of E , at least one of H ( C, L ρ ) and H ( C, L ρ ) vanishes,since C ∼ = P , by Observation 9.2. This implies that R • π ∗ E = (cid:76) ρ ∈ R R • π ∗ L ρ is a complex of vectorbundles . A basic property of virtual fundamental classes ( [5] , Proposition 5.6) now states that [ F (cid:48) β ] vir = e (( R π ∗ E ) C ∗ ) . By definition, I θ ( q, (cid:126) ) is the contribution to the equivariant integral (cid:88) β j q β − β ( θ, γ j (cid:104) γ j ev ∗ [ ∞ ] (cid:105) ,θ,Gr ,β coming from the loci F β of Definition 6.5. By the projection formula, this is the contribution fromthe loci F (cid:48) β to: (cid:88) β,j q β − β ( θ, ι ∗ ( ˜ γ j ) (cid:90) [LGQG , ( X ( θ ) ,β )] vir ev ∗ ( ˜ γ j ⊗ [ ∞ ]) , (24)with ι ∗ ( ˜ γ j ) = γ j , ι ∗ ( ˜ γ j ) = γ j , and (cid:104) Z ∪ ˜ γ j , ˜ γ j (cid:48) (cid:105) = δ j (cid:48) j . We may choose isomorphisms of L x , L y , L z , and L a with the line bundles O P , ( β x ) , O P , ( β y ) , O P , ( β z ) , O P , ( β a ), and write σ as a tuple of sections( σ x ( s, t ) , σ x ( s, t ) , σ x ( s, t ) , σ y ( s, t ) , . . . , σ p y ( s, t ) , σ p z ( s, t )) , where the entries are homogeneous polynomials in s and t of the appropriate degrees, and thedegrees of s and t are 3 and 1, respectively. The fact that σ is C ∗ -fixed implies that σ is of the form σ = ( x s β x − β a , x s β x , x s β x , y s β y − β a , . . . , p y s − β y − , p z s − β z − ) . (25)In particular, this shows Proposition 9.4.
Fix β so that ( β, is θ -effective. The map ev : F (cid:48) β → IX ( θ ) is an embedding. efinition 9.5. On P , , a line bundle L is determined up to isomorphism by its degree β ;the fractional part (cid:104) β (cid:105) determines the multiplicity mult ∞ ( L ) at the orbifold point. In turn,mult ∞ ( L ) determines a component of IX ( θ ), which we denote X ( θ ) (cid:104) β (cid:105) , such that ev factorsthrough X ( θ ) (cid:104) β (cid:105) (cid:44) → IX ( θ ) . We denote the fundamental class of this sector by 1 (cid:104) β (cid:105) . Lemma 9.6.
The image of ev : F (cid:48) β (cid:44) → X ( θ ) (cid:104) β (cid:105) is the substack of X R ( θ ) (cid:104) β (cid:105) := X ( θ ) (cid:104) β (cid:105) ∩ X R ( θ ) cutout by the vanishing of x , if x is a superscript variable and β x − β a ∈ Z < . (Similarly cut out bythe vanishing of y and z .)Proof. An entry of σ in (25) is necessarily zero either of the following holds:(1) The corresponding line bundle L ρ has degree β ρ (cid:54)∈ Z . In this case s β ρ does not make sense.(2) L ρ has degree β ρ ∈ Z < . In this case L ρ has no nonzero global sections.The first case imposes no restriction on X ( θ ) (cid:104) β (cid:105) . In other words, if β ρ (cid:54)∈ Z , then the the correspondingcoordinate vanishes on X ( θ ) (cid:104) β (cid:105) .The second case does impose restrictions. First, if x is a superscript variable, we must have σ p x = 0 . (In fact, we already observed that u factors through X R ( θ ) . ) This shows that Im(ev ) ⊆ X R ( θ ) (cid:104) β (cid:105) .Also, if β x − β a ∈ Z < , then σ x = 0 . (cid:3) Proposition 9.7. If x is a subscript variable, only terms of I θ ( q, (cid:126) ) with β x − β ( θ, ∈ Z < (cid:114) Z are nonzero. If x is a superscript variable, only terms of I θ ( q, (cid:126) ) with β x ∈ Z ≥ are nonzero.Proof. The first claim is immediate from β x − β ( θ, < β x ≥ . If β x (cid:54)∈ Z , by the fact that σ does not vanish at ∞ ∈ P , , we have x = x = 0 in (25). In particular, the sector X R ( θ ) (cid:104) β (cid:105) is supported over thelocus x = x = 0 . Thus for any nonzero term cγ j q β of I θ ( q, (cid:126) ), where c is a scalar and β x (cid:54)∈ Z , wemust have γ j ∈ ι ∗ ( H ∗ ( X R ( θ ) β − )) = 0 . (cid:3) This calculation is adapted from [7].
Proposition 9.8.
The virtual normal bundle to F (cid:48) β inside LGQG , ( X ( θ ) , β ) has C ∗ -equivariantEuler class e C ∗ ( N vir F (cid:48) β ) = ( − (cid:126) ) (cid:81) ρ ∈ R β ρ ≥ (cid:81) ≤ ν ≤(cid:100) β ρ (cid:101)− (ev ∗ D ρ + ( β ρ − ν ) (cid:126) ) (cid:81) ρ ∈ R β ρ < (cid:81) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− (ev ∗ D ρ + ( β ρ − ν ) (cid:126) ) . where the divisors D ρ were defined in Section 3.3.Proof. The factor ( − (cid:126) ) is from moving the marked point on C , and the rest comes from the relativeperfect obstruction theory R • π ∗ E . By Observation 9.3, each summand L ρ of E contributes either( π ∗ L ) mov or − ( R π ∗ L ) mov to the virtual normal bundle, whichever is nonzero, where the superscript‘mov’ denotes the C ∗ -moving invariant subbundle.We see from the form of (25) that for all ρ ∈ R with β ρ ∈ Z ≥ , we have L ρ ∼ = π ∗ ev ∗ L ρ ⊗ (cid:36) ∗ O P , (3 β ρ ) . (26)(Recall that L ρ = u ∗ L ρ and that (cid:36) : U (cid:48) β → P , is the projection.) Claim.
Equation (26) holds, at least up to torsion, for all ρ ∈ R . Proof of Claim.
We check separately for each θ ∈ Θ. For all θ , { a = 0 } ⊆ V uns ( θ ) , so Definition 4.1implies that 3 β a ∈ Z ≥ . Up to torsion, this identifies L a ∼ = π ∗ ev ∗ L (cid:98) t a ⊗ (cid:36) ∗ O P , (3 β a ) . (27)For θ = θ xyza , at least one of β x − β a and β x is a nonnegative integer. By commutativity of tensorproducts and pullbacks, the fact that Equation (26) holds for one of L x and L x ⊗ L ∗ a , together with quation (27), implies that Equation (26) holds for the other. A similar argument works for y and z . It remains to check Equation (26) for ρ = − (cid:98) t x + (cid:98) t R . Since L R ∼ = ω U (cid:48) β /F (cid:48) β , log , the triviality of U (cid:48) β over F (cid:48) β implies L (cid:99) t R ∼ = (cid:36) ∗ O P , ( − . For θ = θ axyz , we must have β − (cid:98) t x + (cid:99) t R = − β x + β (cid:99) t R ∈ Z ≥ . Therefore Equation (26) implies L − (cid:98) t x + (cid:99) t R ∼ = π ∗ ev ∗ L − (cid:98) t x ⊗ (cid:36) ∗ O P , ( − β x + 3 β (cid:98) R )= π ∗ ev ∗ L − (cid:98) t x ⊗ (cid:36) ∗ O P , ( − β x − . Again we have L (cid:99) t R ∼ = (cid:36) ∗ O P , ( − L − (cid:98) t x = L ⊗− x ∼ = π ∗ ev ∗ L − (cid:98) t x ⊗ (cid:36) ∗ O P , ( − β x ) . Thus up to torsion this identifies L x ∼ = π ∗ ev ∗ L (cid:98) t x ⊗ (cid:36) ∗ O P , (3 β x ) . The argument used for θ xyza todescribe L a applies here to to show that Equation (26) holds for L (cid:98) t x − (cid:98) t a . The same argument worksfor the characters (cid:98) t y − (cid:98) t a , (cid:98) t y , (cid:98) t z − (cid:98) t a , and (cid:98) t z . These two arguments together prove the claim for θ xyaz and θ xayz also. (cid:3) Now, by the projection formula, R i π ∗ ( E ) = (cid:77) ρ ∈ R R i π ∗ ( π ∗ ev ∗ L ρ ⊗ (cid:36) ∗ O P , (3 β ρ ))= (cid:77) ρ ∈ R ev ∗ L ρ ⊗ R i π ∗ (cid:36) ∗ O P , (3 β ρ ) . Now R i π ∗ (cid:36) ∗ O P , (3 β ρ ) is trivial with fiber H i ( P , , O P , (3 β ρ )), i.e. R i π ∗ ( E ) = (cid:77) ρ ∈ R ev ∗ L ρ ⊗ H i ( P , , O P , (3 β ρ )) . The C ∗ -action on LGQG , ( Z ( θ ) , β ) is induced from an action on P , via the universal map τ : UQG , ( Z ( θ ) , β ) → P , . Restricting to F (cid:48) β shows that the action on R i π ∗ ( E ) is induced by theprojection (cid:36) : U (cid:48) β → P , . Thus the C ∗ -action on each factor ev ∗ L ρ ⊗ H i ( P , , O P , (3 β ρ )) is thenatural action on H i ( P , , O P , (3 β ρ )).As these groups are identified with the tangent and obstruction spaces at the point σ of Equation(25), the natural C ∗ -action on a section s a t b ∈ H i ( P , , O P , (3 β ρ )) has weight b/ . The sections of H i ( P , , O P , (3 β ρ )) are, explicitly, H ( P , , O P , (3 β ρ )) = (cid:40) C { t β ρ , st β ρ − . . . , s (cid:98) β ρ (cid:99) t (cid:104) β ρ (cid:105) } β ρ ≥ β ρ < H ( P , , O P , (3 β ρ )) = (cid:40) β ρ ≥ − C { s − t β ρ +1) , s − t β ρ +2) . . . , s (cid:98) β ρ (cid:99) +1 t (cid:104) β ρ (cid:105)− } β ρ < − . ecalling the notation D ρ of Section 3.3, we have e C ∗ ( R π ∗ ( E )) = (cid:89) ρ ∈ R β ρ ≥ (cid:89) ≤ ν ≤(cid:98) β ρ (cid:99) ν ∈ Z (ev ∗ D ρ + ( β ρ − ν ) (cid:126) )(28) e C ∗ ( R π ∗ ( E )) = (cid:89) ρ ∈ R β ρ < − (cid:89) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− ν ∈ Z (ev ∗ D ρ + ( β ρ − ν ) (cid:126) ) e C ∗ ( N vir F (cid:48) β ) = ( − (cid:126) ) (cid:81) ρ ∈ R β ρ ≥ (cid:81) ≤ ν ≤(cid:100) β ρ (cid:101)− (ev ∗ D ρ + ( β ρ − ν ) (cid:126) ) (cid:81) ρ ∈ R β ρ < − (cid:81) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− (ev ∗ D ρ + ( β ρ − ν ) (cid:126) )(29) (cid:3) Observation 9.9.
The calculation above also shows that the obstruction bundle R π ∗ E has no C ∗ -fixed part, i.e. [ F (cid:48) β ] vir = [ F (cid:48) β ] . Remark . For ρ such that β ρ ∈ Z , the section s (cid:98) β ρ (cid:99) t (cid:104) β ρ (cid:105) = s β ρ is C ∗ -fixed, and thus is not partof the virtual normal bundle. This explains the difference in indexing between (28) and (29). Themissing terms span the tangent space to F (cid:48) β . Remark . For a fixed θ , the conditions of Definition 4.1 (with m = 1) determine the signs of β x ,β y , β z , and β a . This determines the signs of β ρ for all ρ ∈ R except for ρ ∈ { (cid:98) t x − (cid:98) t a , (cid:98) t y − (cid:98) t a , (cid:98) t z − (cid:98) t a } . Specifically, if β x ≥ , the quantity β x − β a changes sign depending on whether β x ≥ β a or β x < β a . (Similarly for y and z .) Therefore, in view of Proposition 9.8, we will have to treat the case0 ≤ β x < β a separately in what follows. (See Lemma 10.4.) Proposition 9.12. ι R ∗ ( I θ ( q, (cid:126) )) = (cid:88) β q β − β ( θ, (cid:81) ρ ∈ R β ρ < − (cid:81) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− ( D ρ + ( β ρ − ν ) (cid:126) ) (cid:81) ρ ∈ R β ρ ≥ (cid:81) ≤ ν ≤(cid:100) β ρ (cid:101)− ( D ρ + ( β ρ − ν ) (cid:126) ) A x A y A z (cid:104)− β (cid:105) , where ι R ∗ is the embedding Z ( θ ) (cid:44) → X rig R ( θ ) , A x = (cid:40) D x − D a ≤ β x < β a , β x − β a ∈ Z otherwise , and similarly for A y and A z .Proof. Write e C ∗ ( N vir F (cid:48) β ) = ( − (cid:126) ) ev ∗ α from (29). The projection formula gives ι ∗ e C ∗ ( N vir F (cid:48) β ) ∩ [ F (cid:48) β ] = 1 − α (cid:126) ∩ (ev ) ∗ [ F (cid:48) β ] ∈ H ∗ ( IX ( θ )) , o we have Z (cid:88) β,j q β − β ( θ, ˜ γ j (cid:90) [ F (cid:48) β ] vir ev ∗ ( ˜ γ j ⊗ [ ∞ ]) e C ∗ ( N vir F (cid:48) β ) = Z (cid:88) β,j q β − β ( θ, ˜ γ j (cid:90) (ev ) ∗ [ F (cid:48) β ] ( − (cid:126) ) ˜ γ j − α (cid:126) = (cid:88) β,j q β − β ( θ, ( Z ∪ ˜ γ j ) (cid:90) [ X rig R ( θ )] ˜ γ j α ∪ ( A x A y A z (cid:104) β (cid:105) )= (cid:88) β q β − β ( θ, υ ∗ (cid:18) A x A y A z (cid:104) β (cid:105) α (cid:19) = (cid:88) β q β − β ( θ, A x A y A z (cid:104)− β (cid:105) α . The second equality follows from Lemma 9.6 and the last equality follows from the fact that α and A x A y A z are classes on the untwisted sector of IX ( θ ) . (cid:3) Definition 9.13.
Write ι ∗ I θ ( q, (cid:126) ) = (cid:88) β q β I θβ ( (cid:126) ) . The small Givental I θ -function I θ, Giv ( q, (cid:126) ) is defined to be I θ, Giv ( q, (cid:126) ) = (cid:88) β q β + (cid:126) ( H x ,H y ,H z ,H a ) ( − β x +3 β y +3 β z I θβ ( (cid:126) ) . (30)By Section 3.3 we have H a = 0, and H x = 0 if x is a subscript variable. (Similarly for y and z .)In particular, Proposition 9.12 gives I θ, Giv ( q, (cid:126) ) = (cid:88) β q β x + H x / (cid:126) x q β y + H y / (cid:126) y q β z + H z / (cid:126) z q β a a ( − β x +3 β y +3 β z · (cid:81) ρ ∈ R β ρ < − (cid:81) (cid:98) β ρ (cid:99) +1 ≤ ν ≤− ( D ρ + ( β ρ − ν ) (cid:126) ) (cid:81) ρ ∈ R β ρ ≥ (cid:81) ≤ ν ≤(cid:100) β ρ (cid:101)− ( D ρ + ( β ρ − ν ) (cid:126) ) A x A y A z (cid:104) β (cid:105) . Remark . In [7], there is defined a big I -function I ( t, q, (cid:126) ), also on the Lagrangian cone, where t is restricted to the untwisted sectors of H ( θ ) . We may mimic their construction with no modification. I θ, Giv ( q, (cid:126) ) is obtained from the result by restricting t further to only untwisted degree 2 classes,adding the factor ( − β x +3 β y +3 β z , and finally by identifying q = e t . This identification seems mysterious, especially as the symbol q H x / (cid:126) is otherwise meaningless.In fact, the identification arises naturally from the divisor equation in Gromov-Witten theory, seeRemark 3.1.2 of [10]. It is an important part of the formal analytic continuation of Section 10.2.The choice of sign in (30) comes from [6], in which degree d Gromov-Witten invariants of a quinticthreefold with fields (which play a similar role to the LG-quasimap invariants we use) differ fromthe usual Gromov-Witten invariants of the quintic threefold by the sign ( − d +1 . Relating the generating functions of the different quotients
In this section, we use a similar method to that in [10] to relate the I -functions I θ, Giv ( q, (cid:126) ) forvarious θ. The method involves analytic continuation of the Γ-function, and to make sense of thiswe set up some minor formalism. The Γ -function on C × C .Definition 10.1. For s + ξ ∈ C × C , we define the extended Γ -function ˜Γ : C × C → C [[ ξ ]] by˜Γ( s + ξ ) := ∞ (cid:88) k =0 Γ ( k ) ( s ) k ! ξ k . Intuitively, we take ξ to be an extremely small complex number. Observation 10.2.
It is an easy exercise that ˜Γ satisfies the functional equation ˜Γ( s + ξ ) =( s − ξ )˜Γ( s − ξ ) . This agrees with the intuition that s + ξ is a complex number “near” s . It is easy to extend this to C × C n , and we get a map ˜Γ to C [[ ξ , . . . , ξ n ]] . In the next section, weuse the functional equation to rewrite I θ, Giv ( t, q, (cid:126) ) in terms of ˜Γ. This will allow us to carry out(formal) analytic continuation. Notation . Hereafter we drop the tilde from ˜Γ.10.2.
Analytic continuation.
First, we note that the seeming inconsistency in Remark 9.11 canbe conveniently ignored, as follows.
Lemma 10.4.
For β x ≥ , the factor in I θ, Giv ( t, q, (cid:126) ) corresponding to ρ x = (cid:98) t x − (cid:98) t a is equal to (cid:126) −(cid:100) β x − β a (cid:101) Γ(ev ∗ D ρ / (cid:126) + β x − β a − (cid:100) β x − β a (cid:101) + 1)Γ(1 + ev ∗ D ρ / (cid:126) + β x − β a ) . In particular, this holds whether β x < β a or β x ≥ β a . Proof. If β x ≥ β a , we have A x = 1 , and the corresponding factor in I θ, Giv ( t, q, (cid:126) ) is by definition A x (cid:81) ≤ ν ≤(cid:100) β x − β a (cid:101)− (ev ∗ D ρ + ( β x − β a − ν ) (cid:126) ) = (cid:126) −(cid:100) β x − β a (cid:101) (cid:81) ≤ ν ≤(cid:100) β x − β a (cid:101)− (ev ∗ D ρ / (cid:126) + β x − β a − ν ) . Formally using (cid:81) ≤ ν ≤ k ( α − ν ) = Γ(1+ α )Γ( α − k ) , we can write this as (cid:126) −(cid:100) β x − β a (cid:101) Γ(1+ev ∗ D ρ / (cid:126) + β x − β a )Γ(ev ∗ D ρ / (cid:126) + β x − β a −(cid:100) β x − β a (cid:101) +1) = (cid:126) −(cid:100) β x − β a (cid:101) Γ(ev ∗ D ρ / (cid:126) + β x − β a − (cid:100) β x − β a (cid:101) + 1)Γ(1 + ev ∗ D ρ / (cid:126) + β x − β a ) . Meanwhile, if β x < β a , we have A x = D ρ , and the corresponding factor in I θ, Giv ( t, q, (cid:126) ) is (cid:89) (cid:98) β x − β a (cid:99) +1 ≤ ν ≤− (ev ∗ D ρ + ( β x − β a − ν ) (cid:126) ) A x = (cid:89) (cid:100) β x − β a (cid:101)≤ ν ≤− (ev ∗ D ρ + ( β x − β a − ν ) (cid:126) )= (cid:126) −(cid:100) β x − β a (cid:101) Γ(1 + ev ∗ D ρ / (cid:126) + β x − β a − (cid:100) β x − β a (cid:101) )Γ(ev ∗ D ρ / (cid:126) + β x − β a + 1) . (cid:3) ecause of this observation, we no longer need to treat the case 0 ≤ β x < β a separately (andsimilarly for y and z ). We now introduce notation making use of this. Let: I x ( β ) = q β x + H x / (cid:126) x ( − β x (cid:16) Γ(1 − H x / (cid:126) − − H x / (cid:126) − β x ) (cid:17)(cid:16) Γ(1+ H x / (cid:126) + β x − β a )Γ( H x / (cid:126) −(cid:104) β a (cid:105) +1) (cid:17) (cid:16) Γ(1+ H x / (cid:126) + β x )Γ( H x / (cid:126) +1) (cid:17) I x ( β ) = q β x x ( − β x (cid:16) Γ( (cid:104) β x − β a (cid:105) )Γ( β x − β a +1) (cid:17) (cid:16) Γ( (cid:104) β x (cid:105) )Γ( β x +1) (cid:17) (cid:16) Γ( − β x )Γ(1) (cid:17) . We similarly define I y ( β ) , I y ( β ) , I z ( β ) , I z ( β ) . Finally we let I a ( β ) = q β a a (cid:18) β a ) (cid:19) . It is now straightforward to check (if one is very careful with indices of products) that: I xyza ( t, q, (cid:126) ) = (cid:88) β x ,β y ,β z ∈ Z ≥ β a ∈ Z ≥ I x ( β ) I y ( β ) I z ( β ) I a ( β )1 (cid:104) β (cid:105) I xyaz ( t, q, (cid:126) ) = (cid:88) β x ,β y ∈ Z ≥ β a ∈ Z ≥ β z ∈ Z < (cid:114)Z < I x ( β ) I y ( β ) I z ( β ) I a ( β )1 (cid:104) β (cid:105) I xayz ( t, q, (cid:126) ) = (cid:88) β x ∈ Z ≥ β a ∈ Z ≥ β y ,β z ∈ Z < (cid:114)Z < I x ( β ) I y ( β ) I z ( β ) I a ( β )1 (cid:104) β (cid:105) I axyz ( t, q, (cid:126) ) = (cid:88) β a ∈ Z ≥ β x ,β y ,β z ∈ Z < (cid:114)Z < I x ( β ) I y ( β ) I z ( β ) I a ( β )1 (cid:104) β (cid:105) Essentially all we have done is to rewrite these functions formally using the identity (cid:89) ≤ ν ≤ k ( α − ν ) = Γ(1 + α )Γ( α − k ) , and using Lemma 10.4 in exceptional cases. Lemma 10.5.
Fix β y , β z , and β a . Then (cid:88) β x ∈ Z ≥ I x ( β )1 (cid:104) β (cid:105) analytically continues to (cid:88) β x = Z < β x (cid:54)∈ Z β x − β a (cid:54)∈ Z − πi e πi ( β x − H x / (cid:126) ) −
1) Γ( H x / (cid:126) − (cid:104) β a (cid:105) + 1)Γ( H x / (cid:126) + 1) Γ( (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) Γ(3 H x / (cid:126) + 1) I x ( β )1 (cid:104) β (cid:105) roof. We have (cid:88) β x ∈ Z ≥ I x ( β )1 (cid:104) β (cid:105) = (cid:88) β x ∈ Z ≥ q β x + H x / (cid:126) x ( − β x (cid:16) Γ(1 − H x / (cid:126) − − H x / (cid:126) − β x ) (cid:17)(cid:16) Γ(1+ H x / (cid:126) + β x − β a )Γ( H x / (cid:126) −(cid:104) β a (cid:105) +1) (cid:17) (cid:16) Γ(1+ H x / (cid:126) + β x )Γ( H x / (cid:126) +1) (cid:17) (cid:104) β (cid:105) Using that fact that Γ(1 + α )Γ( α − k ) = ( − k +1 Γ(1 − α + k )Γ( − α ) , which can be easily deduced from the function equation, we can write( − β x (cid:18) Γ(1 − H x / (cid:126) − − H x / (cid:126) − β x ) (cid:19) = Γ(1 + 3 H x / (cid:126) + 3 β x )Γ(3 H x / (cid:126) + 1)Now we have (cid:88) β x ∈ Z ≥ I x ( β )1 (cid:104) β (cid:105) = (cid:88) β x ∈ Z ≥ q β x + H x / (cid:126) x Γ(1 + 3 H x / (cid:126) + 3 β x )Γ(1 + H x / (cid:126) + β x − β a )Γ(1 + H x / (cid:126) + β x ) Φ( (cid:104) β a (cid:105) )1 (cid:104) β (cid:105) , where Φ( (cid:104) β a (cid:105) ) = Γ( H x / (cid:126) − (cid:104) β a (cid:105) + 1)Γ( H x / (cid:126) + 1) Γ(3 H x / (cid:126) + 1) . We rewrite this last expression using residues: (cid:88) β x ≥ πi Res s = β x (cid:32) q s + H x / (cid:126) e πis − H x / (cid:126) + 3 s )Γ(1 + H x / (cid:126) + ( s − β a ))Γ(1 + H x / (cid:126) + s ) (cid:33) Φ( (cid:104) β a (cid:105) )1 (cid:104) β (cid:105) . (31)The expression in the large parentheses may be thought of as having simple poles at s ∈ Z and at s + H x / (cid:126) ∈ Z < . (These loci are unions copies of C inside C . The sum of residues can thereforebe written as the integral (cid:32)(cid:90) i ∞− i ∞ q s + H x / (cid:126) e πis − H x / (cid:126) + 3 s )Γ(1 + H x / (cid:126) + ( s − β a ))Γ(1 + H x / (cid:126) + s ) ds (cid:33) Φ( (cid:104) β a (cid:105) )1 (cid:104) β (cid:105) , along a contour in C such that the poles s ∈ Z < and s + H x / (cid:126) ∈ Z < are on one side, andthe poles s ∈ Z ≥ are on the other side. For simplicity, the contour may be chosen inside a slice C × { H (cid:48) } , i.e. we may work with a contour integral in C .Integrals of this form have been well-studied for a long time, see page 49 of [4]. From there wesee that the integral converges to (31) when q (cid:54)∈ R < and | q | < . When q (cid:54)∈ R < and | q | > , theintegral converges to the sum over the remaining poles .First we consider the poles s ∈ Z < . Consider the expressionΓ(1 + 3 H x / (cid:126) + 3 s )Γ(1 + H x / (cid:126) + s − β a )Γ(1 + H x / (cid:126) + s ) (32)as a function of H x / (cid:126) , treating H x / (cid:126) as a (small) complex number, and s as a fixed negative integer.At H x / (cid:126) = 0 , the expression (32) • has a zero of order 1 if (cid:104) β a (cid:105) (cid:54) = 0 (a pole from the numerator and two zeroes from thedenominator), • has a zero of order 1 if β a ≥ s , and • has a zero of order 2 if (cid:104) β a (cid:105) = 0 and β a < s . Lemma 3.3 of [24] instead puts 3 − as the boundary, and [10] agrees. n the first and second cases, H x restricts to zero on F β , see Lemma 9.6. In the last case, we knowthat H x = 0, since H x is the hyperplane class on the 1-dimensional space E . Together, these saythat the residues at the poles s ∈ Z < vanish when multiplied by 1 (cid:104) β (cid:105) . Thus the analytic continuation is the sum (cid:88) ˜ β x ∈ Hx (cid:126) + Z < πi Res s = ˜ β x (cid:32) q s + H x / (cid:126) e πis − H x / (cid:126) + 3 s )Γ(1 + H x / (cid:126) + ( s − β a ))Γ(1 + H x / (cid:126) + s ) (cid:33) Φ( (cid:104) β a (cid:105) )1 (cid:104) β old (cid:105) = (cid:88) β x ∈ Z < πi Res s = β x (cid:18) q s e πi ( s − H x / (cid:126) ) − s )Γ(1 + ( s − β a ))Γ(1 + s ) (cid:19) Φ( (cid:104) β a (cid:105) )1 (cid:104) β old (cid:105) . (33)Here e πi ( s − H x / (cid:126) ) should be interpreted as in Section 10.1, via its expansion at H x / (cid:126) = 0 . Remark . We write β old rather than β to emphasize that it has not changed and in particular isindependent of (cid:104) β x (cid:105) (as it has been all along — β x has been an integer). Later in this section wewill use 1 (cid:104) β (cid:105) to refer to an element of H ( θ (cid:48) ), where θ (cid:48) is obtained by changing x from a superscriptvariable to a subscript variable.What remains is to calculate the residues. When β x ∈ Z or β x − β a ∈ Z , the residue in (33)vanishes because the simple pole in Γ(1 + 3 s ) is canceled by the poles in Γ(1 + ( s − β a )) , or Γ(1 + s ).The residue of Γ(1 + 3 s ) at β x ∈ Z < is ( − β x +1 − β x ) . Thus we rewrite (33) as (cid:88) β x ∈ Z < β x (cid:54)∈ Z β x − β a (cid:54)∈ Z πi q β x e πi ( β x − H x / (cid:126) ) − (cid:16) ( − βx +1 − β x ) (cid:17) Γ(1 + ( β x − β a ))Γ(1 + β x ) Φ( (cid:104) β a (cid:105) )1 (cid:104) β old (cid:105) . Rearranging slightly gives (cid:88) β x ∈ Z < β x (cid:54)∈ Z β x − β a (cid:54)∈ Z πiq β x ( − β x +1 e πi ( β x − H x / (cid:126) ) − Γ( (cid:104) β x − β a (cid:105) )Γ( β x − β a +1) (cid:16) Γ( (cid:104) β x (cid:105) )Γ( β x +1) (cid:17) (cid:16) Γ( − β x )Γ(1) (cid:17) Γ( (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) Φ( (cid:104) β a (cid:105) )1 (cid:104) β old (cid:105) (34) = (cid:88) β x ∈ Z < β x (cid:54)∈ Z β x − β a (cid:54)∈ Z − πi e πi ( β x − H x / (cid:126) ) −
1) Φ( (cid:104) β a (cid:105) )Γ( (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) I x ( β )1 (cid:104) β old (cid:105) . (cid:3) We now use Section 10.1 to expand the factors. That is, again using H x = 0, we have1 e πi ( β x − H x / (cid:126) ) − e πiβ x − πie πiβ x ( e πiβ x − H x / (cid:126) and Φ( (cid:104) β a (cid:105) ) = Γ(1 − (cid:104) β a (cid:105) ) + Γ(1 − (cid:104) β a (cid:105) )( h −(cid:104) β a (cid:105) ) H x / (cid:126) , here h = 0, h − / = π √ − , and h − / = − π √ − . Finally we write (34) as: (cid:88) β x ∈ Z < β x (cid:54)∈ Z β x − β a (cid:54)∈ Z − πi (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) ·· (cid:18) Γ(1 − (cid:104) β a (cid:105) ) e πiβ x − (cid:18) πie πiβ x Γ(1 − (cid:104) β a (cid:105) )( e πiβ x − + Γ(1 − (cid:104) β a (cid:105) )( h −(cid:104) β a (cid:105) ) e πiβ x − (cid:19) H x (cid:126) (cid:19) I x ( β )1 (cid:104) β old (cid:105) . What remains is to make the identification in Section 5.1. Namely, write 1 (cid:104) β old (cid:105) = (1 ⊗ α y ⊗ γ z ) g ∈H ( θ ) , and define 1 (cid:104) β old (cid:105) (cid:55)→ ζ, (cid:104) β old (cid:105) : = (1 ζ ⊗ α y ⊗ γ z ) g ∈ H ( θ (cid:48) ) H x (cid:104) β old (cid:105) (cid:55)→ ζ , (cid:104) β old (cid:105) : = (1 ζ ⊗ α y ⊗ γ z ) g ∈ H ( θ (cid:48) ) . Thus the coefficients − πi (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) · Γ(1 − (cid:104) β a (cid:105) ) e πiβ x − − πi (cid:104) β x − β a (cid:105) )Γ( (cid:104) β x (cid:105) ) · (cid:18) πie πiβ x Γ(1 − (cid:104) β a (cid:105) )( e πiβ x − + Γ(1 − (cid:104) β a (cid:105) )( h −(cid:104) β a (cid:105) ) e πiβ x − (cid:19) define an isomorphism H ( θ ) → H ( θ (cid:48) ). We have proved: Theorem 10.7 (LG/CY correspondence) . This isomorphism identifies the analytically continued I -function I θ, Giv ( q, (cid:126) ) with I θ (cid:48) , Giv ( q, (cid:126) ) . Notation Table
Notation Description1 θ Generator of H ( θ )1 g The fundamental class of g twisted sectors of X (cid:104) β (cid:105) Fundamental class of certain twisted sectors, Section 9 a Coordinates on Vb i Marked point on
CBG
Stack of principal G -bundles, i.e. BG = [Spec C /G ] B , B ∞ Sets of marked points of G-graph quasimap over 0 , ∞ ∈ P β Shorthand for ( β x , β y , β z , β a ) β ( P ) Degree of the basepoint of σ at P , Section 4 β , β ∞ Degrees of LG-graph quasimap supported over 0 , ∞ ∈ P β ( θ, m ) Extremal degree of m -marked LG-quasimaps to X ( θ ) β ρ Degree of L ρ β x , β y , β z , β a , β R Degrees of L x , L y , L z , L a , L R • , ˇ • Points of (cid:98) C mapping to 0 , ∞ ∈ P C m -marked genus zero twisted curve C , C ∞ Components of a graph quasimap over 0 , ∞ ∈ P U (cid:48) β Universal curve over F (cid:48) β C ∗ R Group acting on V C ρ The representation associated to a character ρ Crit( W ) Critical locus of W in V (cid:98) C Parametrized component of a graph quasimap P The order of a point P on CD ρ Toric divisor on X ( θ )ev i Evaluation maps to Z ( θ ) or X ( θ ) E Elliptic curve in P E P × ( G × C ∗ R ) V(cid:15)
Stability parameter, in Q > F β , F (cid:48) β Special components of C ∗ -fixed LG-graph quasimaps to Z ( θ ) , X ( θ ) F B ∞ ,β ∞ B ,β C ∗ -fixed LG-graph quasimaps inducing partitions B (cid:116) B ∞ and β + β ∞ G ( C ∗ ) { γ j } , { γ j } Basis and dual basis for H ( θ ) H Class [ L ] = [ L (cid:48) ] ∈ H ( P /µ ) H x , H y , H z Divisor classes on [ E /µ ] H ( θ ) Compact type state space associated to θ (cid:126) Generator of C ∗ -equivariant cohomology ring of a point I X (Nonrigidified) inertia stack of X I X Rigidified inertia stack of X IX nar , IX nar Narrow components of
IX, IXI θ ( q, (cid:126) ) I -function ι Embedding
Z (cid:44) → X or Z ( θ ) (cid:44) → X ( θ ) κ Isomorphism L R → ω C, log (cid:96) σ ( P ) Length of σ at PL , L (cid:48) Certain lines in [ P /µ ] L ρ Line bundle on X ( θ ) (resp. [ X ( θ ) / C ∗ R ]) corresponding to character ρ of G (resp. G × C ∗ R )LGQ (cid:15) ,m ( X ( θ ) , β ) , LGQ (cid:15) ,m ( Z ( θ ) , β ) Stack of (cid:15) -stable genus zero m -marked LG-quasimaps to X ( θ ) (resp, Z ( θ )) of degree β LGQ (cid:15) ,m ( X ( θ ) , β ) , LGQ (cid:15) ,m ( Z ( θ ) , β ) Stack of LG-graph quasimaps to X ( θ ) (resp. Z ( θ )) L ρ u ∗ ( L ρ ) L x , L y , L z , L a , L R Line bundles on C built from P mult P ( L ) The multiplicity (monodromy) of L at Pm Number of marked points on
Cµ, ν T -fixed points of IX ( θ ) µ d The group of d th roots of unity in C ∗ N vir Virtual normal bundle p x , p y , p z Coordinates on VP Class [ P ] = [ P (cid:48) ] ∈ H ( P /µ ) P , P (cid:48) Certain points in [ P /µ ] P , P with an order 3 orbifold point at [ ∞ ] P Principal G × C ∗ R -bundle π Map from universal curve to moduli stack q, q x , q y , q z , q a Formal parameters keeping track of β, β x , β y , β z , β a R { ρ x , . . . , ρ p z } ρ x , . . . , ρ p z Characters of G × C ∗ R , which define V as a direct sum s Complexification of β x s, t Coordinates on P , σ Section of E σ x , . . . , σ p z Components of σ , sections of L ρ for ρ ∈ R t x , t y , t z , t a ) Element of G (cid:98) t x , (cid:98) t y , (cid:98) t z , (cid:98) t a , (cid:98) t R Characters ( t x , t y , t z , t a , t R ) (cid:55)→ t x , etc., of G × C ∗ R t Coordinates of cohomology ring t R Element of C ∗ R T ( C ∗ ) τ Parametrization map C → P θ, θ xyza , . . . , θ axyz GIT characters of
Gθ, θ xyza , . . . , θ axyz
Lifts of θ, θ xyza , . . . , θ axyz to G × C ∗ R Θ { θ xyza , θ xyaz , θ xayz , θ axyz } u Map C → [ V / ( G × C ∗ R )] V C V ss ( θ ) θ -semistable locus of VV uns ( θ ) θ -unstable locus of V [ V // θ G ] The GIT stack quotient [ V ss ( θ ) /G ] w ( µ, ν ) Tangent weight at µ along curve from µ to νW Function p x ( ax + x + x ) + p y ( ay + y + y ) + p z ( az + z + z ) ω C, log Log canonical bundle of Cx , x , x Coordinates on VX [ V /G ] X ( θ ) [ V // θ G ] X ( θ ) (cid:104) β (cid:105) Component of IX ( θ ), Section 9 X R ( θ ) Points P ∈ [ X ( θ ) / C ∗ R ] with C ∗ R ⊆ G P y , y , y Coordinates on Vz , z , z Coordinates on VZ [Crit( W ) /G ] Z ( θ ) [Crit( W ) ∩ V ss ( θ ) /G ] ζ e πi/ ∈ µ References [1] Dan Abramovich, Tom Graber, and Angelo Vistoli. Gromov-Witten theory of Deligne-Mumford stacks.
AmericanJournal of Mathematics , 130(5):1337–1398, 2008.[2] Dan Abramovich and Angelo Vistoli. Compactifying the space of stable maps.
Journal of the American Mathe-matical Society , 15:27–75, 2002.[3] Pedro Acosta. Asymptotic expansion and the LG/(Fano, general type) correspondence.
ArXiv e-prints , November2014. arXiv:1411.4162 .[4] Harry Bateman and staff of the Bateman Manuscript Project.
Higher Transcendental Functions . McGraw-HillBook Company, Inc., 1953.[5] Kai Behrend and Barbara Fantechi. The intrinsic normal cone.
Inventiones mathematicae , 128(1):45–88, 3 1997.[6] Huai-Liang Chang and Jun Li. Gromov-Witten invariants of stable maps with fields.
International MathematicsResearch Notices , 2012(18):4163–4217, 2012.[7] Daewoong Cheong, Ionut¸ Ciocan-Fontanine, and Bumsig Kim. Orbifold quasimap theory.
Mathematische Annalen ,363(3):777–816, 2015.[8] Alessandro Chiodo, Hiroshi Iritani, and Yongbin Ruan. Landau-Ginzburg/Calabi-Yau correspondence, globalmirror symmetry and Orlov equivalence.
Publications math´ematiques de l’IH ´ES , 119(1):127–216, 2013.[9] Alessandro Chiodo and Jan Nagel. The hybrid Landau-Ginzburg models of Calabi-Yau complete intersections.
ArXiv e-prints , June 2015. arXiv:1506.02989 .[10] Alessandro Chiodo and Yongbin Ruan. Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds viasymplectic transformations.
Inventiones mathematicae , 182(1):117–165, 2010.[11] Alessandro Chiodo and Dmitri Zvonkine. Twisted r -spin potential and Givental’s quantization. Advances inTheoretical and Mathematical Physics , 13(5):1335–1369, 10 2009.[12] Ionut¸ Ciocan-Fontanine and Bumsig Kim. Higher genus quasimap wall-crossing for semi-positive targets.
Journalof the European Mathematical Society . To appear.
13] Ionut¸ Ciocan-Fontanine and Bumsig Kim. Wall-crossing in genus zero quasimap theory and mirror maps.
AlgebraicGeometry , 1(4):400–448, 2014.[14] Emily Clader. Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X , and X , , , . ArXiv e-prints , January 2013. arXiv:1301.5530 .[15] Emily Clader and Dustin Ross. Sigma models and phase transitions for complete intersections.
ArXiv e-prints ,November 2015. arXiv:1511.02027 .[16] Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng. A mirror theorem for toric stacks.
CompositioMathematica , 151:1878–1912, 10 2015.[17] Tom Coates, Hiroshi Iritani, and Yunfeng Jiang. The crepant transformation conjecture for toric completeintersections.
ArXiv e-prints , September 2014. arXiv:1410.0024 .[18] Lance Dixon, Jeffrey Harvey, Cumrun Vafa, and Edward Witten. Strings on orbifolds (II).
Nuclear Physics B ,274:285–314, 9 1986.[19] Igor Dolgachev and Yi Hu. Variation of geometric invariant theory quotients.
Publications math´ematiques del’IH ´ES , 87:5–51, 1998.[20] Huijun Fan, Tyler Jarvis, and Yongbin Ruan. The Witten equation, mirror symmetry, and quantum singularitytheory.
Annals of Mathematics , 178(1):1–106, 2013.[21] Huijun Fan, Tyler Jarvis, and Yongbin Ruan. A mathematical theory of the gauged linear sigma model.
ArXive-prints , June 2015. arXiv:1506.02109 .[22] Maximilian Fischer, Michael Ratz, Jes´us Torrado, and Patrick K.S. Vaudrevange. Classification of symmetrictoroidal orbifolds.
Journal of High Energy Physics , 2013(1):1–53, 2013.[23] Alexander Givental. A mirror theorem for toric complete intersections. In Masaki Kashiwara, Atsushi Matsuo,Kyoji Saito, and Ikuo Satake, editors,
Topological Field Theory, Primitive Forms and Related Topics , pages141–175. Birkh¨auser Boston, Boston, MA, 1998.[24] Richard Paul Horja. Hypergeometric functions and mirror symmetry in toric varieties.
ArXiv Mathematicse-prints , December 1999. arXiv:math/9912109 .[25] Young-Hoon Kiem and Jun Li. Localizing virtual cycles by cosections.
Journal of the American MathematicalSociety , 26:1025–1050, 2013.[26] Y.-P. Lee, Nathan Priddis, and Mark Shoemaker. A proof of the Landau-Ginzburg/Calabi-Yau correspondencevia the crepant transformation conjecture.
ArXiv e-prints , October 2014. arXiv:1410.5503 .[27] Martin Olsson. On (log) twisted curves.
Compositio Mathematica , 143(2):476–494, feb 2007.[28] Nathan Priddis and Mark Shoemaker. A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic.
ArXiv e-prints , September 2013. arXiv:1309.6262 .[29] Dusty Ross and Yongbin Ruan. Wall-crossing in genus zero Landau-Ginzburg theory.
Journal f¨ur die reine undangewandte Mathematik . To appear.[30] Yongbin Ruan. The Witten equation and the geometry of the Landau-Ginzburg model. In Jonathan Block,Jacques Distler, Ron Donagi, and Eric Sharpe, editors,
String-Math 2011 , pages 209–240. American MathematicalSociety, Providence, Rhode Island, 2011.[31] Andrew Schaug. The Gromov-Witten theory of Borcea-Voisin orbifolds and its analytic continuations.
ArXive-prints , June 2015. arXiv:1506.07226 .[32] Michael Thaddeus. Geometric invariant theory and flips.
Journal of the American Mathematical Society , 9:691–723,1996.[33] Hsian-Hua Tseng. Orbifold quantum Riemann-Roch, Lefschetz and Serre.
Geometry and Topology , 14:1–81, 2010.[34] John Voight and David Zureick-Brown. The canonical ring of a stacky curve.
ArXiv e-prints , January 2015. arXiv:1501.04657 .[35] Edward Witten. Phases of N = 2 theories in two dimensions. Nuclear Physics B , 403:259–222, 1993.
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
E-mail address : [email protected]@umich.edu