Big Quantum cohomology of orbifold spheres
BBIG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES
LINO AMORIM, CHEOL-HYUN CHO, HANSOL HONG, AND SIU-CHEONG LAU
Abstract.
We construct a Kodaira-Spencer map from the big quantum cohomology of asphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg poten-tial function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangianand we show that Kodaira-Spencer map is a ring isomorphism.
Contents
1. Introduction 2Acknowledgments 52. Bulk deformed Floer theory of Seidel Lagrangian in P a,b,c P a,b,c and its orbifold quantum cohomology 62.2. Immersed Lagrangian Floer theory 72.3. Orbi-discs and Lagrangian Floer theory for orbifolds 82.4. Bulk deformed Fukaya algebra 103. Weakly unobstructedness for bulk-deformed Fukaya algebra 114. Bulk-deformed potential function and change of variables 144.1. Gauss–Bonnet theorem and convergence 144.2. Fukaya algebra of L KS τ and well-definedness 276.2. Ring homomorphism 297. KS τ is an isomorphism 337.1. Surjectivity 337.2. Jacobian ring of the leading order potential 357.3. Deforming Jac( W τ ) 367.4. Injectivity 388. Calculations 388.1. Euler vector field 388.2. Versality of the potential 39 a r X i v : . [ m a t h . S G ] F e b AMORIM, CHO, HONG, AND LAU KS τ for P , , without bulk-parameters 44Appendix A. Proof of Proposition 6.5 47Appendix B. Proof of Theorem 7.3 (2) 50Appendix C. Proof of Proposition 8.9 54References 571. Introduction
Orbifold projective lines P a,b,c are two-dimensional spheres with three orbifold singularpoints as drawn in Figure 1. They provide a simple yet very interesting class of geometries.Despite low dimensionality, their orbifold Gromov-Witten theory is surprisingly rich. Satake-Takahashi [ST11] computed the Gromov-Witten invariants and Frobenius structures for ellip-tic P a,b,c (where 1 /a + 1 /b + 1 /c = 1), which involves many interesting number theoretic powerseries. Rossi [Ros10] obtained analogous results for spherical P a,b,c (where 1 /a +1 /b +1 /c > P a,b,c (where 1 /a + 1 /b + 1 /c <
1) is isomorphicto the one from their associated affine cusp polynomials. In this paper, we provide a geomet-ric approach to study closed-string mirror symmetry for X = P a,b,c in all three cases, withhelp of Lagrangian Floer theory. Namely, we will construct a Kodaira-Spencer map fromorbifold quantum cohomology of X with bulk deformations to the Jacobian ring of the mirrorpotential function and show that it is an isomorphism.Lagrangian Floer theory has provided a purely mathematical approach to construct andprove mirror symmetry. A typical example is a compact toric manifold, whose mirror canbe nicely constructed from Lagrangian Floer theory. In the Fano case, the second-namedauthor and Yong-Geun Oh [CO06] classified the holomorphic discs bounded by toric fibers andshowed that the LG mirror W can be formulated as the count of these discs. Later Fukaya-Oh-Ohta-Ono [FOOO10, FOOO11, FOOO16b] used Lagrangian deformation theory to constructthe LG mirrors in general. They also constructed the Kodaira-Spencer map (or closed-open map) which produces close-string mirror symmetry for all compact toric manifolds.This provides a mirror construction from the first principle, which has the advantage thatKontsevich’s homological mirror symmetry conjecture [Kon95] can be canonically derived(See [CHL19] for Fano cases).For X = P a,b,c , the Landau-Ginzburg (LG for short) mirrors W were uniformly constructedin [CHL12] based on Lagrangian Floer theory of a certain immersed Lagrangian L , whichwas first used by Seidel [Sei15]. Moreover, homological mirror symmetry for the elliptic andhyperbolic cases was derived by a family version of a Yoneda functor naturally coming withthe construction. In the hyperbolic case, the LG mirror is an infinite series in variables x, y, z .[CHKL17] found an inductive algorithm to compute the explicit expressions in all cases. Inthis article, we consider a bulk-deformed version of such LG mirrors. The deformed potentialshave the same leading order terms as the ones in [CHKL17]. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 3
In this approach to constructing the mirror, it is crucial to find a large space of solutions tothe weak Maurer-Cartan equation. For the immersed Lagrangian L we show (see Proposition3.1) that any linear combination of the odd-degree immersed points gives a solution of theMaurer-Cartan equation. This extends the result in [CHL12] to the case of bulk deformationsby orbi-sectors. The key ingredient is an anti-symplectic involution on P a,b,c , which makesholomorphic polygons appearing in pairs and their contributions to the even-degree immersedsectors cancel.In order to relate the Gromov-Witten invariants of P a,b,c with the Jacobian ring of thebulk-deformed mirror potential, we use the method of Kodaira-Spencer map invented by[FOOO16b], which gives a homomorphism from the quantum cohomology of X to the Jaco-bian ring of the mirror W τ . The following is the main theorem. Theorem 1.1.
Let X = P a,b,c and W τ be its bulk-deformed disc potential by τ ∈ H ∗ ( X, Λ + ) .Let Jac( W τ ) be the completed Jacobian ring over the Novikov field Λ in a certain choice ofcoordinates. Denote the big quantum cohomology of X over Λ with quantum product • τ by QH ∗ orb ( X, τ ) . The Kodaira-Spencer map KS τ : QH ∗ orb ( X, τ ) → Jac( W τ ) is a ring isomorphism. We also show that the map KS τ identifies the Euler vector field on the big quantum cohomol-ogy (see Theorem 8.1 for details) with the Euler vector field on Jac( W τ ), which is the class[ W τ ].The construction of Kodaira-Spencer map [FOOO16b] crucially depends on the existenceof T n -action, hence the definition is still missing in general cases. The above theorem providesthe first class of examples of Kodaira-Spencer map beyond toric manifolds.In fact, there is a crucial difference between our case of P a,b,c and that of toric manifolds.Namely, we need to enlarge the domain of LG potential to make the above theorem hold true.Maurer-Cartan formalism of Lagrangian Floer theory provides a natural set of coordinates˜ x, ˜ y, ˜ z ∈ Λ . Namely, they are the coordinates of the Maurer-Cartan space which are dualto the immersed sectors of L . Given the bulk deformed mirror potential W τ (˜ x, ˜ y, ˜ z ), one candefine the Jacobian ring as in Definition 6.1 as the completed power series ring Λ (cid:28) ˜ x, ˜ y, ˜ z (cid:29) modulo Jacobian ideal of W τ (˜ x, ˜ y, ˜ z ). With this Jacobian ring, KS τ is not an isomorphism ingeneral hence the above theorem fails. In Section 8.3, we give an explicit counter-example.In this paper, we will make the change of variables x = T ˜ x,y = T ˜ y,z = T ˜ z. (1.1)and consider x, y, z ∈ Λ . In terms of old variables, this is equivalent to allowing val (˜ x ) , val (˜ y ) , val (˜ z ) ≥ − . In terms of non-archimedean norm e − val , ˜ x, ˜ y, ˜ z are functions on a disc D (1) of radius1 = e , and x, y, z are functions on a disc D ( e ) which contains D (1). In the above counterexample, critical points of the potential W τ (˜ x, ˜ y, ˜ z ) lie on D ( e ) \ D (1) as shown in Proposition8.6. Thus we need the bigger disc D ( e ) to match the number of critical points with the rankof the quantum cohomology ring. See 8.3 for related discussions. AMORIM, CHO, HONG, AND LAU
However, this necessary enlargement of domain is the main source of complication almostin every steps of the proof of the main theorem. Namely, Lagrangian Floer theory for bound-ing cochains of negative valuation does not work in general and we need to take care ofconvergence issues in each step of the proof.We give another perspective of the above coordinate change. For readers convenience,we first recall the case of toric manifolds briefly. For a compact toric n-fold, which can beunderstood as a compactification of C n , W takes the form z + . . . + z n + ∞ (cid:88) i =1 T A i Z i + h.o.t. where A i > Z i are monomials in z , . . . , z n , and h.o.t. consists of higher-order terms in T . Under the Kodaira-Spencer map, the images of the toric divisors D , . . . , D n , which arecompactifications of the coordinate hyperplanes of C n , are sent to z , . . . , z n , which generate(a suitable completion of ) Λ[ z , . . . , z n ] and hence the Jacobian ring. Thus surjectivity of theKodaira-Spencer map is automatic in this case.On the other hand for P a,b,c , the potential W (with τ = 0) takes the form W (˜ x, ˜ y, ˜ z ) = − T ˜ x ˜ y ˜ z + T a ˜ x a + T b ˜ y b + T c ˜ z c + h.o.t. whereas, in new coordinates, the leading terms of the above become W lead := − T − xyz + x a + y b + z c . The images of the orbifold points [1 /a ] , [1 /b ] , [1 /c ] are T ˜ x, T ˜ y, T ˜ z respectively, but in newcoordinates these orbifold points map to x, y, z which generates Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . This is one ofkey ingredient in proving surjectivity of the KS map. Therefore the coordinate change is alsoquite natural in this perspective as well.Once we establish surjectivity of KS τ , we match the dimension of the Jacobian ring of thebulk-deformed potential with that of QH ∗ orb ( X, τ ) to show that KS τ is injective, where theformer is given as a + b + c −
1. For this, we argue with the deformation invariance of thedimension, as it is relatively easy to analyze the leading order terms. In fact, the rank ofthe Jacobian ring for − T − xyz + x a + y b + z c is already quite nontrivial, as one needs toadditionally take into account the convergence issue when working over Λ. For this reason,the computation for leading order terms is somewhat lengthy which we will see in AppendixB. Then we prove that the leading order terms and the actual potential can be interpolatedby a flat deformation. This involves a delicate induction step together with some nontrivialalgebraic facts.While the necessity of the coordinate change is now clear, it results in the analytic difficultythat we need to insure convergence throughout the construction under this coordinate change,which a priori is not at all obvious. Even though the construction in Floer theory hasautomatic T -adic convergence for bounding cochains in Λ + , this coordinate change has aneffect that our bounding cochains lie in Λ ≥− . Hence we need a better control in areas to haveconvergence. First we will show that in the coordinates x, y, z , every term of W τ has non-negative valuation (Lemma 4.4). Then we use an orbifold version of Gauss-Bonnet theorem(Theorem 4.5) to show that W τ actually converges in T -adic topology. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 5
Theorem 1.2 (Theorem 4.8) . The bulk-deformed potential W τ is a convergent series in newvariables x, y, z as in (1.1) , that is, it is an element of Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . In the orbifold setting, the twisted sectors have fractional degrees. For X = P a,b,c , H < orb ( X )is spanned by the fundamental class X and the twisted sectors [ i/a ] , [ j/b ] , [ k/c ] for 0 < i < a ,0 < j < b , 0 < k < c . The compatibility of KS τ : QH ∗ orb ( X, τ ) → Jac( W τ ) with ring structuresfollows from the standard cobordism argument as in [FOOO16b], but it still requires a carefulanalysis on the associated virtual perturbation scheme in our context. The details will beprovide in 6.2.The main theorem is particularly interesting in the hyperbolic case, which belongs to theclass of general-type manifolds whose mirror symmetry is mostly conjectural. Theorem 1.1together with the result in [CHKL17] provides the first class of manifolds in general-typewhose small quantum cohomology has a presentation which can be explicitly computed.Even in the toric case, W is a highly non-trivial series due to obstructed non-constant spherebubbling with negative Chern number. There is no general algorithm to compute W for toricmanifolds of general type. On the other hand, for hyperbolic P a,b,c with no bulk deformation(that is τ = 0), there is an algorithm to compute the series W τ by [CHKL17], which in turngives an explicit presentation of the small quantum cohomology QH ∗ orb ( X, P a,b,c and so there is no obstruction inthe disc moduli for computing W .)Finally in the last section, we exhibit several interesting properties of the bulk-deformedpotential as well as a few explicit calculations for KS τ . Most importantly, we show thatthe bulk-deformation of the Floer theory of L produces a versal deformation of the mirrorpotential. More specifically, Theorem 1.3 (Theorem 8.2) . Consider P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( P − W lead ) > where W lead = − T − xyz + x a + y b + z c . Then there exist τ (cid:48) ∈ H ∗ orb ( P a,b,c , Λ ) and a coordinatechange ( x (cid:48) , y (cid:48) , z (cid:48) ) such that P ( x (cid:48) , y (cid:48) , z (cid:48) ) = W τ (cid:48) . Note that this is analogous to the versality statement in toric case proven in [FOOO16b,Theorem 2.8.1]. The proof is based on the induction argument on energy, which is similar tothe one used to establish surjectivity of KS τ .The organization of the paper is as follows. In Section 2, we review Floer theory of theLagrangian L in P a,b,c and its bulk-deformation including orbi-sectors. In Section 3, weprove the weakly unobstructedness of L after the bulk-deformation, and in Section 4, westudy the resulting bulk-deformed potential and its convergence after coordinate change.In Section 5, we prove that the bulk-deformed potential changes by an explicit coordinatechange for different choices of cohomology representatives, and hence its well-definednessfollows. Throughout Section 6 and 7, we show that KS τ is a ring homomorphism that issurjective and injective, which proves our main theorem. Finally, we provide some concretecalculations of KS τ , and prove the versality theorem in Section 8. Acknowledgments.
The authors express their gratitude to Kenji Fukaya and Yong-GeunOh for useful discussions on virtual perturbation schemes. C.-H. Cho was supported by theNRF grant funded by the Korea government(MSIT) (No. 2017R1A22B4009488). H. Hong
AMORIM, CHO, HONG, AND LAU is supported by the Yonsei University Research Fund of 2019 (2019-22-0008). S.-C. Lau issupported by Simons Collaboration Grant.2.
Bulk deformed Floer theory of Seidel Lagrangian in P a,b,c In this section, we recall orbifold quantum cohomology and immersed Lagrangian Floertheory mainly to set the notations. In short, we will consider orbifold quantum cohomology byChen-Ruan [CR02] and a de Rham version of immersed Lagrangian Floer theory (defined byAkaho-Joyce [AJ10] and Fukaya[Fuk17]). One can enhance the latter by including orbi-discsfollowing the work of the second author and Poddar [CP14]. This gives bulk deformationsby twisted sectors.2.1. P a,b,c and its orbifold quantum cohomology. Let P a,b,c be an orbifold sphere withthree orbifold points with isotropy groups Z /a , Z /b , Z /c , where a, b, c ≥
2. We take theK¨ahler form ω descended from the universal cover of P a,b,c with constant curvature. For laterconvenience we scale it such that the total area of P a,b,c is 8. The orbifold Euler characteristicis given by χ (cid:0) P a,b,c (cid:1) = 1 a + 1 b + 1 c − . Depending on χ being positive, zero or negative, the universal cover of P a,b,c is S , R or H . We refer to these as the spherical, elliptic and hyperbolic respectively. In all cases, P a,b,c can be constructed as a global quotient of a Riemann surface Σ by a finite group. In thespherical case Σ is a sphere, in the elliptic case Σ is an elliptic curve and in the hyperboliccase Σ is a surface of genus ≥ X , as a vector space, isgiven by the singular cohomology group of the inertia orbifold. In particular, its degree d part (where d ∈ Q ) is given by H dorb ( X ) = (cid:77) g H d − ι ( g ) ( X ( g ) )where the sum is over all twisted sectors g . The degree-shifting ι ( g ) ∈ Q is called the age ofthe twisted sector in literature.For H ∗ orb ( P a,b,c ), we have cohomology classes X , [pt] ∈ H ( P a,b,c , R ), as well as the twistedsectors(2.1) (cid:22) a (cid:23) , . . . , (cid:22) a − a (cid:23) , (cid:22) b (cid:23) , . . . , (cid:22) b − b (cid:23) , (cid:22) c (cid:23) , . . . , (cid:22) c − c (cid:23) where (cid:4) ka (cid:5) has degree ka . Let us denote by H tw ( X ) the span of the twisted sectors.By local computations, the classical part of Chen-Ruan product of (cid:4) ja (cid:5) and (cid:4) ka (cid:5) is (cid:4) j + ka (cid:5) if j + k < a , and is a [pt] if j + k = a and zero otherwise. These are the products from constantorbi-spheres. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 7
There are non-trivial contributions from non-constant orbi-spheres as well for the quantumcohomology QH ∗ orb ( X, τ ). They can be written as follows via the orbifold Poincar´e pairing: (cid:104) X , [pt] (cid:105) P D X = 1 , (cid:104) (cid:22) ja (cid:23) , (cid:22) a − ja (cid:23) (cid:105) P D X = 1 a . Fix τ ∈ H ∗ orb ( X, Λ + ) and for each A, B ∈ H ( X, Λ ) the bulk deformed quantum product A • τ B is defined by (cid:104) A • τ B, C (cid:105)
P D X = ∞ (cid:88) l =0 l ! GW l +3 ( A, B, C, τ, · · · , τ ) . where GW l +3 is the orbifold Gromov-Witten invariant with l + 3 inputs ([CR02]). The abovesum converges over Λ by our choice of τ .2.2. Immersed Lagrangian Floer theory.
Immersed Lagrangian Floer theory was intro-duced by Akaho-Joyce [AJ10] using singular chains, extending the embedded case of Fukaya,Oh, Ohta Ono [FOOO09]. A different version using Morse function (and pearl complex) wasgiven by Seidel [Sei11] and Sheridan [She15, She11]. In our previous work [CHL12], we usedthe definition by Seidel to prove homological mirror symmetry. In this paper, we work withde Rham version of immersed Lagrangian Floer theory (by Fukaya [Fuk17]) since we useKuranishi structures to deal with orbifold quantum cohomology. We refer the readers to theabove references for general definitions.Seidel [Sei11] constructed an immersed circle S (cid:35) P a,b,c with three transversal (double)self-intersections (see Figure 1), and we refer it as the Seidel Lagrangian. We assume that theimage of L is invariant under reflection with respect to the equator (which passes through thethree orbifold points), which is crucial for weakly unobstructedness in the next section. Theimage of L and the equator divide the sphere into eight regions: two triangles and six bigons.We take L such that each of these regions have area 1. The Lagrangian L is equipped witha non-trivial spin structure (this is needed for weakly unobstructedness in Section 3). Thisis given by fixing a point in L (which is not the immersed point) and any holomorphic disccontribution through this point gets a (-1) sign for each A ∞ -operation. Figure 1
AMORIM, CHO, HONG, AND LAU
One associates to the Seidel Lagrangian its Fukaya algebra F ( L ), which is a filtered A ∞ -algebra with the underlying Z -graded vector space F ∗ ( L ) := (cid:32) Ω ∗ ( S ) ⊕ (cid:77) X,Y,Z Λ ⊕ (cid:33) ˆ ⊗ Λ . Here Ω ∗ ( S ) is the classical de Rham cochain algebra of S (the domain of L ) with coefficientsin C . Each of the intersection points gives rise to two generators in F ( L ), one even and oneodd. We denote by X , Y and Z the odd ones and by ¯ X , ¯ Y and ¯ Z the even generators.The A ∞ -operations are defined using the moduli space of pseudo-holomorphic polygons asin Fukaya [Fuk17], to which we refer readers for details. For the case where inputs and theoutput are immersed generators ( X, Y, Z, ¯ X, ¯ Y , ¯ Z ), the corresponding A ∞ -operation is givenby the signed count of rigid immersed polygons in P a,b,c with prescribed (convex) corners. Inthis case, by automatic regularity of (holomorphic) polygons in Riemann surface (see [Sei08,Part II, Section 13]), they are already transversal, and hence it is legitimate to use them forcounting. Also we remark that the interior of a polygon may cover orbifold points of P a,b,c .There exists a manifold cover of P a,b,c , where the lifts of L are embedded Lagrangians, and onemay count polygons in the cover. Orbifold insertions will be considered in bulk deformations,and we explain them in the next subsection.When some of the inputs or the output are differential forms of S (the domain of L ),we follow Fukaya to define A ∞ -operations using pull-back and push-forward of differentialforms over the moduli spaces. In general one needs the technique of a continuous family ofmulti-sections to define push-forwards.2.3. Orbi-discs and Lagrangian Floer theory for orbifolds.
We recall how to incorpo-rate orbi-discs into the story. In our case, the Seidel Lagrangian stays away from the orbifoldpoints of P a,b,c , which can be handled as in the case of toric orbifolds [CP14].Let us first recall the definition of an orbi-disc, adapted for an immersed Lagrangian bound-ary condition. Let T be the index set of inertia components of X , where 0 ∈ T correspondsto the underlying topological space of X . Let R be the index set of the immersed sectors of L , where + ∈ R corresponds to the underlying immersed Lagrangian. Definition 2.1.
Let β ∈ H ( X, L ) be a disc class, γ : { , . . . , k } → R a specification ofimmersed sectors of L and ν : { , . . . , l } → T a specification of twisted sectors of X . Themoduli space M maink +1 ,l ( β ; ν ; γ ) consists of elements of the form (Σ , (cid:126)z + , (cid:126)m, (cid:126)z, u ) such that • (Σ , (cid:126)z + , (cid:126)m ) is a (prestable) bordered orbifold Riemann surface with genus zero, where (cid:126)z + = ( z +1 , . . . , z + l ) ∈ (Σ − ∂ Σ) l is a sequence of interior orbifold marked points whichare not (orbi-)nodes, and (cid:126)m = ( m , . . . , m l ) ∈ N l specifies the multiplicities of theuniformizing chart at these orbifold points. • u : (Σ , ∂ Σ) → ( X, L ) is a holomorphic map on each component, that is, u is a con-tinuous map which is holomorphic in the interior of Σ away from the orbifold points,and around each orbifold point z + i , u can be locally lifted to be a holomorphic mapfrom the uniformizing chart at z + i to a uniformizing chart of X at f ( z + i ) . Moreover, z + i is mapped to the twisted sector X ν ( i ) for i = 1 , . . . , l . • u is good and representable as an orbifold morphism. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 9 • (cid:126)z = ( z , . . . , z k ) ∈ ( ∂ Σ) k +1 is a sequence of boundary marked points obeying the cyclicordering of ∂ Σ . Moreover, z i is mapped to the immersed sector labeled by γ ( i ) for i = 0 , . . . , k . In the case of toric orbifolds, such an orbi-disc with boundary on a Lagrangian torus fiberwas studied and classified. The orbi-disc potential for a toric Calabi-Yau orbifold or a compactsemi-Fano toric orbifold was computed using the mirror map in [CCLT16, CCLT14].To state the dimension formula for the moduli spaces, we use two related notions of theMaslov index. The first one is the desingularized Maslov index µ de (following [CR02]). Givenan orbi-disc with a Lagrangian boundary condition, the pull-back bundle data is an orbi-bundle together with Lagrangian boundary data. This bundle cannot be trivialized due tothe non-trivial orbifold structure. On the other hand there is an associated smooth bundle,called the desingularized bundle, which has the same set of local holomorphic sections. Thelatter property enables us to compute the virtual dimension. The other one is the Chern-WeilMaslov index µ CW . It was shown in [CS16] that(2.2) µ de = µ CW − (cid:88) i ι ( ν ( i ))where ι ( ν ( i )) is the degree shifting number associated to the twisted sector labeled by ν ( i ).Let us also explain how to handle J -holomorphic polygons with transversally intersectingLagrangian boundary conditions. Given two Lagrangian subspaces L i , L i +1 = J L i , there exista positive path of Lagrangian subspaces from L i to L i +1 given by e πJt/ L for t ∈ [0 , µ de . Remark 2.2.
For the definition of the Chern-Weil index, we choose a unitary connection,which asymptotically sends L i to J L i at the puncture along the positive path. Then the relation (2.2) also holds for polygons. We remark that there is an error in [CS16] Proposition 5.6.Namely, the formula (23) in [CS16] holds true for connections which are trivial near thepuncture, but it does not hold for general connections. Rather we have (2.2) with asymptoticconditions given by positive paths.
It is well-known that the Fredholm index of the ∂ operator on discs with smooth Lagrangianboundary condition equals n + µ de . For transversely intersecting Lagrangians, we can glueorientation operators of positive paths (this has index n ) at the punctures to obtain a formula Ind ( ∂ ) + ( k + 1) n = n + µ de (If we had used negative paths instead of positive paths to define the topological Maslovindex, the term ( k + 1) n will disappear. In this sense, it would be more convenient to usenegative paths. We follow the usual convention to use positive paths.)Hence the dimension of the moduli space of J -holomorphic polygons are given by (addingthe effects of l interior and k + 1 boundary marked points and equivalences) Ind ( ∂ ) + 2 l + ( k + 1) − n + µ de − ( k + 1) n + 2 l + k − Figure 2.
Image of [1 /
3] orbi-discs in the quotient space P , , In our case of n = 1 the moduli space M l,k +1 ( β ; ν ; γ ) has virtual dimension µ de + 2 l − µ CW + 2 (cid:88) j (1 − ι ( j )) − l = 0, it is simply given by µ de − Bulk deformed Fukaya algebra.
The Fukaya algebra with bulk deformations bytwisted sectors can be defined as follows. First, we see how to adapt the definition of q operator to our orbifold setting. Let T , . . . , T m denote the twisted sectors in (2.1). For eachmulti-index I = ( i , . . . , i l ) we define the specification of twisted sectors ν I as ν I ( z + j ) = T i j .We denote the corresponding moduli space M maink +1 ,l ( β, ν I , γ ) as M maink +1 ,l ( β, T I , γ ) This spacehas a Z / Z / ev I : M maink +1 ,l ( β, T I , γ ) → k (cid:89) i =1 L ( γ ( i ))as well as ev I where L ( γ ( i )) is the corresponding immersed sector for i (cid:54) = +, and L (+) = L .We can define(2.3) q l,k,β ( T I ; h ⊗ · · · ⊗ h k ) = ( ev I ) ∗ ( ev I ) ∗ ( h × · · · × h k ) q ρl,k ( T I ; h ⊗ . . . ⊗ h k ) = (cid:88) β T β ∩ ω/ π ρ ( ∂β ) q l,k,β ( T I ; h ⊗ · · · ⊗ h k ) . The way to handle the unitary line bundle ρ (on L ) is very standard, and we will omit thesuperscript ρ from now on.Given a cohomology class [ τ ] ∈ H ∗ orb ( X ; Λ ) we pick a representative τ = τ X + τ p + τ tw , where p is a Z -invariant cycle (away from L and the orbi-points) representing [ pt ] ∈ H ( X, Λ ) and τ tw = (cid:80) k τ k T k . We define(2.4) m τk ( h , · · · , h k ) = (cid:88) β exp( τ p ∩ β ) ∞ (cid:88) l =0 T β ∩ ω l ! q l,k,β ( τ ltw ; h , · · · , h k ) IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 11 for k >
0, and m τ = τ · L + (cid:80) β exp( τ p ∩ β ) (cid:80) ∞ l =0 T β ∩ ω l ! q l, ,β ( τ ltw ). Remark 2.3.
Here we are slightly abusing notation. The expression q l,k,β ( τ ltw ; h , · · · , h k ) really stands for (cid:88) I =( i ,...,i l ) τ i · · · τ i l q l,k,β ( T I ; h , · · · , h k )The above formulas define, for each τ , a unital filtered A ∞ -algebra which we denote by F ( L , τ ). Like usual, given a Maurer-Cartan element b ∈ F ( L , τ ) we denote the deformed A ∞ operations by m τ,bk .Recall that the Seidel Lagrangian together with the equator divides X into eight regionswith equal area (say 1). For our convenience, we may take p to be λ times the sum of eightpoints, one in each region (for λ ∈ Q ). We have τ p ∩ β = λ · ( ω ∩ β ) since the area is givenby the number of regions. Then exp( τ p ∩ β ) = t ω ∩ β where t := e λ and ω ∩ β ∈ Z ≥ . We willsee in Proposition 5.2 that the choice of a representative of p does not affect our calculationsignificantly.3. Weakly unobstructedness for bulk-deformed Fukaya algebra
In this section, we show weakly unobstructedness of the Seidel Lagrangian in bulk-deformedFloer theory. The main geometric idea behind this result is the anti-symplectic involutionof the orbifold sphere P a,b,c . Let ι be the anti-symplectic involution on the orbifold sphere.The Seidel Lagrangian i : S (cid:55)→ P a,b,c is chosen so that the immersion i is equivariant (withthe involution on the domain S by π -rotation). Note that ι preserves the orientation andspin structure of the Seidel Lagrangian. The bulk inputs that we will consider are orbifoldcohomology representatives of P a,b,c . The twisted sectors and the fundamental cycle areinvariant under the involution. A representative of the point class will be chosen to beinvariant under ι .Recall that the Seidel Lagrangian is shown to be weakly unobstructed in [CHL12]. Weextend it to the case of bulk deformations. Proposition 3.1.
Fix τ ∈ H ∗ orb ( X, Λ + ) , let ˜ x, ˜ y, ˜ z ∈ Λ + and define b = ˜ xX + ˜ yY + ˜ zZ ∈F ∗ ( L ) . Any such b is a weak Maurer-Cartan element (that is, a weak bounding cochain). Inother words we have m τ,b = (cid:88) k ≥ m τk ( b, . . . , b ) = P ( τ, b ) L , where L is the unit in F ∗ ( L ) and P ( τ, b ) is some element in Λ . Remark 3.2.
Here ˜ x, ˜ y, ˜ z are regarded as a scalar. In later sections they will be regarded asvariables. In particular, we will investigate convergence problems.Proof. The main idea of proof is similar to that of [CHL12], namely, any non-unit outputof m τk ( b, . . . , b ) vanishes due to cancellation from the Z / From Z / m b is given by a linear combination of even-degree immersed generators and the unit L . The weak Maurer-Cartan equation is satisfiedif all outputs except L vanish. The anti-symplectic involution was used to show that anyimmersed output of m k ( b, b, · · · , b ) cancels out with the opposite polygon. In addition, L should be equipped with a non-trivial spin structure which brings the exact cancellation ofsigns (It is not weakly unobstructed with the trivial spin structure).First we will show that sign cancellation still works for the bulk deformed theory, so that m τ,b does not involve even-degree immersed generators. In the de Rham model, this meansthat m τ,b is a zero-form, that is a function on S (the normalization of L ). We will show thatit is simply a constant function on S , which proves the weakly unobstructedness.Since moduli spaces of sphere bubbles attached to the interior of stable discs carry complexorientations, the cancellation for the case without (orbi)-sphere bubbles will imply the generalcases. We will first consider the combinatorial sign rule of Seidel [Sei11] and later argue thatwe may use them for our computation.Let us consider a orbi-polygon P that produces an immersed output of m τk ( b, · · · , b ) (inparticular, such a P should have k + 1 edges). By applying the reflection about the equatorof P a,b,c to P , we get another polygon P op . The A ∞ -operations for P and P op give the sameoutput (in Z / P contributes to m τk ( X , · · · , X k ) then P op contributesto m τk ( X k , · · · , X ). We claim that these two contributions have the opposite signs to eachother.Without loss of generality, let us assume that the boundary orientation of P is coherentwith that of L . Then the boundary operation of P op is opposite (for each edges of P op ) tothat of L since the reflection preserves the orientation of L whereas it reverses boundaryorientations of holomorphic polygons. From the sign rule of [Sei11], There is a sign differenceof ( − k between P and P op . Another source of sign difference is how many times P and P op pass through the point that represents the nontrivial spin structure. Let s and s denotethese numbers, respectively. We now show that s − s , or equivalently, s + s has the sameparity as k + 1. We first claim that ∂P ∪ ∂P op evenly covers L . To see this, let us divide L into 6 minimal arcs, which are edges joining one corner with another without passing throughother corners. We will denote these arcs by (cid:95) XY ± , (cid:95) Y Z ± , (cid:95) ZX ± as in the left of Figure 3. Figure 3
IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 13
Suppose p is a point on the boundary of P that lies in (cid:95) XY ± , then its reflection image, say p (cid:48) , is located on (cid:95) XY ∓ . When p travels along ∂P , the pair ( p, p (cid:48) ) first covers both of (cid:95) XY ± .And then the pair starts covering both of (cid:95) Y Z ± afterward, regardless of having corners at Y or not (see the right of Figure 3. Since p starts at and comes back to the same point whenwe go along ∂P once, we see that ( p, p (cid:48) ) covers L evenly.Having this, let ∂P ∪ ∂P op = s [ L ], which implies s + s = s . If ( k + 1) edges of ∂P consistsof a , a , · · · , a k +1 minimal arcs, we have6 s = 2( a + a + · · · + a k +1 ) ⇒ s = a + a + · · · + a k +1 since ∂P ∪ ∂P op has 6 s minimal arcs. On the other hand, it is easy to see that each edge of ∂P (and ∂P op ) consists of an odd number of minimal arcs i.e., a i are all odd, and hence weconclude that the parity of s is the same as k + 1, which completes the proof of the claim.Now, let us argue that the combinatorial sign of Seidel is compatible with the de Rhammodel we use in this paper. In [Sei08, Part II section 13], it is shown that the sign of an A ∞ -operation defined using Floer theory (orientation operators) and that defined by combinatorialconvention can be identified. Seidel showed that in this surface case, the sign in Floer theoryis local, and hence depends only on absolute indices of intersection points. On the otherhand, there is a combinatorial way of giving sign in this case. Seidel constructs γ ( k ) whichis a linear isomorphism for each Floer group CF k which makes these two signs compatible.We use the existence of this isomorphism to show cancellations. Note that the combinatorialsign only depends on the parity of the intersection points.On the other hand, one can also show that the Floer sign also depends only on the parityof the absolute indices of corners. Note that a choice of path of Lagrangian subspaces from T p L to T p L for p ∈ L ∩ L defines an orientation operator, which can be used to defineits absolute index as well as associated orientation space (determinant of the orientationoperator). Absolute indices from different choice of paths may differ by even integer, andone can fix a canonical isomorphism between two different choices using gluing of discs withLagrangian loop of the difference of paths. It can be shown that this gluing provides acanonical way to relate orientation spaces corresponding to different paths, which gives riseto the same sign for associated polygons. In this way, one can observe that the Floer theoreticsign only depends on the parity of the absolute indices at the intersection points based onthe above isomorphism of orientation spaces.Thus m τk ( b, · · · , b ) does not involve even-degree immersed generators and hence is a zero-form. It remains to show that the non-immersed output is a multiple of the unit (namely aconstant function on S ). Lemma 3.3.
The expression m τk ( b, · · · , b ) is a constant (as a function on x, y, z ) k ≥ .Proof. We proceed by induction on k . Note that m = 0 in the elliptic and hyperbolic cases,and m can be given by the contribution of two hemispheres in the spherical cases [CHKL17,Section 12]. In any cases, it is always a constant multiple of a unit.Let us assume the statement for k = i and prove the case of i + 1. By degree reason, m i +1 ( b, · · · , b ) is even, and hence it is either a smooth function, or an immersed sector. Bythe above reflection argument we know that the output in immersed sectors cancels out. Hence, it is enough to show that the output is a constant function. Observe that m on afunction f is given by df by construction. Hence, in order to show that the output is constant,it is enough to prove that m ( m i +1 ( b, · · · , b )) = 0. This follows from the A ∞ -identity and theinduction hypothesis (and the property of a unit). (cid:3) This proves the weakly unobstructedness of the Seidel Lagrangian ( L , b ). (cid:3) It follows from Lemma 3.3 that each b determines a deformation of F ( L ) with centralcurvature m τ,b . This means that m τ,b is a differential and its cohomology HF ∗ ( L , τ, b ) is analgebra with product m τ,b . We will describe this algebra in Section 4.2.4. Bulk-deformed potential function and change of variables
The previous section asserts that the Lagrangian Floer potential function P ( b ) is a formalpower series in ˜ x, ˜ y and ˜ z with coefficients in the Novikov ring Λ , where b = ˜ xX + ˜ yY + ˜ zZ . Asexplained in the introduction, it is essential for the purpose of studying the Kodaira-Spencermap that we work with the following change of variables. x = T ˜ x,y = T ˜ y,z = T ˜ z. (4.1)with x, y, z ∈ Λ . From now on we denote W τ ( x, y, z ) = P ( τ, b )and call this the potential function. Notice that b on the right hand side is now given by b = T − xX + T − yY + T − zZ .This coordinate change will be essential in our study of Kodaira-Spencer map, and at thesame time it is the main source of complication.After the coordinate change the term of minimal valuation in the potential W τ is T ˜ x ˜ y ˜ z = T − xyz . This negative energy term should be handled in a delicate way as we will see inour proof of the Kodaira-Spencer map being an isomorphism. For this reason, we will needa better control on the energy of the terms appearing in the related Floer operations andalgebraic manipulations.We first examine the potential function and its convergence in new variables. When thereis no bulk deformation (i.e. τ = 0), [CHKL17] gives closed formulas for W in the sphericaland elliptic cases - in these cases, W is simply a polynomial on x, y, z . In the hyperbolic case(again when τ = 0), an algorithm that computes W is given in [CHKL17].4.1. Gauss–Bonnet theorem and convergence.
Recall that b = ˜ xX + ˜ yY + ˜ zZ = T − ( xX + yY + zZ ) where ˜ x, ˜ y, ˜ z are the dual variables to the immersed generators X, Y, Z .Gromov compactness ensures the boundary deformed A ∞ algebra is convergent when val ˜ x ,val ˜ y , val ˜ z >
0. We will show that it is still convergent for val ˜ x, val ˜ y, val ˜ z ≥ − x, val y, val z ≥ IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 15
Definition 4.1.
A convergent power series in x, y, z is a series of the form (cid:88) i,j,k ∈ Z ≥ c i,j,k x i y j z k , with c i,j,k ∈ Λ and lim i + j + k →∞ ν ( c i,j,k ) = + ∞ , where ν is the usual valuation in Λ . We denoteby Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) the ring of convergent power series. Recall that the valuation ν in Λ determines a non-archimedean metric by the formula | ξ | = e − ν ( ξ ) . The condition above then states that the coefficients of the series converge tozero in this norm. Therefore the ring just defined is a special case of the Tate algebra, see[BGR84].Note that any element P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) is indeed convergent (in the unit disc), in the sensethat it determines a map P : Λ −→ Λ.In the remainder of this section we will show that W τ ( x, y, z ) is a convergent power seriesfor each τ with val ( τ ) >
0, and hence, is an element in Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . We begin by establishinga complete classification of the non-positive energy terms of W τ ( x, y, z ). First of all, we have T − xyz from the minimal triangle. In addition, slices of discs for x a , y b , z c -terms give riseto x i , y j , z l with 1 ≤ i ≤ a −
1, 1 ≤ j ≤ b −
1, 1 ≤ l ≤ c − ∗ orb ( X, τ ) to the potential. We give a precisedescription on such discs by the lifting argument below, which is valid for general orbi-discsalthough their liftings are the maps defined on higher genus (bordered) Riemann surfaces inmost of cases.
Lemma 4.2.
Suppose D is an orbifold disc with interior orbifold marked points p , · · · , p k where p i is Z /k i cone point. Consider a map π : U → D between Riemann surfaces withboundary (mapping boundaries to boundaries) and suppose that p i is a branch point of π of multiplicity k i for each i . For any orbifold holomorphic disc u : ( D , ∂ D ) → ( X, L ) , thecomposition u ◦ π : ( U, ∂U ) → ( X, L ) is a holomorphic disc.Proof. Note that u ◦ π is a holomorphic disc away from π − ( p i ) by definition. Near each π − ( p i ), u ◦ π is nothing but the lift to a uniformizing cover, hence it is holomorphic. (cid:3) Applying the lemma to orbi-discs with a single orbi-insertion, we obtain the following.
Corollary 4.3.
A holomorphic orbi-disc u with one orbifold marked point has a holomorphiclift (cid:101) u : U → X , where U is a disc. Therefore we see that all such orbi-discs are slices of the discs that contribute to non-bulk-deformed potential. In particular, the slice of the discs for x a , y b , z c will be called the basicorbi-discs from now on.The following lemma gives a complete description of the low energy orbi-discs contributingto the potential. It is a kind of energy quantization at the corners of the discs. A similarresult in dimension greater or equal than two appears in [PW19, Lemma 4.2]. Lemma 4.4.
Except for the single term corresponding to the minimal triangle, every termof the bulk-deformed orbifold potential in x, y, z variable (4.1) has non-negative T -exponent. Moreover, it has T -exponent being exactly for basic orbi-discs, and the T -exponents arepositive for the rest (except the minimal triangle).Proof. Let u : ( D , ∂ D ) → ( P a,b,c , L ) be a non-constant holomorphic orbi-disc which is not theminimal triangle. We may assume that the counter-clockwise orientation of ∂D agrees withthe orientation of L under u . (The other case can be handled similarly). P a,b,c is decomposedinto 8 pieces by L and the equator. We denote by M u , M l the upper and lower middle trianglepiece and denote by A u , B u , C u (resp. A l , B l , C l ) the triangles with one of their corners at a, b, c orbifold points respectively and lies in the upper (resp. lower) hemisphere. We maydecompose the domain of the orbi-disc D according to the above decomposition under the map u . Suppose u has an immersed corner mapping to X, Y or Z contributing to the monomial ofthe potential. By our choice of orientation, the map u covers the piece M u . (One can checkthat we cannot turn corners at M l in this case). We consider the part of u which maps to M u as in Figure 4. We argue that in the neighborhood of each such corner, we have additionalregions (in D ) of area 2 which are distinct for each corner. Note that the piece M u should beattached to exactly one of A u , B u , C u , say the piece A u , since it involves a corner of the disc.In this case, this A u cannot be attached to any other preimages of M u or M l in D . Now, A u is attached to two of A l pieces A l , A l . Let us further cut these A l pieces into halves. Figure 4
Hence each corner of D at least covers M u , A u , A l , A l , which gives the area 3 (or T ).Given two different corners, the above local pieces do not overlap since A u can be attached tothe only one corner piece M u . This proves the first part of the proposition. Suppose that afterthe coordinate change, it has no T -component. This means that the holomorphic orbi-discconsists of these T pieces only. It is elementary to see that the only orbifold holomorphicdiscs that we can make in this way are the basic ones. This proves the lemma. (cid:3) Next, we will use a version of the Gauss–Bonnet theorem to compute the valuations of themonomials appearing in W τ .Recall that the Seidel Lagrangian L is taken to be symmetric about the equator, and itsubdivides each of the upper and lower hemispheres into four triangles with equal area A (which is set to be 1), which are M u , A u , B u , C u and M l , A l , B l , C l respectively in the proofof Lemma 4.4. Let K be the constant curvature of the orbi-sphere. The equator is taken tobe a union of three geodesics connecting the three orbi-points, and the reflection about theequator is an isometry. Denote by k the geodesic curvature of L . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 17
We arrange L in such a way that the exterior angles of the minimal triangles at X, Y, Z are 2 π (1 /a − (cid:15) ) , π (1 /b − (cid:15) ) , π (1 /c − (cid:15) ) respectively, where (cid:15) is taken such that the anglesare in (0 , π ). (cid:15) can be set to zero in case a, b, c ≥ Theorem 4.5 (Gauss-Bonnet formula for an orbi-polygon) . For an (embedded) orbi-polygon P ⊂ X with exterior angles ∠ i , boundary edges γ j , and ages of interior orbi-points being ι k , (cid:90) P KdA + (cid:88) i ∠ i + (cid:88) j (cid:90) γ j kds + 2 π (cid:88) k (1 − ι k ) = 2 π. Closely related to this, the Maslov-index formula for an orbi-polygon class ( β, α ) in termsof curvature (where α is the collection of immersed generators that the corners hit) is givenas follows (see Remark 2.2, [CS16] or [Pac19]). (Here ∠ X denotes the exterior angles).(4.2) µ CW ( β, α ) = 1 π (cid:32)(cid:90) β KdA + (cid:90) ∂β kds + (cid:88) X ∈ α ∠ X (cid:33) . In our case the Maslov index is given as follows.
Proposition 4.6.
For an orbi-disc bounded by L with corners being only X, Y or Z (thus con-tributing to the potential), denote its area by mA the numbers of X, Y, Z corners by n , n , n respectively. Its Maslov index µ CW equals to (cid:16) n a + n b + n c + ( m − n + n + n )) · χ (cid:17) . In particular the Chern number of an orbi-sphere equals to mχ/ . Recall that χ = − a + b + c is the (orbifold) Euler characteristic of P a,b,c . Proof.
First of all we find the geodesic curvatures of the edges of the minimal triangles. ByGauss-Bonnet formula applied to the upper hemisphere (which is a triangle bounded by threegeodesics segments forming the equator), we have4 AK + (cid:16) π − πa (cid:17) + (cid:16) π − πb (cid:17) + (cid:16) π − πc (cid:17) = 2 π (where A = 1 is the area of the minimal triangle.) Thus(4.3) χ = (cid:18) a + 1 b + 1 c (cid:19) − AK π . Let 2 π k be the total geodesic curvature along the edge connecting the X and Y cornersof the minimal triangle contained in the upper hemisphere. Here, the orientation of the SeidelLagrangian is fixed such that the orientations of the edges agree with the induced ones fromthe minimal triangle contained in the upper hemisphere. k and k are similarly defined. We apply the Gauss-Bonnet formula to the triangle C u which is the triangle in the upperhemisphere having corners at the point X, Y and the orbi-point [1 /c ]. This gives (cid:16) π − πa + π(cid:15) (cid:17) + (cid:16) π − πb + π(cid:15) (cid:17) + (cid:16) π − πc (cid:17) − πk + KA = 2 π. Combining with (4.3), we have k = κ + (cid:15) where κ = − χ/
8. Applying the same argumentfor the other two triangles contained in the upper hemisphere, we obtain k = k = k = − χ (cid:15) = − KA π + (cid:15) = κ + (cid:15). Then the total geodesic curvatures of the edges of the minimal triangle in the lower hemisphere(in the fixed orientation of the Seidel Lagrangian) are (2 π times) k (cid:48) = k (cid:48) = k (cid:48) = − k = − κ − (cid:15). Consider a holomorphic orbi-disc bounded by L with the numbers of X, Y, Z corners being n , n , n respectively. The area is a multiple mA of the area of the minimal triangle for m ∈ Z > . Thus 1 π (cid:90) β KdA = mKAπ = mχ . The edges of the orbi-disc are unions of the edge segments of the two minimal trianglesin the upper and lower hemispheres. By the property of holomorphic orbi-disc, each sidebetween two corners must consist of an odd number of edge segments. The total geodesiccurvatures of the edge segments cancel with each other, except for one edge segment for eachside. Such a segment in each side lie in the same hemisphere, and in the boundary orientationof the holomorphic disc its geodesic curvature is 2 πk = 2 π ( κ + (cid:15) ). Then1 π (cid:90) ∂β kds = 2( n + n + n ) k = − χ · ( n + n + n )4 + 2( n + n + n ) (cid:15). Since the exterior angles of
X, Y, Z are 2 π (1 /a − (cid:15) ) , π (1 /b − (cid:15) ) , π (1 /c − (cid:15) ) respectively, wehave 1 π (cid:88) X ∈ α ∠ X = 2 (cid:16) n a + n b + n c (cid:17) − n + n + n ) (cid:15). Hence the error term 2( n + n + n ) (cid:15) cancels in the sum. Combining the above equations,result follows. (cid:3) Corollary 4.7.
Consider a stable orbi-disc bounded by L which contributes to the disc po-tential. Suppose it has area mA , the interior orbi-insertions have ages ι j , and the numbersof X, Y, Z corners are n , n , n respectively. Then it satisfies (4.4) ( m − n + n + n )) · − χ n a + n b + n c + (cid:88) j (1 − ι j ) − . Proof.
Recall that an orbi-disc contributes to the potential if it is rigid, or equivalently itsMaslov index satisfies µ CW + 2 · (cid:88) j (1 − ι j ) − . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 19
A stable orbi-disc consist of disc and sphere components. Since Maslov index is additive,Proposition 4.6 applied to each component gives the result. (It can also be seen by taking anorbi-smooth disc representative of the class.) (cid:3)
From the corollary, the area of the orbi-disc is given in terms of the numbers of cornersand ages by(4.5) mA = (cid:32) n + n + n ) + 8 n a + n b + n c + (cid:80) j (1 − ι j ) − − (cid:0) a + b + c (cid:1) (cid:33) A when χ (cid:54) = 0, which matches the one given in [CHKL17] when there is no orbi-insertions. If χ = 0 (i.e. elliptic case), we have n a + n b + n c + (cid:80) j (1 − ι j ) = 1.Suppose that T a τ (cid:126)k x n y n z n ∈ Λ[[ τ, x, y, z ]] is one of monomials contained in W τ . Ourdiscussion so far tells us that the exponent a satisfies a = m − n + n + n ) ≥
0, where m is given by (4.5) and (cid:126)k records the ages ι j in the formula. Notice that the coordinate change4.1 is responsible for the term − n + n + n ) in a . By Lemma 4.4, m − n + n + n ) ≥ Theorem 4.8.
The bulk-deformed potential W τ is a convergent series, that is, it is an elementof Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) .Proof. By Gromov compactness, it suffices to show that the area mA is bounded above oncethe exponent m − n + n + n ) of T is bounded. For elliptic case, (4.4) gives n a + n b + n c + (cid:88) j (1 − ι j ) = 1 . There are only finitely many possibilities of n i and orbi-insertions satisfying this. (And hence W τ is a polynomial in x, y, z, τ .) In particular n i are bounded. Once m − n + n + n ) isbounded, m is bounded.For spherical case, m − n + n + n ) ≥ n a + n b + n c + (cid:88) j (1 − ι j ) ≤ . There are only finitely many possibilities of n i and orbi-insertions satisfying this. Thus thereare only finitely many possibilities of m − n + n + n ), and hence m . W τ just consists offinitely many terms.For hyperbolic case, if m − n + n + n ) is bounded above, then n a + n b + n c + (cid:80) j (1 − ι j )is also bounded above. This gives finitely many possibilities of n i and ι j . Hence n + n + n is bounded above, and so is m . (cid:3) Remark 4.9.
Recall that τ = τ X + τ p + τ tw , and exp( τ p ∩ β ) = t ω ∩ β (see the para-graph before Section 3). The above shows that W ( τ , t, τ , . . . , τ m , x, y, z ) is an element of Λ (cid:104)(cid:104) τ , t, τ , . . . , τ m , x, y, z (cid:105)(cid:105) . Similarly, we can show that every m τ,bk ( α , . . . , α k ) is convergent power series and so wehave the following proposition. Proposition 4.10.
The Fukaya algebra F ( L , τ, b ) is convergent over Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) .Proof. Consider an orbi-polygon P with k , k , k numbers of X, Y, Z corners (which have odddegree), and k − , k − , k − numbers of ¯ X, ¯ Y , ¯ Z corners (which have even degree) respectively.Without loss of generality, we assume that in a neighborhood of one of the odd corners, P is contained in the upper hemisphere. (If there is no odd corner, then we assume that in aneighborhood of one of the even corners, P is contained in the lower hemisphere.)Note that for a side of P between odd and even adjacent corners, the number of minimaledge segments is even. Similarly for an odd-odd side or even-even side, the number of minimaledge segments is odd. Thus two corners adjacent to an odd-odd or even-even side remain inthe same hemisphere; an odd-even side connects a corner in the upper hemisphere to a cornerin the lower hemisphere. It implies all corners of P are contained in the upper hemisphere.For an odd-even side, the geodesic curvature of the edge segments cancel among eachother; for an odd-odd edge (resp. even-even edge), the geodesic curvature of all but one edgesegment cancel, and that edge segment lies in the upper (resp. lower) hemisphere. It followsthat the total geodesic curvature of an odd-odd edge, odd-even edge, and even-even edge is2 π times k = κ + (cid:15), , − k = − κ − (cid:15) respectively.We claim that the error term, namely the term which is a multiple of (cid:15) , is zero in the Maslovindex. Recall that the exterior angles of the odd vertices X, Y, Z are 2 π (1 /a − (cid:15) ) , π (1 /b − (cid:15) ) , π (1 /c − (cid:15) ) respectively. As in the proof of Proposition 4.6, each odd vertex contributes − (cid:15) to the error term in the Maslov index. For the even vertices ¯ X, ¯ Y , ¯ Z , the exterior anglesare π − π (1 /a − (cid:15) ) , π − π (1 /b − (cid:15) ) , π − π (1 /c − (cid:15) ) respectively. Each even vertex contributes2 (cid:15) to the error term in the Maslov index.Let l oo , l oe , l ee be the numbers of odd-odd, odd-even, even-even sides respectively. Then thenumbers of odd and even vertices equal ( l oe + 2 l oo ) / l oe + 2 l ee ) / − (cid:15) ( l oe + 2 l oo ) / (cid:15) ( l oe + 2 l ee ) / − (cid:15) ( l oo − l ee ) . The geodesic curvature of an odd-odd (even-even resp.) edge contributes 2 (cid:15) ( − (cid:15) resp.) tothe error; the geodesic curvature of an odd-even edge has no error term. Thus the total errorcontribution from the geodesic curvatures of the sides is2 (cid:15) ( l oo − l ee ) . We see that the above two error contributions cancel among each other and hence there is no (cid:15) -term in the Maslov index.Thus we can throw away the (cid:15) terms. The total geodesic curvature (mod (cid:15) ) of the sidesequal to 2 πκ ( l oo − l ee ), and l oo − l ee equals to the number of odd vertices minus the number ofeven vertices, that is k + k + k − k − − k − − k − . The Maslov index formula in Proposition4.6 generalizes to this situation: µ CW ( P ) = 2 (cid:18) k − k − a + k − k − b + k − k − c + k − + k − + k −
2+ ( m − k + k + k − k − − k − − k − )) · χ (cid:19) . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 21
Now consider a stable orbi-disc contributing to a term of m τ,bk ( α , . . . , α k ) with the monomial T m − n + n + n ) x n y n z n . The corresponding dimension formula is µ CW + δ + 2 · (cid:88) j (1 − ι j ) = 2where δ = 0 if the output is a zero-form or an immersed generator, δ = 1 if the output is aone-form. Combining, we have (cid:32) m − (cid:32) n + n + n + k (cid:88) i =0 s i (cid:33)(cid:33) · − χ n a + n b + n c + k (cid:88) i =0 θ i + δ − (cid:88) j (1 − ι j )where s i = 1 , − , α i for i = 1 , . . . , k being odd generators, evengenerators, or point classes respectively, or the output (for i = 0) being even generators, oddgenerators, or L , pt L respectively; θ i = 1 /a, /b, /c if α i = X, Y, Z, i > α i = ¯ X, ¯ Y , ¯ Z, i =0; θ i = 1 / − /a, / − /b, / − /c if α i = X, Y, Z, i = 0 or α i = ¯ X, ¯ Y , ¯ Z, i > θ i = 0 inall other cases.Then the argument goes in a similar way as the proof of Theorem 4.8. In elliptic case − χ = 0, so the LHS = 0. n i are bounded. The valuation m − n + n + n ) is boundedimplies m is bounded. Similarly, in spherical case − χ < m is automatically bounded above:otherwise mχ/ m − n + n + n ) is bounded above, in hyperbolic case − χ >
0. Then the LHS is bounded. Hence there are just finitely many possibilities of n , n , n and ι j . Then there is only finitely many possibilities of m , and hence the area isbounded above. It follows from Gromov compactness that there are just finitely many termssatisfying the area bound. (cid:3) So far, we have discussed the properties of the potential function W τ . It can be shown asin [FOOO16b] that if we choose a different representative ( Z / τ , we may get a different potential function, but they are equivalent in the Jacobianring.For technical reasons, we choose the representative of the point class away from middleminimal triangle bounded by Seidel Lagrangian for the rest of the paper. Then the result ofthis section shows that Corollary 4.11. W τ is a convergent series. Moreover, if ν ( τ ) > , we have W τ = − T − xyz + x a + y b + z c + W high , where val ( W high ) ≥ λ > for some λ and the representative for [pt] is chosen not to intersectthe minimal triangles. Note that if we choose a representative of point class in the middle triangle, then wemay get bulk deformed contribution of the middle minimal triangle, which have negativevaluation. Nevertheless, the coefficient τ of [pt] has a nonnegative valuation, and hence doesnot affect the valuation of the coefficient of xyz . More specifically, the potential still admitsan expression(4.6) W τ = − ξxyz + x a + y b + z c + W high , and we still have val ( ξ ) = − Fukaya algebra of L . We will carry the argument on the canonical model of F ( L ).That is, we use the homological perturbation lemma to transfer the A ∞ -algebra structure to H ∗ ( L ). In addition, instead of evaluating the A ∞ operations at a specific value of b , we willconsider the canonical model of F ( L ) with values in the ring Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . This will be usefulin section 5.We denote by H ∗ ( L , Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) the canonical model of F ( L ) with the coefficients inthe ring Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) and denote the A ∞ operations by m τ,bk,can . H ∗ ( L , Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) shouldbe thought of as a family of A ∞ -algebras over Λ, and over each ( x, y, z ) in Λ and τ withval ( τ ) > A ∞ -algebra modeled on H ∗ ( L , Λ) (by Proposition 4.10) whose A ∞ -operations are deformed by b = xX + yY + zZ and τ . Lemma 4.12.
Let p be the odd degree generator of H ∗ ( S ) . We have the following identities: m τ,b ,can ( X, Y ) = ¯ ZT + T ξ + T − d L , m τ,b ,can ( Y, Z ) = ¯ XT + T ξ + T − d L , m τ,b ,can ( Z, X ) = ¯
Y T + T ξ + T − d L , m τ,b ,can ( X, ¯ X ) = (1 + c ) p + η , m τ,b ,can ( Y, ¯ Y ) = (1 + c ) p + η , m τ,b ,can ( Z, ¯ Z ) = (1 + c ) p + η , where each ξ i is a linear combination of ¯ X, ¯ Y , ¯ Z with val ( ξ i ) ≥ , val ( d i ) ≥ ; each η i is alinear combination of X, Y, Z and c i is an element of Λ + .Proof. The first term in m τ,b ,can ( X, Y ) is due to the minimal triangle with
X, Y, Z -cornersin counter-clockwise order. Also, the last term is essentially ∂ /∂x∂y applying to the bulkdeformed potential evaluated at ( T x, T y, z ). This is because X = ˜ xX | ˜ x =1 should be in-terpreted as xX | x = T = T X after change of variables (in particular, when computing in x, y, z -variables) and similar for Y . Therefore T − again comes from the minimal triangle,and that is the smallest valuation among the terms in the coefficient of L by Corollary 4.11.Finally, T ξ comes from the polygons apart from the minimal triangle, where one of theircorners are used as outputs. Since the output (one of ¯ X, ¯ Y , ¯ Z ) in this case has valuationzero unlike variables x, y, z , we have additional T (which we would lose if the correspondingcorner was not an output). The valuations in m τ,b ,can ( Y, Z ) and m τ,b ,can ( Z, X ) can be estimatedin a similar way.For m τ,b ,can ( X, ¯ X ), constant triangle contributes to p which gives the first term. In general, m τ,b ,can ( X, ¯ X ) should be of odd degree. In fact, there does not exist a polygon whose corners are X, Y, Z ’s except one of ¯ X, ¯ Y , ¯ Z corner due to orientation of Lagrangian. Thus any non-trivialpolygon contributing to m τ,b ,can ( X, ¯ X ) should have an output in X, Y, Z . (cid:3) IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 23
Lemma 4.13.
Let R ( X, Y, Z ) be the subring of H ∗ ( L , Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) generated by X, Y, Z .There exists r, s, t in the closure of R ( X, Y, Z ) and q , q , q ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) such that ¯ X = r + q L , ¯ Y = s + q L , ¯ Z = t + q L . Proof.
The proof of the three statements is identical, we prove the last one. First we proveby induction, that for each integer k ≥ t k ∈ R ( X, Y, Z ) and c k ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) suchthat E k = ¯ Z − t k − c k L is a linear combination of ¯ X, ¯ Y , ¯ Z and val ( ¯ Z − t k − c k L ) ≥ k, val ( t k − t k − ) ≥ k − , val ( c k − c k − ) ≥ k − . The first equation in Lemma 4.12, gives the case k = 1 with t = T − m τ,b ,can ( X, Y ) and c = T − d . Assuming the statement for k , we have E k = T k (cid:0) α ¯ X + β ¯ Y + γ ¯ Z (cid:1) , where val ( α, β, γ ) ≥
0. Now using the formulas for ¯ X, ¯ Y , ¯ Z given by the first three equationsin Lemma 4.12 we can write E k = T k (cid:0) T − R + d L + T J (cid:1) , where R ∈ R ( X, Y, Z ), J is a linear combination of ¯ X, ¯ Y , ¯ Z and val ( R ) ≥ val ( d ) ≥ − val ( J ) ≥
0. Therefore we have by induction¯ Z = t k + T k − R + ( c k + dT k )1 L + T k +2 J. Hence we can take t k +1 = t k + T k − R and c k +1 = c k + dT k , satisfying the conditions required.Finally we simply take the limits t = lim k t k and c = lim k c k . (cid:3) We are now ready to prove the main proposition in this subsection.
Proposition 4.14.
The image of m τ,b ,can is contained in the Jacobian ideal. That is, we havethe following Im (cid:16) m τ,b ,can (cid:17) ⊂ (cid:104) ∂ x W τ , ∂ y W τ , ∂ z W τ (cid:105) · H ∗ ( L , Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) . Proof.
In order to prove this proposition one first differentiates the Maurer-Cartan equation ∂ P ∂ ˜ x ( τ, b ) · L = (cid:88) k ,k ≥ m τk + k +1 ( k (cid:122) (cid:125)(cid:124) (cid:123) b, . . . , b, X, k (cid:122) (cid:125)(cid:124) (cid:123) b, . . . , b ) = m τ,b ( X ) . We have analogous identities for Y and Z . Taking the change of variables into account wehave m τ,b ( X ) = T ∂W τ ∂x ( b ) L , m τ,b ( Y ) = T ∂W τ ∂y ( b ) L and m τ,b ( Z ) = T ∂W τ ∂z ( b ) L . By the previous lemma, m τ,b ,can ( ¯ X ) = m τ,b ,can ( r ) + q m τ,b ,can (1 L ) = m τ,b ,can ( r ) , Now, given the Leibniz rule for m τ,b ,can and m τ,b ,can , we have that m τ,b ,can ( R ( X, Y, Z )) is con-tained in the Jacobian ideal. Recall from [BGR84, Section 5.2.7], that the Jacobian ideal,like any ideal in the Tate algebra is closed. Therefore m τ,b ,can of the closure of R ( X, Y, Z ) isalso in the Jacobian ideal. Hence m τ,b ,can ( ¯ X ) is in the Jacobian ideal. The same is true for ¯ Y and ¯ Z .Finally, from the fourth equation in Lemma 4.12 we have m τ,b ,can ( p ) = (1 + c ) − (cid:16) m τ,b ,can ( m τ,b ,can ( X, ¯ X )) + m τ,b ,can ( η ) (cid:17) , since 1 + c is invertible. Again, it follows from the Leibniz rule that the first term on theright is in the Jacobian ideal. Recall, by construction η is a linear combination of X, Y, Z and therefore we conclude that m τ,b ,can ( η ) is in the Jacobian ideal, which completes the proof. (cid:3) Proposition 4.15.
The cohomology HF ∗ ( L , τ, b ) is nonzero if and only if ( x, y, z ) (corre-sponding to b ) is a critical point of W τ .In this case, HF ∗ ( L , τ, b ) is isomorphic to H ∗ ( L ) := (cid:32) H ∗ ( S ) ⊕ (cid:77) X,Y,Z Λ ⊕ (cid:33) ⊗ Λ , as a vector space.Proof. From the previous proposition we have m τ,b ( X ) = T ∂W τ ∂x ( b ) L , m τ,b ( Y ) = T ∂W τ ∂y ( b ) L and m τ,b ( Z ) = T ∂W τ ∂z ( b ) L . Since L is the identity in HF ∗ ( L , τ, b ), this immediately implies that HF ∗ ( L , τ, b ) is zeroif ( x, y, z ) is not a critical point of W τ .For the converse, note that Proposition 4.14 implies that Im( m τ,b ) = 0, when ( x, y, x ) is acritical point. This implies that HF ∗ ( L , τ, b ) is isomorphic to H ∗ ( L ). (cid:3) Moreover, one can show that when b is a critical point, HF ∗ ( L , τ, b ) is isomorphic, as aring, to the Clifford algebra associated to the Hessian of W τ at the point b . But we will notmake use of this fact.5. Dependence of the potential on chain level representatives of bulk
In this section, we prove that if we change the Z / τ p for the H ( P a,b,c , Λ )component of the bulk deformation, the associated potentials are related by a coordinatechange. We start with the definition of this notion. Definition 5.1.
A coordinate change is a map ϕ : Λ (cid:104)(cid:104) x (cid:48) , y (cid:48) , z (cid:48) (cid:105)(cid:105) → Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) of the form x (cid:48) → c x + u , y (cid:48) → c y + u , z (cid:48) → c z + u , where c i ∈ C ∗ and u i ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) satisfy val ( u i ) > , for i = 1 , , . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 25
Proposition 5.2.
Let τ and τ (cid:48) be two Z / -invariant representatives of the same class in H ( P a,b,c , Λ ) . Their associated potentials W τ and W τ (cid:48) are related by a coordinate change x (cid:48) = exp( k X ) x, y (cid:48) = exp( k Y ) y, z (cid:48) = exp( k Z ) z, with k X , k Y , k Z ∈ Λ , i.e., W τ ( x (cid:48) , y (cid:48) , z (cid:48) ) = W τ (cid:48) ( x, y, z ) . The coefficients k X , k Y , k Z are givenin Lemma 5.3. The main ingredient in the proof of this result is the following topological lemma.
Lemma 5.3.
Suppose τ and τ (cid:48) are two reflection-invariant, cohomologous cycles, and let Q = τ − τ (cid:48) . Given an (orbi-)polygon β contributing to the potential W τ , denote by β ( X ) , β ( Y ) and β ( Z ) the numbers of X , Y and Z corners respectively. Then there exist k X , k Y and k Z such that Q ∩ β = k X β ( X ) + k Y β ( Y ) + k Z β ( Z ) for any (orbi-)polygon β contributing to the potential W τ .Proof. Since Q = τ − τ (cid:48) is cohomologous to zero we can choose a 1-(co)cycle R such that ∂R = Q . Here, we are abusing notations for cycles and cocycles via Poincar´e duality (seeFigure 5 (a)). Moreover we can choose R os that it is reflection invariant and avoids X, Y, Z .Then we have Q ∩ β = − R ∩ ∂β for any (orbi-)polygon class β for the potential.Let (cid:95) XY + denote the minimal segment of L between X and Y lying on the upper hemisphereand (cid:95) XY − denote its reflection image, see Figure 5 (a). We analogously define (cid:95) Y Z ± and (cid:95) ZX ± . Define i, j, k by i := R ∩ (cid:95) XY + , j := R ∩ (cid:95) Y Z + and k := R ∩ (cid:95) ZX + . Since R is reflectioninvariant R ∩ (cid:95) XY − = − R ∩ (cid:95) XY + , therefore R ∩ ∂β equals(5.1) ± (cid:16) i (cid:16) β (cid:16) (cid:95) XY + (cid:17) − β (cid:16) (cid:95) XY − (cid:17)(cid:17) + j (cid:16) β (cid:16) (cid:95) Y Z + (cid:17) − β (cid:16) (cid:95) Y Z − (cid:17)(cid:17) + k (cid:16) β (cid:16) (cid:95) ZX + (cid:17) − β (cid:16) (cid:95) ZX − (cid:17)(cid:17)(cid:17) where β (cid:16) (cid:95) XY ± (cid:17) is the number of (cid:95) XY ± -segments in ∂β (and analogously for Y Z and ZX ).The plus or minus depends on β , having its boundary orientation match with that of L or itsopposite. The former was called a positive polygon in [CHKL17], and the latter a negativepolygon for this reason.Let us assume that β is positive, we claim that(5.2) β (cid:16) (cid:95) XY + (cid:17) − β (cid:16) (cid:95) XY − (cid:17) = β ( X ) + β ( Y ) − β ( Z ) . We consider the loop p β in π ( L ) obtained by attaching three consecutive minimal segmentsfor each corner of ∂β . Namely, p β near a corner of β is parameterized in such a way thatwe walk past the corner without turning and coming back to the same vertex, rather thanjumping to another branch at the corner. See Figure 5 (b).Since p β is an integer-multiple of [ L ], we have(5.3) p β (cid:16) (cid:95) XY + (cid:17) − p β (cid:16) (cid:95) XY − (cid:17) = 0 . Thus β (cid:16) (cid:95) XY + (cid:17) − β (cid:16) (cid:95) XY − (cid:17) can be computed from the extra minimal segments that areattached to β to obtain p β . For instance, the extra segments attached to X -corner of β Figure 5 consists of (cid:95) XY − , (cid:95) Y Z + and (cid:95) ZX − , and hence removing these from p β increases (5.3) by 1.Likewise, for each Y -corner (resp. Z -corner) of β , removing extra segments from p β increases(resp. decreases) (5.3) by 1. Therefore we conclude the claim.When β is negative, formula (5.2) still holds but now with a minus sign on the right-handside. There are analogous formulas for the arcs Y Z and ZX . Combining all of these with(5.1) we obtain R ∩ ∂β = ( i − j + k ) β ( X ) + ( i + j − k ) β ( Y ) + ( − i + j + k ) β ( Z ) , which gives the desired result. (cid:3) Proof of Proposition 5.2.
Let us expand our bulk-deformed potential as W τ ( x, y, z ) = (cid:88) β exp( τ ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) where the sum is taken over the set of all (orbi-)polygon classes. Here, the potential alsodepends on other bulk parameters as the coefficient c β,τ tw shows, but we wrote W τ to highlightits dependence on [pt] which is what we want to analyze. W τ (cid:48) can be written analogously. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 27
Using Lemma 5.3, one can compute W τ ( x (cid:48) , y (cid:48) , z (cid:48) ) as follows W τ ( x (cid:48) , y (cid:48) , z (cid:48) ) = (cid:88) β exp( τ ∩ β ) c β,τ tw x (cid:48) β ( X ) y (cid:48) β ( Y ) z (cid:48) β ( Z ) T ω ( β ) = (cid:88) β exp( τ ∩ β ) c β,τ tw exp( k X β ( X ) + k Y β ( Y ) + k Z β ( Z )) x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) = (cid:88) β exp( τ ∩ β + R ∩ ∂β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) = (cid:88) β exp( τ ∩ β − Q ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) = (cid:88) β exp( τ (cid:48) ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) = W τ (cid:48) ( x, y, z ) , which is the desired result. (cid:3) The Kodaira–Spencer map
In this section we will define the Kodaira-Spencer map in our setting. This is a map KS τ :QH ∗ orb ( X, τ ) −→ Jac( W τ ) from the quantum cohomology of P a,b,c to the Jacobian ring of W τ ,constructed geometrically using J -holomorphic discs. The only previously known constructionof KS τ map is the case of toric manifolds by Fukaya-Oh-Ohta-Ono [FOOO16b]. We will followthe line of their construction. Their construction heavily uses the T n -action on the modulispace of holomorphic discs. In our construction, Z / KS τ map is well-defined (independent of the choice of cohomology representative)and KS is a ring homomorphism.6.1. Definition of KS τ and well-definedness. We start by defining the Jacobian ring of W τ . Recall that we use the convergent power series ring Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) defined in Definition 4.1. Definition 6.1.
Consider P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . We define the Jacobian ring of P as the ring Jac( P ) = Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) < ∂ x P, ∂ y P, ∂ z P > .
We would like to point out that there is no need to take closure of the ideal, since inΛ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) (as a Tate algebra) all ideals are closed, see [BGR84, Section 5.2.7].
Remark 6.2.
We saw in Section 5 that W τ is well defined up to a change of variables. Sincea change of variables induces a ring isomorphism on the corresponding Jacobian rings, wesee that the Jacobian ring Jac( W τ ) is well defined up to isomorphism. Let w , · · · , w B be coordinates of τ with respect to the basis { f i } Bi =0 (i.e. τ = (cid:80) i w i f i ). Thepotential function W τ can be regarded as a function W τ ( w , · · · , w B , x, y, z ) with w i ∈ Λ + .Regarding τ as an element of H ∗ orb ( X, Λ), we identify the tangent space T τ H ∗ orb ( X, Λ) at τ with QH ∗ orb ( X, τ ). Definition 6.3.
We define the Kodaira-Spencer map KS τ : QH ∗ orb ( X, τ ) −→ Jac( W τ ) by the formula KS τ ( ∂∂w i ) = ∂W τ ∂w i There is an ambiguity of the choice of representatives f i in QH ∗ orb ( X, τ ). In our case, thetwisted sectors as well as the fundamental cycle have canonical representatives. Hence weonly need to consider the choice of τ . Lemma 6.4.
The map KS τ is well-defined. In other words, if τ = ∂R then KS τ ( ∂∂τ ) = 0 in Jac( W τ ) .Proof. As in the proof of Proposition 5.2 we write W τ ( x, y, z ) = (cid:88) β exp( τ ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) . Then, by definition we have KS τ ( ∂∂τ ) = (cid:88) β ( τ ∩ β ) exp( τ ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) . By assumption, τ = ∂R , then by Lemma 5.3, there are k X , k Y , k Z such that Q ∩ β = k X β ( X ) + k Y β ( Y ) + k Z β ( Z ). Therefore KS τ ( ∂∂τ ) = (cid:88) β ( k X β ( X ) + k Y β ( Y ) + k Z β ( Z )) exp( τ ∩ β ) c β,τ tw x β ( X ) y β ( Y ) z β ( Z ) T ω ( β ) = k X x ∂W τ ∂x + k Y y ∂W τ ∂y + k Z z ∂W τ ∂z . Therefore KS τ ( ∂∂τ ) = 0 in Jac( W τ ) (cid:3) Here is an alternative description of KS , for i such that f i is a twisted sector. The derivative ∂∂w i has the effect of removing w i in one of the τ = (cid:80) i w i f i insertions on the disc. Thereforewe have the following expression KS τ ( f i ) = (cid:88) β,k exp( τ p ∩ β ) ∞ (cid:88) l =0 T β ∩ ω l ! q l +1 ,k,β ( f i , τ ltw ; b, . . . , b ) . We would like to have an analogous description for the cases of the fundamental and pointclasses in P a,b,c . More concretely, let Q be a Z / P a,b,c , and define themoduli spaces M maink +1 ,l +1 ( β, Q , τ tw , γ ) = M maink +1 ,l +1 ( β, τ tw , γ ) × X Q . Using these spaces and their evaluation maps, analogously to (2.3) we can define maps q l +1 ,k,β ( Q , τ ltw , − ). Then we have the following statement. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 29
Proposition 6.5.
Let Q be the cycle representing the fundamental cycle or the point classin P a,b,c . Then KS τ ( Q ) = (cid:88) β,k exp( τ p ∩ β ) ∞ (cid:88) l =0 T β ∩ ω l ! q l +1 ,k,β ( Q , τ ltw ; b, . . . , b ) , in Jac( W τ ) . This proposition essentially asserts that the q maps are unital and satisfy a version of thedivisor axiom in Gromov-Witten theory. Both properties are related to the compatibility ofthe Kuranishi structures (and perturbations) on the moduli spaces of discs with forgettinginterior marked points. It turns out that ensuring this compatibility for all moduli spacesseems a rather complicated problem. We will avoid tackling that problem by taking homo-topy between the usual Kuranishi structures on M maink +1 ,l +1 ( β, Q , τ tw , γ ) and one constructedspecifically to ensure this compatibility. Therefore the equality in the statement holds only inthe Jacobian ring, but not necessarily at chain-level. We will postpone this proof to AppendixA.6.2. Ring homomorphism.
In this subsection we will prove the following
Theorem 6.6.
The map KS τ : QH ∗ orb ( X, τ ) −→ Jac( W τ ) is a ring homomorphism. This map is rather surprising in that it identifies complicated quantum multiplication witha standard multiplication of polynomials in Jacobian ring. The geometric idea behind thismap is rather well-known. Namely, the closed-open maps in topological conformal field theoryare ring homomorphisms from the closed theory to Hochschild cohomology of the open theory.They are explored in Seidel [Sei12], Biran-Cornea [BC13], Fukaya-Oh-Ohta-Ono [FOOO09].A benefit of this construction is that Hochschild cohomology of the Fukaya category is veryheavy object to handle, whereas the construction of KS τ map is rather direct and simple. Proof.
As before, we will follow the line of proof of [FOOO16b] Theorem 2.6.1 and we will usetheir notation freely to shorten our exposition. The proof is based on a cobordism argument.Consider two cohomology representatives
A, B in QH ∗ orb ( X, τ ). Consider the forgetful mapfor the moduli space introduced in Section 2.4 forget : M maink +1 ,l +2 ( β, A ⊗ B ⊗ τ ⊗ ltw , γ ) → M main , which forgets maps and shrinks resulting unstable components if any, followed by the forgetfulmap M maink +1 ,l +2 → M main , forgetting the boundary marked points, except the first one andforgetting the interior marked points except the first two. In Lemma 2.6.3 [FOOO16b], M main , is shown to be topologically a disc with some stratification, so that the above forget is a continuous and stratified smooth submersion.The idea of proof is to consider a line segment in M main , which connects two point strataof D . Σ is a stratum where two interior marked points lie on a sphere bubble, and Σ is a component where there are two disc bubbles each of which contains one of the interiormarked points. We will see that integration over forget − (Σ ) and forget − (Σ ) correspondto KS τ ( A • τ B ) , KS τ ( A ) KS τ ( B ) respectively and the pre-image of line segment will define thedesired cobordism relation between them. There is a technical issue in that the map forget is only a stratified submersion. We will explain below how to handle this issue following[FOOO16b].We first have to construct Kuranishi structures and continuous family of multi-sections on(neighborhoods of) the spaces forget − (Σ ) and forget − (Σ ). To describe the neighborhoodnear Σ , we consider the following moduli space. For α ∈ H ( X, Z ), let M l ( α ) the modulispace of stable maps from genus zero closed Riemann surface with l -marked points and ofhomology class α . M l +3 ( α, A ⊗ B ⊗ τ l tw ) = M l +3 ( α ) ( ev , ··· ,ev l ) × X l ( A ⊗ B ⊗ τ l tw )Then ev l +3 defines an evaluation map from the above moduli space to X . Define the modulispace M k +1 ,l ,l ( α, β ; A, B, K ) to be the fiber product (cid:0) M l +3 ( α, A ⊗ B ⊗ τ l + ) × M maink +1 ,l +1 ( β, τ l + ) (cid:1) × ( IX × IX ) K for a chain K in IX × IX .Let us consider the case that K = ∆ (cid:48) defined as(6.1) ∆ (cid:48) = { ( x, g ) , ( x, g − ) } ⊂ IX × IX for the inertia orbifold IX . The following is an analogue of Lemma 2.6.9 [FOOO16b], towhich we refer readers for the proof. Lemma 6.7.
There exist a surjective map (6.2)
Glue : (cid:91) α(cid:93)β (cid:48) = β (cid:91) l + l = l M k +1 ,l ,l ( α, A, B, ∆ (cid:48) ) → forget − (Σ ) which defines an isomorphism outside codimension 2 strata as a space with Kuranishi struc-ture. The
Glue map gives a way to describe an element of forget − (Σ ) as a fiber product of sphereand disc moduli space. In this way, it corresponds to first taking the quantum multiplicationand then taking the Kodaira-Spencer map. In the case that there are several sphere bubblesattached to a disc component (codimension higher than 2), there may be several ways of suchdescription. Namely, Glue map image may overlap in codimension two strata.The more important issue is the compatibility of Kuranishi perturbations to be chosen,where there are differences between toric and our cases. Recall that the proof of [FOOO09]uses T n -action on moduli space of holomorphic discs in an essential way. Because finite groupsymmetry is much easier to handle than T n -symmetry, many of the arguments simplify inour Z / Z / J -holomorphic discs or spheres.Therefore, we may consider the following Kuranishi structure on forget − (Σ ). We choose acomponent-wise Z / M maink +1 ,l +2 ( β, A, B, τ ltw ), following [Fuk10]. Herecomponent-wise means that the Kuranishi structure is compatible with the fiber productdescription of each of the strata of disc-sphere stratification. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 31
Lemma 6.8.
There exist component-wise Z / -equivariant Kuranishi structures and CF per-turbations on the moduli spaces M maink +1 ,l +2 ( β, A, B, τ ltw ) . This Kuranishi structure and per-turbations induce Kuranishi structures and perturbations on forget − (Σ ) . These Kuranishistructures and perturbations coincide with the ones induced by the Glue map (6.2) that coin-cide with each other on the overlapped part. Remark 6.9.
Recall that T n -equivariant analogue of this lemma near Σ is given at Lemma2.6.23 for toric cases. But this does not hold near Σ for toric cases because the moduli spaceof J -holomorphic spheres cannot be made T n -equivariant. Therefore, the construction for Σ is much more involved than that of Σ in toric cases, but since we can impose Z / -symmetryeven for sphere moduli spaces, we can treat both cases in the same way.Proof. We can choose Z / ev can be made submersive (see Lemma 3.1 [Fuk10]). (cid:3) The multi-section in the neighborhood of forget − (Σ ) can be constructed in a similar way.There exist a surjective map Glue : (cid:91) β (0) + β (1) + β (2) = β (cid:0) ( M k +1 ,l +1 ( β (1) , A ⊗ τ l + ) × M k +1 ,l +1 ( β (2) , B ⊗ τ l + )) ( ev ,ev ) × ( ev i ,ev j ) M k +3 ,l ( β (0) , τ l + ) (cid:1) → forget − (Σ )The relationship of Kuranishi structures under the Glue map is the same as that of Lemma2.6.22 [FOOO16b] (we consider Z / T n -equivariance), and we canchoose Z / ∈ M main , , consider forget − (Σ). Following (2.3), we use theevaluation map ev : forget − (Σ) → (cid:81) ki =1 L ( α ( i )) to define (cid:101) Z Σ b,τ = (cid:88) k,β,l T ω ∩ β l ! ( ev ) ∗ ( ev ∗ b ∧ · · · ∧ ev ∗ k b ) . Lemma 6.10.
We have (cid:101) Z Σ b,τ = z Σ b,τ L , for some z Σ b,τ ∈ Λ Proof.
The proof is the same as that of Proposition 3.1. Namely, the output given by im-mersed sectors vanishes by the reflection argument. Hence, the output is in Ω ( L ), i.e. afunction on L . One can consider the boundary configuration of forget − (Σ) to conclude thatthe output is m τ,b -closed. But m τ,b on a function on L is given by a total derivative. Henceit is a constant function. (cid:3) Now, pick Σ ,i (resp. Σ ,i ) very close to Σ (resp. Σ ) in M main , which converges to Σ (resp. Σ ) as i → ∞ . We have the following analogue of Lemma 2.6.27 of [FOOO16b]. Proposition 6.11. lim i →∞ ( ev ) ∗ ( forget − (Σ ,i )) = ( ev ) ∗ ( forget − (Σ ))lim i →∞ ( ev ) ∗ ( forget − (Σ ,i )) = ( ev ) ∗ ( forget − (Σ )) Proof.
The proof in our case is easier than that of [FOOO16b], because of Lemma 6.8.Namely, in our case, Glue map is compatible with Z / , Σ . Therefore, we can just apply Lemma 4.6.5 [FOOO16b] which claims the C -convergence of perturbations as i → ∞ . We remark that this type of convergence wasextensively studied in [FOOO16a]. (cid:3) We will now show that Z Σ ,i b,τ and Z Σ ,i b,τ are equal in the Jacobian ring. For this purpose weintroduce an additional moduli space: choose a smooth curve ψ on the open stratum of M , connecting Σ ,i to Σ ,i and define N k +1 ,l +2 ( β ) = forget − ( ψ ) ⊂ M maink +1 ,l +2 ( β, A ⊗ B ⊗ τ ⊗ ltw , α ).Since forget is a weakly smooth submersion when restricted to the open stratum, the Kuranishistructure defined in Lemma 6.8 induces a Kuranishi structure on N k +1 ,l +2 ( β ). Using theevaluation map ev : N k +1 ,l +2 ( β ) → (cid:81) ki =1 L ( α ( i )), as before, we define (cid:101) Y b,τ = (cid:88) k,β,l T ω ∩ β l ! ( ev ) ∗ ( ev ∗ b ∧ · · · ∧ ev ∗ k b ) . By the homological perturbation lemma there is an A ∞ -quasi-isomorphism from F ( L ) toits canonical model H ∗ can ( L ). Denote by Π b,τ the arity one (or linear) component of thisquasi-isomorphism, by definition we have Π b,τ ◦ m τ,b = ( m τ,b ) can ◦ Π b,τ and Π b,τ ( L ) = L . Wedefine Y b,τ := Π b,τ ( (cid:101) Y b,τ ) . The following lemma can be proved in the same way as Proposition 4.10 for the maps of m τ,bk . Lemma 6.12. (cid:101) Y b,τ (resp. Y b,τ ) is convergent series, more precisely it is as an element of F ( L, Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) (resp. H ∗ can ( L , Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) ) . Proposition 6.13.
We have the following relation in H ∗ can ( L ) : ( m τ,b ) can ( Y b,τ ) = (cid:101) Z Σ ,i b,τ − (cid:101) Z Σ ,i b,τ . Therefore Z Σ ,i b,τ = Z Σ ,i b,τ in the Jacobian ring Jac( W τ ) .Proof. First note that the second statement follows from the first together with the fact,proved in Proposition 4.14 that Im (cid:16) m τ,b ,can (cid:17) is contained in the Jacobian ideal. Second notethat the first statement is equivalent to m τ,b (cid:16) (cid:101) Y b,τ (cid:17) = (cid:101) Z Σ ,i b,τ − (cid:101) Z Σ ,i b,τ , by definition of Π b,τ .In order to prove this relation we describe the boundary of N k +1 ,l +2 ( β ). As a space withKuranishi structure the boundary of N k +1 ,l +2 ( β ) is the union of forget − (Σ ,i ), forget − (Σ ,i )and the fiber products(6.3) N k +1 ,l ( β ) ev × ev i M k +2 ,l ( β , τ l tw ) and M k +2 ,l ( β , τ l tw ) ev × ev i N k +1 ,l ( β ) , where β + β = β , k + k = k, l + l = l and i ∈ { , . . . , k + 1 } . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 33
Now, using Stokes theorem [FOOO11, Lemma 12.13] and summing over all β, k, l (like inLemma 2.6.36 [FOOO16b]), we see that the first product in (6.3) gives m τ,b (cid:16) (cid:101) Y b,τ (cid:17) . The secondproduct in (6.3) contributes as zero since m b,τ is a multiple of the unit and the perturbationin N k,l ( β ) is compatible with forgetting boundary marked points. Finally, by definition, forget − (Σ ,i ) and forget − (Σ ,i ) give (cid:101) Z Σ ,i b,τ and (cid:101) Z Σ ,i b,τ respectively. Now the desired relationfollows from the Stokes theorem. (cid:3) Now we need to relate cohomological intersection product and geometric intersection. Let { f i } mi =1 be basis of H ∗ orb ( X ), g ij = (cid:104) f i , f j (cid:105) P D and ( g ij ) be its inverse matrix. On this basis, wewrite A • τ B = (cid:80) i c i f i . Let R be a chain in IX × IX such that ∂R = ∆ (cid:48) − (cid:88) ij g ij f i × f j , and consider the moduli space M k +1 ,l ,l ( α, β ; A, B, R ) defined above. Using the boundaryevaluation maps on these moduli spaces we define (cid:101) Ξ( A, B, K, b ) = (cid:88) α,β,l ,l ,k T ω ∩ ( α(cid:93)β ) ( l + l )! ( ev ) ∗ ( ev ∗ b ∧ · · · ∧ ev ∗ k b ) . The following lemma can be proved exactly as Lemma 2.6.36 in [FOOO16b].
Lemma 6.14. (cid:88) i c i KS τ ( f i ) L − Z Σ b,τ = m τ,b ( (cid:101) Ξ( A, B, R, b ))Please note that here we are using the description for KS τ provided by Proposition 6.5. Proposition 6.15. KS ( A • τ B ) equals Z Σ b,τ modulo the Jacobian ideal.Proof. As before we can show that (cid:101) Ξ( A, B, R, b ) is convergent. Then we apply Π b,τ to theequation in the previous lemma to conclude that KS τ ( A • τ B ) L and Z Σ b,τ L differ by anelement in the image of ( m τ,b ) can . The result now follows from Proposition 4.14. (cid:3) Proposition 6.16 (c.f. Lemma 2.6.29 [FOOO16b]) . We have KS τ ( A ) · KS τ ( B ) = Z Σ b,τ . This proposition is completely analogous to Lemma 2.6.29 [FOOO16b]. Now combiningPropositions 6.16, 6.15, 6.13 and 6.11 we obtain the proof of Theorem 6.6.7. KS τ is an isomorphism Surjectivity.
In this subsection we show that KS τ is surjective. We start with compu-tations on lower energy contributions. Lemma 7.1.
There is λ > (depending on τ ) such that: KS τ (cid:32)(cid:22) a (cid:23) • τ i (cid:33) = x i mod T λ , KS τ (cid:32)(cid:22) b (cid:23) • τ j (cid:33) = y j mod T λ , (7.1) KS τ (cid:32)(cid:22) c (cid:23) • τ k (cid:33) = z k mod T λ , KS τ (8pt) = − T − xyz + 3 ax a + 3 bx b + 3 cz c mod T λ , where ≤ i < a , ≤ j < b and ≤ k < c .Proof. The first order term follows from direct computation. For example, a -slice of the disccontributing to x a in W τ produces x in the first equation (see Corollary 4.3 and the precedingdiscussion for the precise description for these orbi-discs). Thus it suffices to show that all thehigher order terms in the above equations have strictly positive powers in T . This directlyfollows from Lemma 4.4. (cid:3) Lemma 7.1 together with the fact that KS τ is a ring map, is enough to establish surjectivity. Proposition 7.2.
The map KS τ is surjective.Proof. Since any element in Jac( W τ ) can be written as T − (cid:15) R such that (cid:15) > R only haspositive powers in T , it is enough to prove that any R ∈ Jac( W τ ) with Λ -coefficients is inthe image of KS τ . Let λ be the minimum of powers of T appearing in the higher order termsin (7.1). We claim that for any such R there exists ρ with R − KS τ ( ρ ) = T λ U where U is also an element in Jac( W τ ) with Λ -coefficients only. To see this, write R as R = N (cid:88) l =1 a l T λ l x i l y j l z k l + T λ ˜ U where ˜ U has T with positive powers (either of summands could be zero even for nonzero R ).We take ρ to be as follows ρ = N (cid:88) l =1 a l T λ l (cid:22) a (cid:23) • τ i l • τ (cid:22) b (cid:23) • τ j l • τ (cid:22) c (cid:23) • τ k l . Using the fact that KS τ is a ring homomorphism, Lemma 7.1 implies that the valuation of R − KS τ ( ρ ) is no less than λ .We next use this inductively to prove the surjectivity. For R ∈ Jac( W τ ) only with Λ -coefficients, there exists ρ such that R − KS τ ( ρ ) = T λ R . Applying the same to R , we get ρ such that R − KS τ ( ρ + T λ ρ ) = T λ ( R − KS τ ( ρ )) = T λ R IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 35
Inductively, one sees that (cid:80) i T ( i − λ ρ i maps to R under KS τ . (cid:3) Jacobian ring of the leading order potential.
From Corollary 4.11, we can write T W τ = W lead + W + , where(7.2) W lead = − xyz + T ( x a + y b + z c )and W + = T W high . In particular, we have val ( W + ) = λ > λ . The coefficientof xyz in W lead depends on the choice of a representative of [pt], but we will only considerthe case of (7.2) in this section to make our exposition simpler. In general, one can have W lead = − ˜ ξxyz + T ( x a + y b + z c ) for some ˜ ξ with val (cid:16) ˜ ξ (cid:17) = 0 (see (4.6) where ˜ ξ = T ξ ),but the argument below will still apply for any ˜ ξ without much change, since what essentiallymatters is its valuation. We set the following notation g = ∂ x W lead , g = ∂ y W lead , g = ∂ z W lead . Moreover let γ , · · · , γ N denote the following set of elements in Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) :(7.3) 1 , x, x , · · · , x a − , y, · · · , y b − , z, · · · , z c − , xyz. Theorem 7.3.
Let A be the Jacobian ring of W lead , that is A := Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) / (cid:104) g , g , g (cid:105) .We have the following: (1) { γ , . . . , γ N } forms a linear basis of A over Λ . (2) Any ρ ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( ρ ) ≥ can be written as (7.4) ρ = N (cid:88) i =1 c i γ i + (cid:88) j =1 t j g j where c i ∈ Λ with val ( c i ) ≥ − and t j ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( t j ) ≥ − . Note that the ideal (cid:104) g , g , g (cid:105) is closed since Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) is a Tate algebra [BGR84, Section5.2.7], as are other ideals appearing in earlier sections.It requires a clever usage of relations in the Jacobian ring to see that the condition (2) ofTheorem 7.3 holds true, and the argument varies for different types of ( a, b, c ). We will providea detailed proof in Appendix B. It implies that the monomials in (7.3) form a generating setfor the Λ-vector space A . Thus, in order to complete the proof of Theorem 7.3, it onlyremains to show that they are linearly independent. Proposition 7.4.
The rank of A is a + b + c − .Proof. We need to show that1 , x, x , . . . , x a − , y, . . . , y b − , z, . . . , z c − , xyz. An analogous statement over C is well-known, but here, we additionally need a careful estimate on thevaluation to prove this over Λ. are linearly independent in A = Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) / (cid:104) g , g , g (cid:105) . As in (7.3), we write them as γ , · · · , γ N . Suppose we have the following equation in Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) :(7.5) N (cid:88) i =1 c i γ i + f g + f g + f g = 0where c i ∈ Λ, f j ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . It is enough to show that c = · · · = c N = 0. From theexpression of g , g , g it is easy to see that f i g i cannot have terms like1 , x, . . . , x a − , y, . . . , y b − , z, . . . , z c − . Thus we find that the coefficients on these monomials should vanish, and the equation (7.5)can be written as(7.6) c x x a − + c y y b − + c z z c − + c N xyz + f g + f g + f g = 0 . If c x ∈ Λ is non-zero, then f must have a nontrivial constant term f ∈ Λ in order to cancel c x x a − making use of f g = f ( yz + T x a − ). However, the monomial f yz cannot appearin other expressions of (7.6). Thus f = 0, and hence c x = 0. In the same way c y = c z = 0,and the equation (7.6) can be written as(7.7) c N xyz + f g + f g + f g = 0 . If c N (cid:54) = 0, then one of f , f , f should have a term of monomial x, y, z respectively. Suppose f has a monomial f x . Then, f x a cannot appear in other expressions of (7.7) and thus f = 0. Similarly f and f cannot have monomials in y and z respectively, which implies c N = 0. Therefore all the coefficients c , . . . , c N must vanish, as desired. (cid:3) This completes the proof of Theorem 7.3.7.3.
Deforming
Jac( W τ ) . In this subsection we will construct a flat family of rings interpo-lating between Jac( W lead ) and Jac( W τ ).Recall T W τ = W lead + W + , for some W + with val ( W + ) = λ >
8. We define W ( s ) = W lead + sW + ∈ Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) , and denote f = ∂ x W ( s ) , f = ∂ y W ( s ) , f = ∂ z W ( s ) . Note that f i = g i + sh i , for some h i ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( h i ) ≥ λ > Proposition 7.5.
Let A = Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) / (cid:104) f , f , f (cid:105) . Then A is a finitely generated Λ (cid:104)(cid:104) s (cid:105)(cid:105) -module.Proof. We will show that (7.3) forms a generating set of A . It is enough to show that anyconvergent series in x, y, z with non-negative valuation is in the Λ (cid:104)(cid:104) s (cid:105)(cid:105) -span of (7.3). Let ρ be such series with val ( ρ ) ≥
0, by Theorem 7.3 we have ρ = N (cid:88) i =1 c i γ i + n (cid:88) j =1 t j g j , with c i ∈ Λ, t j ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) and val ( c i ) , val ( t j ) ≥ −
8. Rearranging we get
IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 37 ρ = N (cid:88) i =1 c i γ i + n (cid:88) j =1 t j ( f j − sh j )= N (cid:88) i =1 c i γ i + n (cid:88) j =1 t j f j − T λ − s n (cid:88) j =1 t (cid:48) j h (cid:48) j with val ( t (cid:48) j h (cid:48) j ) ≥
0. By setting ρ = (cid:80) nj =1 t (cid:48) j h (cid:48) j , we can repeat the argument to prove that ρ = N (cid:88) i =1 c i γ i + n (cid:88) j =1 t j f j − T λ − sρ , for some ρ ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with non-negative valuation. Combining the two we obtain ρ = N (cid:88) i =1 ( c i − T λ − sc i ) γ i + n (cid:88) j =1 ( t j − T λ − t j s ) f j + T λ − sρ . Note that this process increases the valuation of the error term (each time by λ − > ρ = N (cid:88) i =1 ˜ c i γ i + n (cid:88) j =1 ˜ t j f j with ˜ c i ∈ Λ (cid:104)(cid:104) s (cid:105)(cid:105) and ˜ t j ∈ Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) , which implies the result. (cid:3) Proposition 7.6. A is a flat Λ (cid:104)(cid:104) s (cid:105)(cid:105) -module.Proof. Flatness of A is equivalent to flatness of the localizations Λ (cid:104)(cid:104) s (cid:105)(cid:105) n → A m for all maximalideals m of A , with n = m ∩ A , (see [Mat80, Section 3.J]). Note that, since A is a PID, n = (cid:104) s − s (cid:105) for some s ∈ Λ .Now, since Λ (cid:104)(cid:104) s (cid:105)(cid:105) n is a regular local ring of dimension 1, A m is flat over it if and only if s − s is not a zero-divisor in A m , by Lemma 10.127.2 in [Aut20].By the previous proposition rk Λ ( A / n ) < ∞ , which implies that dim( A / n ) = 0. Thereforedim Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) m (cid:48) (cid:104) f , f , f , s − s (cid:105) = 0 , where m (cid:48) is the ideal of Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) corresponding to m . Which implies that f , f , f , s − s is a system of parameters of Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) m (cid:48) . Since this is a regular local ring, Theorem 31 in[Mat80], shows that f , f , f , s − s is a regular sequence in Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) m (cid:48) . This immediatelyimplies that s − s is not a zero-divisor in A m , which gives the desired result. (cid:3) Corollary 7.7. A is a free, finite dimensional Λ (cid:104)(cid:104) s (cid:105)(cid:105) -module.Proof. This is an immediate consequence of the two previous propositions, since Λ (cid:104)(cid:104) s (cid:105)(cid:105) is aPID. (cid:3) Remark 7.8.
It follows from our argument that (7.3) forms a basis of A , since this is thecase for s = 0 . In addition to this, it follows from the proof of Proposition 7.5, that any ρ ∈ A with non-negative valuation, can be written as ρ = N (cid:88) i =1 c i γ i + (cid:88) j =1 t j f j with c i ∈ Λ , val ( c i ) ≥ − and t j ∈ Λ (cid:104)(cid:104) s, x, y, z (cid:105)(cid:105) , val ( t j ) ≥ − . Injectivity.
Now we are ready to prove injectivity of the Kodaira-Spencer map.
Proposition 7.9.
The Kodaira-Spencer map KS τ : QH ∗ orb ( X, τ ) → Jac( W τ ) is injective, andhence a ring isomorphism.Proof. We have already established that KS τ is a surjective ring homomorphism, so we needonly to compare the ranks of the quantum cohomology and the Jacobian ring, to proveinjectivity. It follows from the definition that H ∗ orb (cid:0) P a,b,c (cid:1) has rank a + b + c − Λ Jac( W lead ) = dim Λ A / (cid:104) s (cid:105) = dim Λ A / (cid:104) s − (cid:105) = dim Λ Jac( W τ ) . We know, by Proposition 7.4, that Jac( W lead ) has rank a + b + c −
1, therefore Jac( W τ ) alsohas rank a + b + c −
1, which implies the result. (cid:3)
Remark 7.10.
In fact we have shown that (7.3) forms a basis of
Jac( W τ ) . Moreover, asexplained in Remark 7.8, it follows that any ρ ∈ Jac( W τ ) with non-negative valuation, can bewritten as ρ = N (cid:88) i =1 c i γ i + t ∂ x W τ + t ∂ y W τ + t ∂ z W τ . with c i ∈ Λ , val ( c i ) ≥ − and t j ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) , val ( t j ) ≥ . Calculations
Euler vector field.
Let χ be the Euler characteristic of P a,b,c . We have c ( P a,b,c ) = χ [pt]. Theorem 8.1.
Under the Kodaira-Spencer map KS τ : QH ∗ orb ( X, τ ) → Jac( W τ ) , KS τ (cid:32) χ [pt] + (cid:88) k (1 − ι k ) τ k T k (cid:33) = [ W τ ] where T k form a basis of twisted sectors and τ k are the corresponding coordinates.Proof. Recall that KS τ (pt) is defined by counting discs with one interior point passing throughpt and one boundary output point to the unit. Since the total area of P a,b,c equals to 8 A , theimage of pt equals to 18 A T · ∂ W τ (˜ x, ˜ y, ˜ z ) ∂T written in terms of the geometric variables (˜ x, ˜ y, ˜ z ) corresponding to the immersed generators. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 39
Let T mA (cid:16)(cid:81) j τ l j j (cid:17) ˜ x n ˜ y n ˜ z n be a term in W τ (˜ x, ˜ y, ˜ z ). Using Proposition 4.6, (cid:32) χ · A · T ∂∂T + (cid:88) j (1 − ι j ) τ j ∂∂τ j (cid:33) · T mA (cid:32)(cid:89) j τ l j j (cid:33) ˜ x n ˜ y n ˜ z n = (cid:32) mχ (cid:88) j l j (1 − ι j ) (cid:33) T mA (cid:32)(cid:89) j τ l j j (cid:33) ˜ x n ˜ y n ˜ z n = (cid:18) − n a − n b − n c + 3 χ · ( n + n + n ) (cid:19) T mA (cid:32)(cid:89) j τ l j j (cid:33) ˜ x n ˜ y n ˜ z n = (cid:18)(cid:18) χ − a (cid:19) ∂∂x + (cid:18) χ − b (cid:19) ∂∂y + (cid:18) χ − c (cid:19) ∂∂z (cid:19) · T mA (cid:32)(cid:89) j τ l j j (cid:33) ˜ x n ˜ y n ˜ z n + T mA (cid:32)(cid:89) j τ l j j (cid:33) ˜ x n ˜ y n ˜ z n Hence (cid:32) χ · A · T ∂∂T + (cid:88) j (1 − ι j ) τ j ∂∂τ j (cid:33) · W τ (˜ x, ˜ y, ˜ z )= W τ (˜ x, ˜ y, ˜ z ) + (cid:18)(cid:18) χ − a (cid:19) ∂∂ ˜ x + (cid:18) χ − b (cid:19) ∂∂ ˜ y + (cid:18) χ − c (cid:19) ∂∂ ˜ z (cid:19) · W τ (˜ x, ˜ y, ˜ z ) . Changing back to the variables x = T ˜ x, y = T ˜ y, z = T ˜ z , the left hand side is KS τ (cid:32) χ [pt] + (cid:88) k (1 − ι k ) τ k T k (cid:33) . The right hand side equals to W τ ( x, y, z )+ (cid:16)(cid:0) χ − a (cid:1) ∂∂x + (cid:0) χ − b (cid:1) ∂∂y + (cid:0) χ − c (cid:1) ∂∂z (cid:17) W τ ( x, y, z )which is in the same class of W τ ( x, y, z ) in the Jacobian ideal. (cid:3) Versality of the potential.
The goal of this section is to prove the following statement,which describes the power series (up to a coordinate change) that can appear as the bulkdeformed potential W τ . Theorem 8.2.
Consider P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( P − W lead ) > . Then there exist τ (cid:48) ∈ H ∗ orb ( P a,b,c , Λ ) and a coordinate change ( x (cid:48) , y (cid:48) , z (cid:48) ) such that P ( x (cid:48) , y (cid:48) , z (cid:48) ) = W τ (cid:48) . In order to prove this proposition we first need three lemmas.
Lemma 8.3 (Refined surjectivity) . For any P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( P ) ≥ , there is ρ ∈ QH ∗ orb ( X, τ ) with val ( ρ ) ≥ such that KS τ ( ρ ) = P + t ∂ x W τ + t ∂ y W τ + t ∂ z W τ , for t , t , t ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( t , t , t ) ≥ . Proof.
Take λ > W τ = W lead mod T λ . We will show thereis ρ ∈ QH ∗ orb ( X, τ ) and t , t , t ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) of non-negative valuation such that(8.1) KS τ ( ρ ) − P − t ∂ x W τ − t ∂ y W τ − t ∂ z W τ = T λ Q, for some Q ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) of non-negative valuation. It is enough to consider the case P = x i y j z k .If at least two of ( i, j, k ) are non zero we take ρ = (cid:4) a (cid:5) • τ i • τ (cid:4) b (cid:5) • τ j • τ (cid:4) c (cid:5) • τ k . Then val ( KS τ ( ρ )) ≥
0. In fact, it follows from Lemma 4.4, that KS τ has non-negative valuationon the standard basis of QH ∗ orb ( X, τ ) except on the point class pt, in which case it is − ρ , hence val ( ρ ) ≥ val ( KS τ ( ρ )) ≥ KS τ ( ρ ) − KS τ (cid:0)(cid:4) a (cid:5)(cid:1) i KS τ (cid:0)(cid:4) b (cid:5)(cid:1) j KS τ (cid:0)(cid:4) a (cid:5)(cid:1) k has non-negative valuation and as KS τ is a ring map, it is in the Jacobian ideal. It then follows from Remark 7.10 that there are t , t , t ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) of non-negative valuation such that KS τ ( ρ ) − KS τ (cid:18)(cid:22) a (cid:23)(cid:19) i KS τ (cid:18)(cid:22) b (cid:23)(cid:19) j KS τ (cid:18)(cid:22) a (cid:23)(cid:19) k = t ∂ x W τ + t ∂ y W τ + t ∂ z W τ . It follows from Lemma 4.4, that KS τ (cid:0)(cid:4) a (cid:5)(cid:1) i KS τ (cid:0)(cid:4) b (cid:5)(cid:1) j KS τ (cid:0)(cid:4) a (cid:5)(cid:1) k = x i y j z k + T λ Q , whichproves (8.1), in this case.If i < a , j = k = 0 we take ρ = (cid:4) a (cid:5) • τ i . Using the fact that (cid:4) a (cid:5) • τ i = (cid:4) ia (cid:5) mod T and thesame argument as above we find t , t , t ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) of non-negative valuation such that KS τ ( ρ ) − KS τ (cid:18)(cid:22) a (cid:23)(cid:19) i = t ∂ x W τ + t ∂ y W τ + t ∂ z W τ . Again it follows from Lemma 4.4, that KS τ (cid:0)(cid:4) a (cid:5)(cid:1) i = x i + T λ Q , which proves (8.1), in thiscase.If i = a , we take ρ = pt /a . Recall that y∂ y W τ = − T − xyz + by b mod T λ and z∂ z W τ = − T − xyz + cz c mod T λ . Using Lemma 4.4, we easily compute KS τ ( ρ ) + 58 a x∂ x W τ − a y∂ y W τ − a z∂ z W τ = x a mod T λ , which is equivalent to (8.1). Here, we used the computation in the proof of Theorem 8.1which tells us that KS τ ( ρ ) = 1 a KS τ (pt) ≡ x a + 3 b a y b + 3 c a z c − T − a xyz mod T λ . Finally, if i = na + i , j = k = 0, with n > , i < a , we take ρ = (pt /a ) (cid:63) τ n • τ (cid:4) a (cid:5) • τ i .Only non-constant spheres contribute to the product defining ρ so we can argue as in thefirst case. The remaining cases follow by the exact same arguments.Now that we have established Equation (8.1), we can proceed by induction, as in the proofof Proposition 7.2, to complete the proof. (cid:3) Lemma 8.4.
Let P ∈ Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) with val ( P − W lead ) ≥ and define G ( s, x, y, z, τ ) := W τ ( x, y, z ) + s ( P ( x, y, z ) − W lead ) ∈ Λ (cid:104)(cid:104) s, x, y, z, τ (cid:105)(cid:105) . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 41
There exist c i ∈ Λ (cid:104)(cid:104) s, τ (cid:105)(cid:105) , and t , t , t ∈ Λ (cid:104)(cid:104) s, x, y, z, τ (cid:105)(cid:105) with val ( c i ) , val ( t j ) > such that ∂G∂s = (cid:88) i c i ∂G∂τ i + t ∂G∂x + t ∂G∂y + t ∂G∂z . Proof.
We will prove the following more general statement: given Q ∈ Λ (cid:104)(cid:104) x, y, z, τ, s (cid:105)(cid:105) with val ( Q ) ≥ c i and t j as in the statement with valuation greater or equal than zerosuch that(8.2) Q = (cid:88) i c i ∂G∂τ i + t ∂G∂x + t ∂G∂y + t ∂G∂z . Which easily implies the lemma. It is enough to consider the case of Q ∈ Λ (cid:104)(cid:104) x, y, z, τ (cid:105)(cid:105) . Weproceed in two steps:Step 1: At s = 0, ∂G∂τ i = KS τ ( e i ) and ∂ x G = ∂ x W τ , ∂ y G = ∂ y W τ , ∂ z G = ∂ z W τ . ApplyingLemma 8.3 to Q we obtain t j and ρ = (cid:80) i c i e i satisfying Equation (8.2).Step 2: (Similar to Proposition 7.5) By Step 1 ,there are c (0) i and t (0) j satisfying Equation(8.2). By definition, ∂ x G = ∂ x W τ + sT α f for some α > f with valuation greater orequal than zero. Similarly for y and z . Hence Q = (cid:88) i c (0) i ∂G∂τ i + t (0)1 ∂G∂x + t (0)2 ∂G∂y + t (0)3 ∂G∂z − sT α (cid:88) j t (0) j f j . Next, we apply Step 1 to Q (1) := (cid:80) j t (0) j f j gives c (1) i and t (1) j which allows us to rewrite Q as (cid:88) i ( c (0) i − sT α c (1) i ) ∂G∂τ i +( t (0)1 − sT α t (1)1 ) ∂G∂x +( t (0)2 − sT α t (1)2 ) ∂G∂y +( t (0)3 − sT α t (1)3 ) ∂G∂z + s T α Q (2) By induction, we construct c ( n ) i and t ( n ) j and define c i := (cid:80) n c ( n ) i s n T nα and t j := (cid:80) n t ( n ) j s n T nα .From the construction, it is clear these satisfy Equation (8.2). (cid:3) The next lemma is a general result about the existence of coordinate changes by inte-grating a vector field in our non-archimedean setting. It should be well known to experts.We include a proof for completeness. We will use the short-hand notation Λ (cid:104)(cid:104) x, τ (cid:105)(cid:105) :=Λ (cid:104)(cid:104) x , . . . , x n , τ , . . . τ m (cid:105)(cid:105) . Lemma 8.5.
Consider A j ∈ Λ (cid:104)(cid:104) s, x, τ (cid:105)(cid:105) and B i ∈ Λ (cid:104)(cid:104) s, τ (cid:105)(cid:105) with valuations ≥ (cid:15) > and let X be the vector field X := (cid:88) j A j ∂∂x j + (cid:88) i B i ∂∂τ i . Then there exists a coordinate change Φ( s, x, τ ) = ( s, ψ ( s, x, τ ) , ϕ ( s, τ )) , with Φ(0 , x, τ ) =(0 , x, τ ) and d Φ ds ( s, x, τ ) = X (Φ( s, x, τ )) . Proof.
Simplifying the notation, we have to show that there is ( ψ s ( x, τ ) , ϕ s ( τ )) such that dds (( ψ s ( x, τ ) , ϕ s ( τ ))) = ( A ( s, ψ s ( x, τ ) , ϕ s ( τ )) , B ( s, ϕ s ( τ ))) and ( ψ ( x, τ ) , ϕ ( τ )) = ( x, τ ). This is equivalent to(8.3) ( ψ s ( x, τ ) , ϕ s ( τ )) − ( x, τ ) = (cid:18)(cid:90) s A ( u, ψ u ( x, τ ) , ϕ u ( τ )) du, (cid:90) s B ( u, ϕ u ( τ )) du ) (cid:19) . We define a sequence Φ k := ( ψ k , ϕ k ) inductively as ( ψ , ϕ ) = ( x, τ ) and( ψ k +1 s ( x, τ ) , ϕ k +1 s ( τ )) = (cid:18)(cid:90) s A ( u, ψ ku ( x, τ ) , ϕ ku ( τ )) du, (cid:90) s B ( u, ϕ ku ( τ )) du ) (cid:19) . By assumption there is (cid:15) > val ( A ) , val ( B ) ≥ (cid:15) . We claim that val (Φ k − Φ k − ) ≥ k(cid:15) . We prove it by induction on k . First note that val (Φ k − Φ k − ) = val ( (cid:90) s F ( u, Φ k − ) − F ( u, Φ k − ) du ) , where F = ( A, B ). Then by induction Φ k − = Φ k − + T ( k − (cid:15) ρ for some ρ of non-negativevaluation. Hence F ( u, Φ k − ) − F ( u, Φ k − ) = T (cid:15) T ( k − (cid:15) ˜ ρ , for some ˜ ρ of non-negative valuation,which conclude the induction step.This argument shows that ( ψ, ϕ ) := ( ψ , ϕ ) + (cid:80) k ≥ ( ψ k , ϕ k ) − ( ψ k − , ϕ k − ) converges. Byconstruction, it is a coordinate change. Obviously ( ψ, ϕ ) = lim k ( ψ k , ϕ k ) and therefore itsolves (8.3). (cid:3) Proof of Theorem 8.2.
Given P , define G as in Lemma 8.4 and let t j , c i be the series providedby that lemma. Let X be the vector field X := (cid:88) j − t j ∂∂x j + (cid:88) i − c i ∂∂τ i , and let Φ( s, x, τ ) be the coordinate change provided by Lemma 8.5. By construction X · G = − ∂G∂s which implies dds ( G (Φ( s, x, y, z, τ ))) = 0. Hence G (0 , x, y, z, τ ) = G (Φ(1 , x, y, z, τ )).Using the notation Φ( s, x, y, z, τ ) = ( s, ψ s ( x, y, z, τ ) , ϕ s ( τ )), this is equivalent to G (0 , ψ − ( x, y, z, τ ) , ϕ − ( τ )) = G (1 , x, τ ) . Evaluating at τ = 0 we obtain W ϕ − (0) ( ψ − ( x, y, z, W lead ( x, y, z ) + P ( x, y, z ) − W lead ( x, y, z ). Denoting ϕ − (0) = τ (cid:48) and ( x (cid:48) , y (cid:48) , z (cid:48) ) = ψ ( x, y, z,
0) we get the desired equality W τ (cid:48) ( x, y, z ) = P ( x (cid:48) , y (cid:48) , z (cid:48) ) . (cid:3) Valuation of critical points.
Recall that instead of working with geometric variablesfor the immersed sectors (˜ x, ˜ y, ˜ z ), we switched to new variables ( x, y, z ) which were definedby x = T ˜ x, y = T ˜ y, z = T ˜ z . The Jacobian ring for W τ ( x, y, z ) and W τ (˜ x, ˜ y, ˜ z ) are a prioridifferent. We show that indeed, there is an example that the Kodaira-Spencer map is not anisomorphism if we consider the map to the Jacobian ring of W τ (˜ x, ˜ y, ˜ z ).This can happen because of the following. Before bulk deformation, at critical pointsof W τ (˜ x, ˜ y, ˜ z ), we have val (˜ x ) , val (˜ y ) , val (˜ z ) ≥
0. But after bulk deformations by twistedsectors τ = τ tw with small valuations, critical points of W τ (˜ x, ˜ y, ˜ z ) may begin to have negativevaluations, hence moving away from the disc of Novikov convergence Λ , (˜ x, ˜ y, ˜ z ) . We illustrate IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 43 such a phenomenon in the example ( a, b, c ) = (2 , , r ). In fact, in this example all the criticalpoints will escape. Proposition 8.6.
Consider the case X = P , ,r . There exists τ ∈ H ∗ orb ( X, Λ + ) such thatany critical point of the bulk-deformed potential W τ ( x, y, z ) has at least one coordinate whosevaluation is smaller than . Proposition 8.6 implies that every critical point of W τ (˜ x, ˜ y, ˜ z ) have at least one coordinatewith a negative valuation, and hence the conventional boundary deformation of L would notcapture these points. Proof.
Let us take τ = T λ (cid:4) (cid:5) a + T λ (cid:4) (cid:5) b supported on the two orbi-points with Z / λ < min (cid:8) , r − (cid:9) . The associated bulk-deformation only adds twoterms T λ x + T λ y to the non-bulk-deformed potential. To see this, recall that if the term x n y n z n appears in the potential, the contributing polygon satisfies the inequality n n n r + (cid:88) j (1 − ι j ) ≤ . by the area formula together with Lemma 4.4. In our case ι j = , and hence the polygoncan have at most one orbi-insertion. Any such polygon lifts to a holomorphic disc in theuniversal cover of P , ,r by Corollary 4.3, and the lift must be invariant under the groupaction corresponding to either (cid:4) (cid:5) a or (cid:4) (cid:5) b . It is easy to see that the 2-gons responsiblefor x and y are only such (see diagrams in [CHKL17, Section 12]), and their halves areorbi-discs producing T λ x + T λ y in W τ .More concretely, the resulting bulk-deformed potential is given as W τ ( x, y, z ) = − T − xyz + x + y + z r + f ( z ) + T λ x + T λ y where f ( z ) is a polynomial in z of the form f ( z ) = c T z r − + c T z r − + · · · = (cid:98) r (cid:99) (cid:88) k =1 c k T k z r − k for some combinatorially defined integers c k . The precise value of c k , which can be found in[CHKL17, Theorem 12.2], is not important to us, but we will use the fact that val ( c k ) = 0 inthe argument below.Given the formula, critical points of W τ satisfy(8.4) − T − yz + 2 x + T λ = 0 − T − xz + 2 y + T λ = 0 − T − xy + rz r − + f (cid:48) ( z ) = 0 . Subtracting the second equation from the first gives( x − y )( T − z + 2) = 0 , and hence, either z = − T or x = y . (i) If z = − T , then 2( x + y ) = − T λ and T − xy = CT r − for some constant C with val ( C ) = 0. Thus x and y are solutions of the quadratic equation (in s ) s + 12 T λ s + CT r − = 0 , which are − T λ ± T λ (cid:112) − CT r − − λ . In particular, one of x and y always has valuation λ , which is smaller than 3.(ii) Consider the case of x = y . The second equation in (8.4) reads x ( − T − z + 2) = − T λ . If val ( z ) ≥
8, then val ( x ) = λ <
3, we are done.Now suppose val ( z ) <
8, which implies val ( x ) + val ( z ) = λ + 8. Therefore(8.5) val ( x ) = λ + 8 − val ( z )On the other hand, the third equation of (8.4) gives(8.6) T − xy = rz r − + f (cid:48) ( z ) = rz r − + (cid:98) r − (cid:99) (cid:88) k =1 ( r − k ) c k T k z r − k − , and the valuation of monomials in f (cid:48) ( z ) can be estimated as val ( T k z r − k − ) = 16 k + ( r − k − val ( z )= ( r − val ( z ) + 2 k (8 − val ( z )) > ( r − val ( z ) . Therefore the right hand side of (8.6) has valuation ( r − val ( z ), and we obtain(8.7) − val ( x ) = ( r − val ( z )Combining (8.5) and (8.7), we conclude that val ( z ) = λ +8 r +1 , which is again smallerthan 3.Therefore, at least one coordinate of any critical point ( x, y, z ) of W τ has valuation smallerthan 3, and this proves the claim. (cid:3) In ( x, y, z ) coordinates, these critical points still have val ( x ) , val ( y ) , val ( z ) ≥ val ( τ ) ≥
0, so it does not violate the isomorphism KS τ : QH ∗ orb ( X, τ ) ∼ = Jac( W τ ( x, y, z )). Remark 8.7.
Although geometric variables ˜ x, ˜ y, ˜ z with negative valuations are not legiti-mate in Floer theory of L , we speculate that such a deformation can be replaced by anotherLagrangian using the gluing procedure explained in [CHL18] . Explicit computation of KS τ for P , , without bulk-parameters. In this section,we give an explicit computation of the Kodaira-Spencer map for P , , without bulk-insertions.Namely, we set τ = 0 throughout the section. For notational simplicity, let us write X for P , , from now on.We use the following notations for generators of QH ∗ orb ( X ) := QH ∗ orb ( X, X tobe the unit class, and denote twisted sectors by ∆ / i and ∆ / i for i = 1 , , i indicatesorbifold points. We introduce these new notations to avoid the potential confusion due tothe coincidence a = b = c = 3 in this case. In the previous terminology, they all happen to IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 45
Figure 6. P , , and its coverbe (cid:4) i (cid:5) . ∆ / i has degree shifting number ι (∆ / i ) = 1 /
3, and ∆ / i has degree shifting number ι (∆ / i ) = 2 / , · · · , pt on P , , which are not orbifoldsingular points, as shown in Figure 6. Then we define the point class pt to be the average ofthese 8 points. i.e. pt := (cid:80) i pt i . This is to make the calculation of KS ([pt]) easier. Noticethat such a choice makes the number of [pt]-insertions proportional to the symplectic area ofthe contributing polygons. Then X , [pt] , ∆ / i , ∆ / i for i = 1 , , QH ∗ orb ( X ).On the other hand, in [CHL17], a Morse model was adopted for CF ( L , L ) together withthe combinatorial sign rule in [Sei11], which produces the explicit formula for the potential W given as(8.8) W = ˜ φ ( T )(˜ x + ˜ y + ˜ z ) − ˜ ψ ( T )˜ x ˜ y ˜ z where T is the (exponentiated) area of the minimal triangle as before and(8.9) ˜ φ ( T ) = ∞ (cid:88) k =0 ( − k (2 k + 1) T (6 k +3) ˜ ψ ( T ) = T + ∞ (cid:88) k =1 ( − k (cid:16) (6 k + 1) T (6 k +1) − (6 k − T (6 k − (cid:17) See [CHL17] for detailed computation . We then make the change of coordinate (4.1) toobtain W ( x, y, z ) = φ ( T )( x + y + z ) − ψ ( T ) xyz with φ ( T ) = (cid:80) ∞ k =0 ( − k (2 k + 1) T k +36 k ψ ( T ) = T − (cid:16) (cid:80) ∞ k =1 ( − k (cid:16) (6 k + 1) T k +12 k − (6 k − T k − k (cid:17)(cid:17) . To be more precise, (8.8) is obtained by changing ˜ x and ˜ z to − ˜ x and − ˜ z in the formula therein. ˜ x ,˜ y , ˜ z did not appear to be symmetric in [CHL17], due to some asymmetric choice of the input data for thecombinatorial sign rule. Recall that the map KS : QH ∗ orb ( X ) → Jac( W ) is defined by sending a cycle C to thepolynomial class in Jac( W ) represented by(8.10) (cid:88) β (cid:88) k ≥ q β n k ( β ; C ; b, . . . , b )where b = xX + yY + zZ (= T ˜ xX + T ˜ yY + T ˜ zZ ). (8.10) is a series in x, y, z in general,but we will see from explicit calculations that it is just a polynomial (over the Novikov field)for C = X , pt , ∆ / i , ∆ / i , i = 1 , ,
3. By dimension counting (2.3) restricted to this case, n k ( β ; ∆ / i ; b, . . . , b ) (resp. n k ( β ; ∆ / i ; b, . . . , b )) is non-zero only when µ CW = 2 / µ CW = 4 / X has a manifold cover ˜ X , (which is the unique elliptic curve E thatadmits Z / (cid:101) L of L in ˜ X is a Z / (cid:101) L , which arerepresented as dotted lines along three different directions in Figure 6. Recall that any orbi-disc in our interest should lift to ˜ X by Corollary 4.3. We will need the following classificationof orbi-discs for explicit calculations: Proposition 8.8.
Let u be a holomorphic orbi-disc in X bounded by L of Maslov index / with one interior orbi-marked point passing through the twisted sector ∆ / (or ∆ / , ∆ / ),and suppose that u only passes through the immersed sectors X, Y, Z (but not ¯ X, ¯ Y , ¯ Z ). Then,it can be lifted to a holomorphic disc in ˜ X bounded by L of Maslov index two whose boundarypasses through the (preimage of ) immersed sector X (or Y , Z respectively) three times.If u is a holomorphic orbi-disc in X bounded by L of Maslov index / with one interiororbi-marked point passing through the twisted sector ∆ / (or ∆ / , ∆ / ), it can be lifted toa holomorphic disc in ˜ X bounded by (cid:101) L of Maslov index four whose boundary either passesthrough X (or Y , Z respectively) six times or passes through both ( Y, Z ) for three times (or ( X, Z ) , ( X, Y ) respectively).Proof. Let u be a holomorphic orbi-disc in X bounded by L of Maslov index 2 / / . The orbi-marked pointin the domain of u must have isotropy group Z / u can be lifted to a Z / u : (∆ , ∂ ∆) → ( ˜ X, (cid:101) L )with the domain of ˜ u covers the domain of u by the map ζ = ˜ ζ . Since u has Maslov index2 /
3, ˜ u has Maslov index two. Moreover ˜ u only passes through the immersed sectors X, Y, Z .Each of these immersed sectors contribute 2 / u can only passthrough three of them. By the Z / u pass through are all X .The proof for the second statement is similar. Consider the uniformizing disc ˜ u of the orbi-disc of Maslov index 4 /
3. ˜ u can only pass through six immersed sectors by the constraint onMaslov index. Unless ˜ u is constant, Z / u , and hence the six corners of˜ u can only pass through at most two distinct immersed sectors. Thus the immersed sectors ˜ u passes through are either all X (or all Y or all Z ) or three copies of Y and Z (or three copiesof X and Y , or three copies of Z and X ). (See Figure 8 for the shape of these orbi-discs.)To see this, we first fix the twisted sector ∆ / , and choose one of its pre-images in C , say p a . Pick any point (cid:101) Y ∈ C corresponding to the immersed sector Y . Then, if ˜ u pass through IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 47 p a (at the orbifold point) and has an immersed boundary at (cid:101) Y , then we know that ˜ u alsopasses through Z / (cid:101) Y with respect to p a , say (cid:101) P , (cid:101) Q . These two pointsstill corresponds to immersed sector Y . Then, remaining immersed sectors should connecttwo points out of (cid:101) Y , (cid:101)
P , (cid:101) Q along the lifts of the Lagrangian. It is not hard to check thatsuch immersed sector correspond to Z . The remaining cases are similar and we omit thedetails. (cid:3) By calculating contribution from each orbi-disc in the above classification, we can explicitlycalculate the map KS : QH ∗ orb ( X ) → Jac( W ) as follows. Proposition 8.9.
The map KS from the orbifold quantum cohomology of X to the Jacobianring of W defined in (8.10) is given by X (cid:55)→ , [pt] (cid:55)→ T ∂∂T W, ∆ / (cid:55)→ P ( T ) x ∆ / (cid:55)→ P ( T ) y ∆ / (cid:55)→ P ( T ) z , ∆ / (cid:55)→ Q ( T ) x + R ( T ) yz ∆ / (cid:55)→ Q ( T ) y + R ( T ) zx ∆ / (cid:55)→ Q ( T ) z + R ( T ) xy . where P ( T ) = ∞ (cid:88) k =0 ( − k (2 k + 1) T k +12 k ,Q ( T ) = ∞ (cid:88) k =0 (2 k + 1) T k +24 k + ∞ (cid:88) k =1 k − (cid:88) i =0 ( − k − i (6 k − i + 2) T k +36 k − i − i ,R ( T ) = ∞ (cid:88) k =1 k − (cid:88) i =0 ( − k − i T − (cid:16) (6 k − i ) T k +12 k − i − i − (6 k − i − T k − k − i − i (cid:17) . The proof will be given in Appendix C.
Remark 8.10.
Satake-Takahashi [ST11] gave an explicit description of the genus zero Gromov-Witten potential of P , , , which, in particular, determines the structure constants for theproduct structure on QH ∗ orb ( P , , ) . For instance, one of the interesting quantum products isgiven by ∆ / • ∆ / = f ( q )∆ / , where f ( q ) given in [ST11] is an expression involving Dedekind eta function. This gives acomplicated looking identity on Jac( W ) -side through our explicit map, which is a priori highlynontrivial. Appendix A. Proof of Proposition 6.5
Proposition 6.5 is equivalent to the following equalities in the Jacobian ring from theDefinition 6.3 of Kodaira-Spencer map and the potential.(A.1) (cid:88) β,k,l exp( τ p ∩ β ) T β ∩ ω l ! q l +1 ,k,β ( X , τ ltw ; b k ) = L (A.2) (cid:88) β,k,l exp( τ p ∩ β ) T β ∩ ω l ! q l +1 ,k,β ( Q , τ ltw ; b k ) = (cid:88) β,k,l exp( τ p ∩ β ) T β ∩ ω l ! ( Q ∩ β ) q l,k,β ( τ ltw ; b k )The proof of both statement is similar, we will start with the second one. The maintechnical issue is that the Kuranishi structure which is used to define bulk deformation aswell as Kodaira-Spencer map may not be compatible with the operation of forgetting interiormarked point. To overcome this problem, we will construct Kuranishi structures and CFperturbations on the spaces M parak +1 ,l +1 ( β, Q , τ tw , γ ) := M k +1 ,l +1 ( β, Q , τ tw , γ ) × [0 , ∂ M parak +1 ,l +1 ( β, Q , τ ltw , γ ) × [0 ,
1] = M k +1 ,l +1 ( β, Q , τ ltw , γ ) × { } (cid:91) M k +1 ,l +1 ( β, Q , τ tw , γ ) × { } (cid:91) β + β = βk + k = k, ≤ i ≤ k +1 M parak +1 ,l +1 ( β ; Q , τ l tw , γ ) ev × ev i M k +2 ,l ( β , τ l tw , γ )(A.3) (cid:91) β + β = βk + k = k, ≤ i ≤ k +1 M k +1 ,l ( β , τ l tw , γ ) ev × ev i M parak +2 ,l +1 ( β ; Q , τ l tw , γ )On the factors with no Q insertion in the second and third line in Equation A.3 the Kuran-ishi structure and CF perturbations coincide with the ones used to define the m τk operations.On the component M k +1 ,l +1 ( β, Q , τ tw , γ ) × { } the Kuranishi structure and CF perturbationscoincide with ones used to define the maps q l +1 ,k,β and are used in Section 6.2 and cruciallyrespect the decomposition in (6.2). Finally, on the component M k +1 ,l +1 ( β, Q , τ ltw , γ ) × { } we require compatibility with the map that forgets the interior marked point where we insertthe divisor Q π : M k +1 ,l +1 ( β, Q , τ tw , γ ) → M k +1 ,l ( β, τ ltw , γ ) . The notion of compatibility we use here is a variation of the notions considered in [Amo17,Def. 3.1] and [FOOO09, Sec 7.3.2]. We say the Kuranishi structures are compatible (withrespect to π ) if for every u ∈ M k +1 ,l +1 ( β, Q , τ ltw , γ ) and v = π ( u ), there is a map between theKuranishi neighborhoods ( V u , E u , Γ u , s u , ψ u ) and ( V v , E v , Γ v , s v , ψ v ) satisfying the following • h uv : Γ u → Γ v is an injective homomorphism; • V u ∼ = V v × B where B is a ball in C and ϕ uv : V u → V v is h uv -equivariant continuousmap, strata-wise smooth; • an isomorphism E u (cid:39) ϕ ∗ uv E v ⊕ N where N is a rank two bundle; • the ϕ ∗ uv E v component of s u equals ϕ ∗ uv s v ; • ϕ ◦ ψ u = ψ v ◦ ϕ uv on s − u (0) / Γ u . Lemma A.1.
There are Kuranishi structures on the moduli spaces M parak +1 ,l +1 ( β, Q , τ ltw , γ ) which respect the boundary decomposition (A.3) and have the compatibilities described above.Proof. With the exception of the compatibility with the forgetful map π , the construction ofsuch Kuranishi structure is standard by now, see [Fuk10] for example. To ensure compatibilitywith π we proceed as follows. Given the Kuranishi neighborhood ( V v , E v , Γ v , s v , ψ v ) we take V u ∼ = V v × B where B parameterizes the position of the additional marked point z +1 . Thenthe map ϕ uv is locally modeled on a forgetful map Π : M k +1 ,l (cid:48) +1 → M k +1 ,l (cid:48) , see the proof ofProposition 4.2 in [Amo17] for an analogous argument. Then taking the obstruction bundle IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 49 ϕ ∗ uv E v would give a Kuranishi neighborhood in M k +1 ,l +1 ( β, τ ltw , γ ), that is, without incidencerelation with Q . We include this restriction by identifying a neighborhood of ev z +1 ( u ) (whichincludes no other component of Q ) in P a,b,c with a ball in R ∼ = N . Then the N componentof the obstruction map s u ( x ) is ev z +1 ( x ) − ev z +1 ( u ). It is not hard to see this satisfies all theproperties. (cid:3) Remark A.2.
As explained in Appendix A.1.4 in [FOOO09] , when constructing Kuranishistructures on moduli spaces of discs one has to take a special smooth structure near nodaldiscs. Due to this choice, forgetful maps are continuous but smooth only when we restrict toa stratum of the stratification according to combinatorial type of the underlying disc. This isthe reason ϕ uv is only strata-wise smooth. Now consider a continuous family of multisections ( U α , W α , S α ) α ∈ I on M parak +1 ,l +1 ( β, Q , τ ltw , γ ).We say it is compatible with π if its restriction to M k +1 ,l +1 ( β, Q , τ ltw , γ ) × { } is the pull-back of a continuous family of multisections ( V β , W β , S β ) β ∈ J on M k +1 ,l ( β, τ ltw , γ ). By pull-back wemean there are maps of Kuranishi neighborhoods from U α to V β ( α ) , W α = W β ( α ) , θ α = θ β ( α ) and ϕ αβ induces a covering map S − α,i,j (0) → S − β,i,j (0). Lemma A.3.
There are continuous families of multisections ( U α , W α , S α ) α ∈ I on the modulispaces M parak +1 ,l +1 ( β, Q , τ ltw , γ ) which, given the decomposition of the boundary A.3 , the restric-tion of the multisections to the boundary agrees with the fiber product of multisections on theright-hand side of
A.3 . Moreover they are compatible with π and the evaluation at the 0-thboundary marked point maps ( ev ) α | S − α (0) are submersions.Proof. Once again the proof follows the strategy in [Fuk10] and [Amo17, Proposition 4.4].We take the continuous family of multisections on M k +1 ,l ( β, τ ltw , γ ) and define W α = W β ( α ) , θ α = θ β ( α ) . On the ϕ ∗ αβ E β component we take the map S α = S β ◦ ϕ αβ . On the N componentwe take a generic perturbation of Q so that it becomes transversal to the image of ev z +1 . Withthis choice, ϕ αβ induces a natural covering map S − α,i,j (0) → S − β,i,j (0). Please note if considerthe union over all α over a fixed β the resulting covering map has exactly Q ∩ β sheets. Theremainder of the proof follows the usual induction argument on energy, see [Fuk10]. (cid:3) Equipped with these CF of perturbations on M parak +1 ,l +1 ( β, Q , τ ltw , γ ) we can use the evalu-ation maps at the boundary marked points to define new operations on the Fukaya algebraand set F b,τ = (cid:88) k,β,l T ω ∩ β l ! ( ev ) ∗ ( ev ∗ b ∧ · · · ∧ ev ∗ k b ) . Now we apply the Stokes theorem [FOOO11, Lemma 12.13] to M parak +1 ,l +1 ( β, Q , τ ltw , γ ).Please note that the terms coming from the second line in (A.3) contribute with zero since m τ,b is unital. Also the terms in the first line of (A.3) give respectively the left and right-handside of Equation A.2, by the previous proof. Therefore we conclude that m τ,b ( F b,τ ) is exactlythe difference between the two side of (A.2). Combining this with Proposition 4.14 provesthe desired statementThe proof of (A.1) is entirely analogous. The main difference is that in this case whendescribing compatibility with π there is no extra component N is the obstruction bundle. Therefore the induced map S − α,i,j (0) → S − β,i,j (0) is a submersion of positive codimension andthe corresponding operation gives zero (see [Amo17, Proposition 3.7]). There is one exception,when β = 0 and k = l = 0 in which case it is easy to see that we obtain L . This proves thatthe two sides of A.1) differ by a m τ,b coboundary which proves the result. Appendix B. Proof of Theorem 7.3 (2)
We give a proof of Theorem 7.3 (2), here. First observe that we have the following basicrelations in A from the ideal (cid:104) g , g , g (cid:105) : aT x a − = yz, bT y b − = zx, cT z c − = xy. We will refer to these as the Jacobi relations.
Definition B.1.
Given a monomial in Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) , the operation replacing yz, zx, xy in themonomial by T x a − , T y b − , T z c − will be referred to as type I replacement, and replacing x a − , y b − , z c − by T − yz, T − zx, T − xy will be referred to as type II replacement. Each ofindividual replacements (as well as their corresponding relations in (cid:104) g , g , g (cid:105) ) will be calledby I x , I y , I z , II x , II y , II z , respectively. Hence, if we perform type I replacement a -times and type II replacement b times, then thethe original exponent of T is increased by 8 a − b . We will use the following properties of (cid:104) g , g , g (cid:105) , frequently. Lemma B.2.
If an expression x p y q z r − T m x p + i y q + j z r + k lies in the ideal (cid:104) g , g , g (cid:105) with p, q, r, i, j, k ≥ and m > , then so does x p y q z r itself.Proof. We have x p y q z r − T m x p + i y q + j z r + k = x p y q z r (1 − T m x i y j z k )and (1 − T m x i y j z k ) is invertible in Λ (cid:104)(cid:104) x, y, z (cid:105)(cid:105) . Hence the lemma follows. (cid:3) Lemma B.3.
For p, q, r, i, j, k ≥ , if an expression x p y q z r transforms to another expression x i y j z k by performing type I or II replacements then their difference lies in the ideal (cid:104) g , g , g (cid:105) .i.e. x p y q z r − x i y j z k = (cid:88) j =1 t j g j for some t j with val ( t j ) ≥ s , where s is the minimum valuation of the intermediate expressions(including both ends of the operation sequence).Proof. It directly follows from the fact that both of replacements are trivial modulo relationsin the ideal (cid:104) g , g , g (cid:105) . (cid:3) Let us now begin the proof of Theorem 7.3 (2). We divide the proof into a few differentcases (Lemma B.4, B.5, B.6 and B.7 below) depending on the type of ( a, b, c ). Lemma B.4. If a, b, c ≥ , then Theorem 7.3 (2) holds. IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 51
Proof.
Consider a monomial x i (cid:48) y j (cid:48) z k (cid:48) (with i (cid:48) , j (cid:48) , k (cid:48) ≥ i (cid:48) ≥ j (cid:48) ≥ k (cid:48) . First we consider the case that k (cid:48) (cid:54) = 0. Then i (cid:48) ≥ x i (cid:48) y j (cid:48) z k (cid:48) would be xyz ⊂ { γ , · · · , γ N } . Thus we can write i (cid:48) = i + 2 , j (cid:48) = j + 1 , k (cid:48) = k (with i, j, k ≥ z , I y , I x successively, we have x i y j z k ≡ c x i y j z k ( x ( T z c − )) ≡ c x i y j z k + c − T ( zx ) ≡ ( bc ) x i y j z k + c − T y b − ≡ ( bc ) x i y j + b − z k + c − T yz ≡ ( abc ) x i + a − y j + b − z k + c − T Hence the difference x i y j z k − ( abc ) x i + a − y j + b − z k + c − T = x i y j z k (1 − ( abc ) x a − y b − z c − T )lies in the ideal. Lemma B.3 tells us that the term in the right hand side lies in (cid:104) g , g , g (cid:105) .Therefore x i (cid:48) y j (cid:48) z k (cid:48) (= x i y j z k ) with k (cid:48) (cid:54) = 0 belongs to (cid:104) g , g , g (cid:105) by Lemma B.2.Now, let us consider the case for k (cid:48) = 0. If i (cid:48) ≥ , j (cid:48) ≥
1, then we can apply exactly thesame argument as above to prove that x i (cid:48) y j (cid:48) lies in the ideal. If i (cid:48) = 1 , j (cid:48) = 1, then we have x i (cid:48) y j (cid:48) = xy ≡ cT z c − ⊂ cT { γ , . . . , γ N } and thus the claim holds.We are left with the case when j (cid:48) = k (cid:48) = 0 and i (cid:48) ≥ a . If i (cid:48) = a , then x a = a T − xyz andhence the claim still holds. If i (cid:48) = a + i with i ≥
1, then x i (cid:48) = x a + i ≡ (1 /a ) T − xyz · x i = (1 /a ) T − x i +1 yz We have already shown that x i +1 yz is an element in the ideal which can be written as (cid:80) nj =1 t j g j with val ( t j ) ≥
0. Therefore, x i (cid:48) = (1 /a ) T − x i +1 ( g (cid:122) (cid:125)(cid:124) (cid:123) aT x a − − yz ) + T − (cid:88) j =1 t j g j = (cid:88) j =1 t (cid:48) j g j with val ( t (cid:48) j ) ≥ − (cid:3) Lemma B.5. If ( a, b, c ) = (2 , , c ) ( c ≥ , then Theorem 7.3 (2) holds.Proof. For simplicity, we represent the monomial x i y j z k by its exponent vector ( i, j, k ) inwhat follows. Our argument splits into a few different cases depending on which entries ofthe vector vanish. Below, i, j, k are all assumed to be positive integers.( i, , k ): By first applying I y and later I x (or I z ), we can make it into ( i, , k − i − , , k ))Repeating the procedure, we can reduce it to one of the basis element (by type Ireplacements only).(0 , j, k ): This follows from the previous case by symmetry of (2 , , n ). Again, we only needtype I replacements.(0 , j, j ≥
3. We first apply II y to get (1 , j − , z to get(0 , j − , c ). Since we have applied each of type I and II exactly once, the exponentof T remains zero, and we can now apply the previous case of (0 , j, k ). The sameargument can be used for ( i, , , , k ): We may assume k ≥ c + 1. We can proceed as(0 , , k ) ∼ II z (1 , , k − c + 1) ∼ I y (0 , , k − c )to go back to one of the previous cases. ( i, j, i ≥
2. We have ( i, j, ∼ I z ( i − , j − , c − ∼ I y ( i − , j, c − I z as many times as needed to get ( ∗ , , ∗ ) or (0 , ∗ , ∗ ) and we goback to one of the previous cases.( i, j, k ): We use induction on i + j + k and ( i, j, k ) ∼ I x ( i + 1 , j − , k − i + 1 , j − , k −
1) if all entries are non-zero or one of theabove steps otherwise. (cid:3)
Lemma B.6. If ( a, b, c ) = (2 , , c ) , then Theorem 7.3 (2) holds.Proof. This is the most elaborate case. We claim that a given type of monomial is eitherequivalent to a basis element or to zero element modulo (cid:104) g , g , g (cid:105) by applying type I and IIreplacements. Since we also need to control the valuation of t j in (7.4), the type II replacementshould be applied carefully. It will be always coupled with the type I to compensate the energy.Here, we only consider c ≥ c = 2 has already been covered by LemmaB.5. i, j, k are all assumed to be positive integers, below.(0 , j, k ): We further divide the case into two.(i) j ≤
2: The lowest possibly non-basis element is (01 , , , ∼ I x (1 , , k ≥
3, observe that(0 , , k ) ∼ I x (1 , , k − ∼ I y (0 , , k − , , k )for k ≥
1, for which we have(0 , , k ) ∼ I x (1 , , k − ∼ I x (2 , , k − ∼ I y (1 , , k − ∼ I z (0 , , k + c − ∼ I x (1 , , k + c − ∼ I y (0 , , k + c − . (For k = 1 , c ≥
6, by applyingLemma B.2 to the first and the last term, we obtain the claim. If c = 3 , , , k ) ∼ (0 , , k − , , k ) ∼ (0 , , k −
2) or (0 , , k ) ∼ (0 , , k − , , , (0 , , , (0 , ,
2) which was covered inthe first step. Note that we only uses type I in this case.(ii) j ≥
3: If j = 3, then (0 , , k ) ∼ I x (1 , , k + 1) ∼ I y (0 , , k ) and we are done by (i).Consider j ≥
4. The same argument as above shows that (0 , j, k ) ∼ (0 , j, k + c − c ≥
6, this shows the vanishing of the monomial modulo the relations byLemma B.2. Thus it is enough to consider the case that 3 ≤ c ≤
5. For c = 3,we run an induction on j to get (0 , j, , (0 , j, , (0 , j,
2) as in (i). Finally,(0 , j, ∼ II y (1 , j − , ∼ I z (0 , j − , c ) , (0 , j, ∼ II y (1 , j − , ∼ I z (0 , j − , c + 1) , (0 , j, ∼ II y (1 , j − , ∼ I z (0 , j − , c + 2) , and inductively, we go back to the case j ≤
3. The other case c = 4 , , j, , j, ∼ II y (0 , j − , c ) uses type II, but thelatter can be reduced without further energy loss. So this proves the claim.(0 , , k ): We can transform it to the first case since(0 , , k ) ∼ II z (1 , , k − c + 1) ∼ I y (0 , , k − c ) . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 53 ( i, , k ): If i ≥
3, ( i, , k ) ∼ I y ( i − , , k − ∼ I z , I z ( i − , , a − c − , so we can run induction on i to make i = 1 or i = 2. The case with i = 1 , i = 2 canbe handled easily as follows.(1 , , k ) ∼ I y (0 , , k − , (2 , , k ) ∼ I y (1 , , k − ∼ I z (0 , , c + k − . ( i, , i, , ∼ II x ( i − , , ∼ I z ( i − , , c )where the last term was covered in the previous step.( i, j, c ≥ i, j, ∼ I z ( i − , j − , c − ∼ I x ( i, j − , c − ∼ I y ( i − , j, c − ∼ I x ( i, j − , c − ∼ I x ( i + 1 , j − , c − ∼ I y ( i, j, c − . Thus, for c ≥
6, we have the vanishing of the monomial modulo (cid:104) g , g , g (cid:105) by com-paring two ends. If c = 3, then ( i, j, ∼ I z , I x , I y ( i − , j, c −
3) = ( i − , j, i . For c = 4 ,
5, we similarly run induction on j .( i, j, k ): We run induction on ( i + j + k ) for ( i, j, k ). Since ( i, j, k ) ∼ I x ( i + 1 , j − , k − j or k vanish by applying this operation repeatedly. (cid:3) Lemma B.7. If ( a, b, c ) = (2 , b, c ) with b, c ≥ , then Theorem 7.3 (2) holds.Proof. We again divide the argument by the type of the exponent of a monomial. Like before, i, j, k below are positive integers.(0 , j, k ): Note that (0 , j, k ) ∼ I z (1 , j − , k − ∼ I z (0 , j + b − , k − ∼ I x (1 , j + b − , k − ∼ I z (0 , j + b − , k + c − , and since b, c ≥
4, this shows that the monomial (0 , j, k ) is trivial modulo (cid:104) g , g , g (cid:105) if i ≥ , j ≥
3. The remaining case can be handled by(0 , , ∼ I x (1 , , , (0 , j, ∼ I x (1 , j − , ∼ I z (0 , j − , c − , (0 , j, ∼ I x (1 , j − , ∼ I z (0 , j − , c ) . (0 , j, j ≤ b , then it is a basis element, so we only consider j ≥ b + 1. In this case, wehave (0 , j, ∼ II y (1 , j − b + 1 , ∼ I z (0 , j − b, c ).(0 , , k ): We only need to consider k ≥ c +1 for which (0 , , k ) ∼ II z (1 , , k − c +1) ∼ I y (0 , b, k − c ).( i, , k ): Let us use induction on i . Observe that ( i, , k ) ∼ I y ( i − , b − , k − z (which adds ( − , − , c )) until either the first or the second entry become 0,depending on the relative sizes of i and b . In the former, we have (0 , ∗ , ∗ ) which wasalready covered. In the latter case, we obtain ( i − b, , k − b − c − i, , i ≥
3, in which case ( i, , ∼ II x ( i − , , ∼ I z ( i − , , c ). ( i, j, i, j, ∼ I z ( i − , j − , c − ∼ I x ( i, j − , c − ∼ I y ( i − , j + b − , c − ∼ I x ( i, j + b − , c − . Thus we can say that ( i, j,
0) is equivalent to 0 if i ≥ , j ≥
2. Also note that(2 , , ∼ I z (1 , , c − i, j, k ): It can be handled by using induction on ( i + j + k ) based on the relation ( i, j, k ) ∼ I x ( i + 1 , j − , k − (cid:3) Appendix C. Proof of Proposition 8.9
Proposition 8.9 can be shown by directly counting the contributing orbi-discs, which istedious, but elementary. Below is a reformulation of Proposition 8.9 in ˜ x, ˜ y, ˜ z -variables whichare more accessible in actual disc counting. In addition, we choose(C.1) b = − ˜ xX + ˜ yY − ˜ zZ in order to make the signs in the formula more symmetric. It is not difficult to check thatProposition C.1 is equivalent to the original statement in Proposition 8.9. Proposition C.1.
The map KS from the orbifold quantum cohomology of X to the Jacobianring of W (˜ x, ˜ y, ˜ z ) defined in (8.10) is given by X (cid:55)→ (cid:55)→ A T ∂∂T W (˜ x, ˜ y, ˜ z );∆ / (cid:55)→ ˜ x ∞ (cid:88) k =0 ( − k (2 k + 1) φ k ( T );∆ / (cid:55)→ ˜ y ∞ (cid:88) k =0 ( − k (2 k + 1) φ k ( T );∆ / (cid:55)→ ˜ z ∞ (cid:88) k =0 ( − k (2 k + 1) φ k ( T ); IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 55 ∆ / (cid:55)→ ˜ x ∞ (cid:88) k =0 (2 k + 1) φ k ( T ) + ˜ x ∞ (cid:88) k =1 k − (cid:88) i =0 ( − k − i (6 k − i + 2) φ k ( T ) φ i ( T )+ ˜ y ˜ z ∞ (cid:88) k =1 k − (cid:88) i =0 (cid:18) ( − k − i (6 k − i ) ψ + k ( T ) φ i ( T ) + ( − k − i − (6 k − i − ψ − k ( T ) φ i ( T ) (cid:19) ;∆ / (cid:55)→ ˜ y ∞ (cid:88) k =0 (2 k + 1) φ k ( T ) + ˜ y ∞ (cid:88) k =1 k − (cid:88) i =0 ( − k − i (6 k − i + 2) φ k ( T ) φ i ( T )+ ˜ z ˜ x ∞ (cid:88) k =1 k − (cid:88) i =0 (cid:18) ( − k − i (6 k − i ) ψ + k ( T ) φ i ( T ) + ( − k − i − (6 k − i − ψ − k ( T ) φ i ( T ) (cid:19) ;∆ / (cid:55)→ ˜ z ∞ (cid:88) k =0 (2 k + 1) φ k ( T ) + ˜ z ∞ (cid:88) k =1 k − (cid:88) i =0 ( − k − i (6 k − i + 2) φ k ( T ) φ i ( T )+ ˜ x ˜ y ∞ (cid:88) k =1 k − (cid:88) i =0 (cid:18) ( − k − i (6 k − i ) ψ + k ( T ) φ i ( T ) + ( − k − i − (6 k − i − ψ − k ( T ) φ i ( T ) (cid:19) where φ k ( T ) = T k +12 k +3 , ψ + k ( T ) = T (6 k +1) , ψ − k ( T ) = T (6 k − .Proof. KS ( X ) = 1 by the unital property of KS , and KS ([pt]) was already computed in theproof of Theorem 8.1. It only remains to compute the image of ∆ i/ for i = 1 , L with the minimum e which serves as the unit class in CF ( L , L ). In addition,we choose a generic point which is close to e that represents a nontrivial structure put on L .Readers may consult [CHL17, 3.4] for more details on the disc counting in this setting.(1) KS (∆ / ) : From our earlier lifting argument, the holomorphic triangles counted for thepotential W can be regarded as uniformizing covers of [1/3] orbi-discs which contribute to KS (∆ / ), as shown in Figure 7. Comparing with (8.9), there are sequences ∆ x,k and ∆ opx,k ofsuch orbi-discs with sizes φ k ( T ), k = 0 , , , · · · . Here, we set ∆ x,k to be a positive triangle,and ∆ opx,k a negative one.We also need to count the number of times in which the discs meet the minimum e .By direct counting, we see that for ∆ x,k and ∆ opx,k of size φ k ( T ), there are k + 1 and k many e ’s on their boundaries, respectively. Taking signs into account ( s (∆ x,k ) = ( − k +1 , s (∆ opx,k ) = ( − | X | ( − k = ( − k +1 ), the element ∆ / of QH ∗ orb ( X ) maps to˜ x ∞ (cid:88) k =0 ( − k (2 k + 1) φ k ( T ) = φ ( T )˜ x as desired. Notice that ( − k +1 in s (∆ x,k ) = s (∆ opx,k ) = ( − k +1 turns into ( − k due to (C.1). Figure 7. [1 /
3] orbi-discs
Figure 8. [2 /
3] orbi-discs ∆ xyz, , + \ ∆ x, and its tripled image (lifting)(2) KS (∆ / ) : From Proposition 8.8, there are two types of such orbi-discs, corresponding toeither ˜ x or ˜ y ˜ z . The images of liftings of orbi-discs can be triangles or immersed hexagons asdepicted in Figure 8. We first consider the case when the images are triangles, which occursonly for ˜ x -type orbi-discs. Similarly to (1), we have two sequences ∆ x ,k and ∆ opx ,k for suchdiscs. Namely, we can take two third of such triangles to get desired orbi-discs, and theircount is given by ˜ x ∞ (cid:88) k =0 (2 k + 1) φ k ( T ) . Here, the two triangles ∆ x ,k and ∆ opx ,k have the common size φ k ( T ). Also, they have 2 k + 2and 2 k many e ’s on their boundaries respectively, but because of the rotation symmetry(which gives an automorphism on the moduli) we should count them as k + 1 and k . Signsof contribution are given by s (∆ x ,k ) = ( − k +2 = 1 , s (∆ opx ,k ) = ( − | X | + | X | ( − k = ( − k +2 = 1 . IG QUANTUM COHOMOLOGY OF ORBIFOLD SPHERES 57
We next consider orbi-discs whose liftings become immersed hexagons. Again, there aretwo types of such orbi-discs corresponding to either ˜ x or ˜ y ˜ z .(i) ˜ x : In this case, we count the orbi-discs ∆ x ,k \ ∆ x,i ( i = 0 , · · · , k −
1) of size φ k ( T ) φ i ( T ) , whichhas 3 k + 3 − ( i + 1) = 3 k − i + 2 many e ’s on its boundary, and s (∆ x ,k \ ∆ x,i ) =( − k − i +2 . Its reflection image (∆ x ,k \ ∆ x,i ) op has 3 k − i many e ’s on the boundary,and s ((∆ x ,k \ ∆ x,i ) op ) = ( − | X | + | X | ( − k − i = ( − k − i . In total, they produce˜ x k − (cid:88) i =0 (cid:0) ( − k − i +2 (3 k − i + 2) + ( − k − i (3 k − i ) (cid:1) φ k ( T ) φ i ( T )= ˜ x k − (cid:88) i =0 ( − k − i (6 k − i + 2) φ k ( T ) φ i ( T ) . (ii) ˜ y ˜ z : Denote the two positive triangles contributing to the k -th terms in 8.9 by ∆ xyz,k, ± .The only contribution to ˜ y ˜ z is from the count of ∆ xyz,k, ± \ ∆ x,i ( i = 0 , , · · · , k − ψ ± k ( T ).∆ xyz,k, + \ ∆ x,i has 3 k + 1 − ( i + 1) = 3 k − i many e ’s along the boundary, and s (∆ xyz,k, + \ ∆ x,i ) = ( − k − i . For (∆ xyz,k, + \ ∆ x,i ) op , we have 3 k − i many e ’s, and s ((∆ xyz,k, + \ ∆ x,i ) op ) = ( − | Y | + | Z | ( − k − i = ( − k − i . So, these two discs give˜ y ˜ z (cid:0) ( − k − i (3 k − i ) + ( − k − i (3 k − i ) (cid:1) ψ + k ( T ) φ i ( T )Similarly, ∆ xyz,k, − \ ∆ x,i and its reflection image contribute˜ y ˜ z (cid:0) ( − k − i − (3 k − i −
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