Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras
KKODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORYAND ORBIFOLD JACOBIAN ALGEBRAS
CHEOL-HYUN CHO AND SANGWOOK LEEA
BSTRACT . A version of mirror symmetry predicts a ring isomorphism between quan-tum cohomology of a symplectic manifold and Jacobian algebra of the Landau-Ginzburgmirror, and for toric manifolds Fukaya-Oh-Ohta-Ono constructed such a map calledKodaira-Spencer map using Lagrangian Floer theory. We discuss a general constructionof Kodaira-Spencer ring homomorphism when LG mirror potential W is given by J - holomorphic discs with boundary on a Lagrangian L : We find an A ∞ -algebra B whose m -complex is a Koszul complex for W under mild assumptions on L . Closed-openmap gives a ring homomorphism from quantum cohomology to cohomology algebra of B which is Jacobian algebra of W .We also construct an equivariant version for orbifold LG mirror ( W , H ). We constructa Kodaira-Spencer map from quantum cohomology to another A ∞ -algebra ( B (cid:111) H ) H whose cohomology algebra is isomorphic to the orbifold Jacobian algebra of ( W , H ) un-der an assumption. For the 2-torus whose mirror is an orbifold LG model given by Fer-mat cubic with a (cid:90) /3-action, we compute an explicit Kodaira-Spencer isomorphism.
1. I
NTRODUCTION
Let X be a symplectic manifold with a possibly immersed Lagrangian submanifold (cid:76) on it. Fukaya, Oh, Ohta and Ono defined an A ∞ -algebra for (cid:76) in [17] (Akaho-Joyce [2]for immersed case) as well as a potential function W (cid:76) : Y → Λ on its deformation space Y using J -holomorphic discs. Suppose (cid:76) is essential in X in the sense that it generatesFukaya category of X , or more informally (cid:76) is a representing Lagrangian object for theskeleton of X \ D for a choice of anti-canonical divisor D of X . Then W (cid:76) may be taken asa mirror Landau-Ginzburg model for X ([3]). Main examples are Lagrangian torus fibersin toric manifolds [14], [17], and Seidel Lagrangian in orbi-sphere (cid:80) a , b , c [28],[20], [9].Higher dimensional analogue of the latter example is an immersed sphere constructedby Sheridan in higher dimensional pair of pants [29] or in Fermat type hypersurfaces[30] although weakly unobstructedness of (cid:76) in these cases are not yet proved in general.Once W (cid:76) is constructed from Maurer-Cartan equation of (cid:76) ⊂ X , the first author withHong and Lau constructed a canonical A ∞ -functor from Fukaya category of X to thematrix factorization A ∞ -category of W , called a localized mirror functor [9, 10]. Thisfunctor is constructed as a curved version of Yoneda embedding relative to a fixed La-grangian (cid:76) , called reference Lagrangian. Hence the construction of the functor is basedon Fukaya category operations and provides a geometric way to understand homologi-cal mirror symmetry between these mirror pairs at hand.Let us turn our attention to closed string mirror symmetry. Historically, closed stringmirror symmetry between quantum cohomology and Jacobian algebra of W (or theirFrobienius manifolds) has been proved much earlier (Batyrev [4], Givental [21], Iritani[23] and so on) than homological mirror symmetry. These approaches are based onstudy of Gromov-Witten invariants and Picard-Fuchs equations. a r X i v : . [ m a t h . S G ] J u l CHEOL-HYUN CHO AND SANGWOOK LEE
Fukaya, Oh, Ohta and Ono [18] introduced a Lagrangian Floer theoretic approach tostudy closed string mirror symmetry. Namely, they constructed a geometric map, calledKodaira-Spencer map from quantum cohomology to Jacobian algebra of W in the caseof toric manifolds, and T n -action plays an essential role there. Here Jacobian algebra of W (which is also called Milnor ring) is a polynomial algebra modulo the ideal generatedby the partial derivatives of W (cid:76) . Recently, Amorim, Hong, Lau and the first author [1]generalized their construction to the case of (cid:80) a , b , c orbifold by using (cid:90) /2-action insteadof T n -action.In this paper, we define Kodaira-Spencer ring homomorphism in much more gener-ality, including the case of any weakly unobstructed Lagrangian torus (cid:76) in a symplec-tic manifold M with Lagrangian Floer potential W (cid:76) . The main idea is to replace Jaco-bian ring with the Koszul complex of W (cid:76) , which can be geometrically constructed fromMaurer-Cartan theory of the Lagrangian submanifold.Kodaira-Spencer map provides a nice geometric explanation for the isomorphismbetween complicated quantum multiplication and a trivial multiplication in Jacobianalgebra, which can be easily extended for bulk deformations. Also, as shown in [18],this provides a natural setting to compare Poincaré duality pairing of symplectic mani-folds and residue pairing of W (cid:76) in complex geometry. More recently, we have shown in[13] that these two pairings are conformally equivalent with conformal factor given bythe ratio of quantum volume form and classical volume form of (cid:76) . The results of [13]was restricted to the case of toric manifolds and (cid:80) a , b , c due to the absence of generalKodaira-Spencer map. Therefore, the construction in this paper can be used to prove ageneralization of the results in [13] for the cases of ( M , (cid:76) ) satisfying Cardy relations.Often mirror symmetry involves a finite group symmetry. In the very first example ofFermat quintic 3-folds [8] as well as in more recent works of Seidel [27], [28] and Sheri-dan [29] such symmetry plays an important role for the proof of HMS conjecture. Inthis paper, we will consider a closed string mirror symmetry between Q H ∗ ( X ) and orb-ifold Jacobian algebra Jac( W , (cid:98) G ). As far as the authors know, there has been no knownexamples, partly because the product structure of Jac( W , (cid:98) G ) is just beginning to be un-derstood. Let us suppose that (cid:98) G is a finite abelian group acting linearly on variables and W is a (cid:98) G -invariant polynomial. Orbifold Jacobian algebra Jac( W , (cid:98) G ) is easy to describeas a module, which is given by the Jacobian algebra of W restricted to the fixed part of χ -action for each χ ∈ (cid:98) G (see Definition 3.2). But it has quite an interesting and mysteri-ous product structure, which has been the focus of several interesting recent works (see[6], [7], [22], [32]).In this paper, we will define an equivariant version of Kodaira-Spencer map, namely aring homomorphism from quantum cohomology to the orbifold Jacobian algebra undersome assumption on the Lagrangian. We will describe an explicit equivariant Kodaira-Spencer isomorphism for T at the end.Let us give more detailed summary of our construction and precise theorems. Wefirst recall the idea behind the construction of Kodaira-Spencer map by Fukaya-Oh-Ohta-Ono. Recall that closed-open map in A -model usually defines a map from quan-tum cohomology of X to a Hochschild cohomology of Fukaya category of X . Fukaya-Oh-Ohta-Ono observed that instead of the full Hochschild cochains, one may focuson the cochains with no input and one output, coming from holomorphic discs withboundary on a Lagrangian torus fiber (cid:76) . Since Lagrangian (cid:76) is given by an orbit of T n -action, there is a free T n -action on the moduli space of holomorphic discs with at leastone boundary marked point. Given α ∈ Q H ∗ ( X ), the (virtual) evaluation image on (cid:76) of ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 3 J -holomorphic discs which intersect α ∈ Q H ∗ ( X ) in the interior is always a multiple offundamental class [ (cid:76) ] (from T n -action) and ks : Q H ∗ ( X ) → Jac( W (cid:76) ) is defined by read-ing the coefficient of [ (cid:76) ]. But unfortunately simple dimension counting of closed-openmap shows that the image of a closed-open map is not expected to become a multipleof [ (cid:76) ] in general and this was the main obstacle to define Kodaira-Spencer map beyondthe above examples.A key idea is to replace Jacobian algebra by an associated Koszul complex and to ob-serve that MC formalism of (cid:76) provides such a complex under a mild assumption onLagrangian (cid:76) . More precisely, given an A ∞ -algebra A of (cid:76) , we define a new A ∞ -algebra B by tensoring Λ [ x , ··· , x n ] and by considering an A ∞ -operation m bk for the family ofweak MC elements b = (cid:80) x i X i . The cohomology of B is isomorphic to Jacobian ring of W (cid:76) under a simple assumption 4.4, which states that for A ∞ -algebra A , degree one co-homology classes X , ··· , X n generate the cohomology algebra H ∗ ( A ) of rank 2 n . Thisassumption is satisfied for any Lagrangian torus in a symplectic manifold or Seidel La-grangian in (cid:80) a , b , c .With this setup, we use the closed-open map with boundary deformation by MC ele-ments b to define a ring homomorphism ks : Q H ∗ ( X ) → H ∗ ( B ) alg .Here we denote by H ∗ ( B ) alg the cohomology algebra of the complex ( B , m b ) with anassociative product v · w : = ( − | v | m b ( v , w ).Our setup enables us to apply the construction of Fukaya, Oh, Ohta and Ono [18]( small modification of [1]) to define a general (localized) Kodaira-Spencer map. Notethat we do not require any symmetry on (cid:76) other than its weakly unobstructedness. Theorem 1.1 (Theorem 8.1) . Let L be an object of Fukaya category of M and let A ( L ) beits Fukaya algebra, with bounding cochain b and potential function W . Denote by B ( L ) the A ∞ -algebra constructed in Definition 4.1. There is a Kodaira-Spencer map which is aring homomorphism ks : Q H ∗ ( M , Λ ) → H ∗ ( B ( L )) alg . Under the Assumption 4.4, H ∗ ( B ( L )) alg is isomorphic to Jac( W L ) by Proposition 4.8. We remark that further geometric investigation is needed to show that ks is an iso-morphism. In the case of [17], the images of toric divisors under ks is analyzed toprove an isomorphism property. In [1], authors had to enlarge the domain of Landau-Ginzburg mirror (valuations of variables in a Novikov field) to make ks an isomorphism.Let us move on to the equivariant cases. Let X be a symplectic manifold with a finitesymmetry group G . Then the quotient [ X / G ] is a symplectic orbifold. Suppose we havea possibly immersed Lagrangian submanifold L on [ X / G ] such that it has embeddedLagrangian lifts (cid:101) L in X . By applying the previous construction on L ⊂ [ X / G ], we obtaina Fukaya A ∞ -algebra for L with a potential function W L : Y → Λ . Dual group (cid:98) G natu-rally acts on Y leaving W invariant and we obtain the mirror Landau-Ginzburg orbifold( W L , (cid:98) G ). Here the action of χ ∈ (cid:98) G on the mirror variable x i is determined by the changeof branchs of (cid:101) L for the corresponding Lagrangian intersection X i (Definition 6.16).Now, suppose ¯ L is essential in [ X / G ] and then the mirror of the orbifold [ X / G ] shouldbe the LG model W : = W L . Also the mirror of X should be the orbifold LG model( W , (cid:98) G ). In this setup, the first author together with Hong and Lau constructed homolog-ical mirror symmetry functor ([9],[12]). Namely, there exist a localized mirror functor CHEOL-HYUN CHO AND SANGWOOK LEE F ¯ L : Fu([ X / G ]) → MF A ∞ ( W ). Here we define Fu([ X / G ]) to be G -invariant part Fu( X ) G .There is also an upstairs localized mirror functor F (cid:101) L : Fu( X ) → MF A ∞ (cid:98) G ( W )to the (cid:98) G -equivariant matrix factorization category of W (see [12] for more details).In this paper, we find a direct connection of orbifold Jacobian algebra to symplecticgeometry. Namely, we define a Kodaira-Spencer map from quantum cohomology to theorbifold Jacobian algebra, which is an equivariant version of Theorem 1.1. We will definean A ∞ -algebra B ( L ) (cid:111) (cid:98) G and an additional (cid:98) G -action on it and define an equivariantversion of Kodaira-Spencer map. Theorem 1.2 (Theorem 8.2) . Suppose the Lagrangian L ⊂ [ X / G ] and its potential func-tion W is given as above. There is a Kodaira-Spencer map which is a ring homomorphism ks : Q H ∗ ( M , Λ ) → H ∗ (cid:161) ( B ( L ) (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg . (1.1)We prove that under suitable assumption, H ∗ (cid:161) ( B ( L ) (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg is isomorphic to orb-ifold Jacobian algebra. Theorem 1.3 (Theorem 7.1) . We haveH ∗ (cid:161) ( B ( L ) (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg ∼= Jac( W , (cid:98) G ). under the following Assumption 5.1 on a reference Lagrangian L (which is L in this case)withpotential function W ( x ) : (cid:161) C F ( L , L ) ⊗ Λ [ x , ··· , x n , y , ··· , y n ], m b ( x ), b ( y )1 (cid:162) (1.2) defines a matrix factorization of W ( y ) − W ( x ) and it is isomorphic to the Koszul matrixfactorization for the regular sequence ( y − x , ··· , y n − x n ) . Let us first explain the Assumption 5.1. Conjecturally, the Assumption 5.1 for an es-sential Lagrangian L ⊂ M may be formulated as a correspondence between geometricdiagonal ∆ ⊂ M − × M and an algebraic diagonal of W ( y ) − W ( x ) which is the kernel ofidentity functor of matrix factorization category MF ( W ). Therefore, we expect this as-sumption to hold in such cases. We hope to discuss this in more detail elsewhere. Inthis paper, we show that monotone Lagrangian tori in a symplectic manifold as well asSeidel Lagrangian in (cid:80) a , b , c satisfies this assumption 5.1 more directly.Let us also briefly explain the idea of the proof of the above theorem 7.1. The semi-direct product A ∞ -algebra ( B ( L ) (cid:111) (cid:98) G ) concerns Maurer-Cartan data of the Lagrangianlifts (cid:101) L , while keeping the same variables ( x , ··· , x n ). We apply localized mirror functorrelative to the reference ( L , b ( y )) to obtain the matrix factorization (1.2) of W ( y ) − W ( x ).The rest of the construction is to consider the additional (cid:98) G -action to compare it withthe equivariant Kernel ∆ (cid:98) G × (cid:98) GW for matrix factorization category.We give an explicit example. It turns out that ks for a symplectic torus T is alreadynon-trivial. Recall that Polishchuk-Zaslow [25] explained the HMS of T from the pointof view of Strominger-Yau-Zaslow formalism[33] that mirror pairs are obtained as dualtorus fibrations. In [9] (following the work of Seidel [28]), the (cid:90) /3-quotient of T and aSeidel Lagrangian L ∈ [ T / (cid:90) /3] was considered. In [9], L is shown to be weakly unob-structed with a potential function W = − φ ( q ) i ( x + x + x ) + ψ ( q ) i x x x . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 5
Here φ , ψ are power series in q given in (10.1). Localized mirror functor provides anexplicit HMS equivalence in this case.We prove the following theorem, by computing the Kodaira-Spencer map and com-paring with the algebraic generators of orbifold Jacobian algebra by Shklyarov [32]. Theorem 1.4.
A Kodaira-Spencer ring isomorphism ks : Q H ∗ ( T , Λ ) → Jac( W , (cid:90) /3) isgiven by pt T (cid:55)→ q ∂ W ∂ q , l χ : = − i γ · [ C h ] − ˇ χ [ C v ]ˇ χ − (cid:55)→ ξ χ , l χ : = − i γ · [ C v ] − ˇ χ [ C h ]ˇ χ − (cid:55)→ ξ χ . Here γ is a modular form γ = (cid:88) k ∈ (cid:90) ( − k i q (6 k + .The organization of this paper is as follows. We first review the notion of a ker-nel for a functor between categories of matrix factorizations in Section 2. The kernel ∆ W for the identity functor id : MF ( W ) → MF ( W ) is a Koszul matrix factorization andHom( ∆ W , ∆ W ) describes the Hochschild cohomology of MF ( W ), or the Jacobian ringof W . We review the algebraic definition of orbifold Jacobian algebras in Section 3 fol-lowing Shklyarov [32]. In section 4, we explain how to use Maurer-Cartan formalism of A ∞ -algebra A by Fukaya-Oh-Ohta-Ono to define a new A ∞ -algebra B . We describeAssumption 4.4 which implies that the cohomology algebra of B is isomorphic to Ja-cobian algebra of the superpotential W of A . And we define Floer theoretic kernel ∆ Fl using assumption 5.1. In section 5, we show that monotone Lagrangian torus as well asSeidel Lagrangian in (cid:80) a , b , c satisfies this assumption 5.1 (which implies 4.4). In section 6,we give an equivariant analogue of the construction in section 4. From an A ∞ -algebra (cid:102) A with finite group G -action, we define a new A ∞ -algebra B (cid:111) (cid:98) G , by studying equivari-ant version of Maurer-Cartan formalism. In section 7, we show that H ∗ (cid:161) ( B (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg isquasi-isomorphic to orbifold Jacobian algebra under the assumption 5.1. For the proof,we use the notion of equivariant Kernel ∆ (cid:98) G × (cid:98) GW and show that the A ∞ -algebra ( B (cid:111) (cid:98) G ) (cid:98) G isshown to be A ∞ -quasi-isomorphic to hom MF A ∞ ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ) whose coho-mology algebra (in dg sign convention) is isomorphic to the orbifold Jacobian algebra.In section 8, we construct Kodaira-Spencer map from quantum cohomology to coho-mology algebra of B or ( B (cid:111) (cid:98) G ) (cid:98) G . In section 9, we review the construction of Shklyarovso that we can match our construction to that of [32]. In section 10, we illustrate our con-struction to compute the Kodaira-Spencer map for T using geometric construction. InAppendix A, we write the algebraic computation of the orbifold Jacobian algebra for themirror of T . Notation 1.5.
We introduce a few notations. Here k is a base field. For mirror symmetry k is Novikov field Λ over (cid:67) which is algebraically closed with a valuation ν : Λ → (cid:82) . R : = k [ x , ··· , x n ], R e : = k [ x , ··· , x n , y , ··· , y n ], CHEOL-HYUN CHO AND SANGWOOK LEE Λ : = (cid:110) ∞ (cid:88) j = a j T λ j | a j ∈ (cid:67) , λ j ∈ (cid:82) ≥ , lim j →∞ λ j = ∞ (cid:111) , Λ : = Λ [ T − ], Λ + : = (cid:110) ∞ (cid:88) j = a j T λ j ∈ Λ | λ j > (cid:111) , ν ( ∞ (cid:88) j = a j T λ j ) : = min j ( λ j ).Here R and R e are suitably completed when k = Λ . For example R (for k = Λ ) is theconvergent power series ring which is the set of elements (cid:88) i , ··· , i n ≥ c i , ··· , i n x i ··· x i n n such that the elements c i , ··· , i n ∈ Λ satisfies lim i +···+ i n →∞ ν ( c i , ··· , i n ) = ∞ . Acknowledgements.
We would like to thank Lino Amorim, Yong-Geun Oh, David Faveroand Yoosik Kim for very helpful discussions. C.-H. Cho was supported by the NRF grantfunded by the Korea government(MSIT) (No. 2017R1A22B4009488). S. Lee was sup-ported by an Individual Grant(MG063902) at Korea Institute for Advanced Study andby the Fields Institute during the Thematic Program on Homological Algebra of MirrorSymmetry. 2. K
ERNELS FOR MATRIX FACTORIZATIONS
Let us recall our basic set-up for matrix factorization category. Let W be an elementin R = k [ x , ··· , x n ] for an algebraic closed field k of characteristic 0. We assume that W has only isolated singularity at the origin. Definition 2.1. A matrix factorization of W is a (cid:90) /2-graded projective R -module P = P ⊕ P together with a morphism d = ( d , d ) of degree 1 such that d = W · id.A morphism φ : ( P , d ) → ( Q , d (cid:48) ) of degree j ∈ (cid:90) /2 is a map of (cid:90) /2-graded R -modules ofdegree j . Define D φ : = d (cid:48) ◦ φ − ( − | φ | φ ◦ d and define the composition of morphisms in the usual sense. This defines a dg-categoryof matrix factorizations ( MF ( W ), D , ◦ ).This dg category can be turned into an A ∞ -category MF A ∞ ( W ) as follows: Definition 2.2.
The A ∞ -category MF A ∞ ( W ) has the same set of objects as ( MF ( W ), D , ◦ ). hom MF A ∞ ( W ) ( E , F ) : = hom MF ( W ) ( F , E ), m ( Φ ) : = D Φ , m ( Φ , Ψ ) : = ( − | Φ | Φ ◦ Ψ , (2.1)and m k = k .We still denote a morphism Φ ∈ hom MF A ∞ ( W ) ( E , F ) as Φ : F → E . Remark . Conversely, given an A ∞ -algebra C with m =
0, its m -cohomology H ∗ ( C )has an induced associative algebra structure ◦ given by Φ ◦ Ψ : = ( − | Φ | m ( Φ , Ψ ).To emphasize that we need this additional sign, we will denote such cohomology alge-bra as H ∗ ( C ) alg . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 7
There is a class of matrix factorizations, called
Koszul matrix factorizations , whichplay a central role in the theory. Let I be an ideal of R generated by a regular sequence( b , ··· , b n ), and suppose that W ∈ I . Let ( a , ··· , a n ) be elements of R such that W = n (cid:88) i = a i · b i . Definition 2.4.
The n-th graded Clifford algebra isCl n : = k [ θ , ··· , θ n , ∂ θ , ··· , ∂ θ n ], | θ i | = − | ∂ θ i | = i together with relations θ i θ j = − θ j θ i , ∂ θ i ∂ θ j = − ∂ θ j ∂ θ i , ∂ θ i θ j = − θ j ∂ θ i + δ i j . (2.2)From the regular sequence { (cid:126) b }, the associated Koszul complex is defined as follows. K • = (cid:161) k [ θ , ··· , θ n ] ⊗ R , s (cid:162) , with s = (cid:88) j b j ∂ θ j . (2.3)The Koszul matrix factorization for { (cid:126) a , (cid:126) b } is defined as ( K • , δ (cid:162) , where δ = s + s , with s ( α ) = n (cid:88) j = a j θ j · α .It is a result of Eisenbud that this Koszul matrix factorization corresponds to the stabi-lization of R / W -module R / I (see Proposition 2.3.1 of [26] for example). Note that corre-sponding module R / I does not depend on the choice of a regular sequence (cid:126) b generating I nor (cid:126) a .We recall a few results using derived Morita theory of matrix factorizations. Definition 2.5 ([34]) . Let A be a dg category. The Hochschild cochain complex of A isdefined as C ∗ ( A , A ) : = hom RHom c ( A , A ) (Id A ,Id A )where R Hom c ( A , A ) is the dg category of continuous functors from A to itself.By modifying results in [34] for differential (cid:90) /2-graded categories and applying themto matrix factorizations, Dyckerhoff showed the following. Theorem 2.6 ([15]) . There is a quasi-equivalence between two dg categoriesR Hom c ( MF ( R , W ), MF ( R (cid:48) , W (cid:48) )) (cid:39) MF ( R ⊗ k R (cid:48) , − W ⊗ + ⊗ W (cid:48) ). (2.4) In the case that ( R (cid:48) , W (cid:48) ) = ( R , W ) , identity functor Id MF ( R , W ) corresponds via the abovequasi-equivalence to a matrix factorization, that is given by a resolution ofR ∼= R e /( y − x , ··· , y n − x n ) as an R e /( W ( y , ··· , y n ) − W ( x , ··· , x n )) -module. Definition 2.7.
For F ∈ R Hom c ( MF ( R , W ), MF ( R (cid:48) , W (cid:48) )), the corresponding matrix fac-torization under the quasi-equivalence (2.4) is called a kernel for F .We remark that a kernel for identity functor is not unique. We take the followingKoszul matrix factorization and denote it by ∆ W . For j = ··· , n , define polynomials ∆ j W ∈ R e (also written as ∇ x → ( x , y ) j ( W ) later) by ∆ j W : = W ( x , ··· , x j − , y j , ··· , y n ) − W ( x , ··· , x j , y j + , ··· , y n ) y j − x j . CHEOL-HYUN CHO AND SANGWOOK LEE
Then we have W ( y , ··· , y n ) − W ( x , ··· , x n ) = n (cid:88) j = ∆ j W · ( y j − x j ), (2.5)From now on, we write W ( x ) = W ( x , ··· , x n ), W ( y ) = W ( y , ··· , y n ) for simplicity. Corollary 2.8.
We haveC ∗ ( MF ( R , W ), MF ( R , W )) (cid:39) hom MF ( R e , W ( y ) − W ( x )) ( ∆ W , ∆ W ).Now, let us recall the notion of equivariant matrix factorizations. Let H be a finitegroup, and suppose H acts on R and that W is invariant under this action. Definition 2.9. An H-equivariant matrix factorization of W is a matrix factorization( P , d ) where P is equipped with an H -action, and d is H -equivariant. An H-equivariantmorphism φ between two H -equivariant matrix factorizations ( P , d ) and ( Q , d (cid:48) ) is an H -equivariant morphism of (cid:90) /2-graded R -modules P and Q .It is well-known that H -equivariant matrix factorizations with H -equivariant mor-phisms also form a differential (cid:90) /2-graded category MF H ( W ). Example 2.10.
Let R = (cid:67) [ z ] with the polynomial W = z n . Define a (cid:90) / n -action by setting[1] · z = ξ z for an n th root of unity ξ and [1] ∈ (cid:90) / n , which preserves W . For any k (where0 ≤ k ≤ n ), we can define a matrix factorization given by P = R 〈 e 〉 and P = R 〈 o 〉 with d ( e ) = z k o , d ( o ) = z n − k e . For any 0 ≤ l < n , we can set[1] · e = ξ k + l e , [1] · o = ξ l o to give a (cid:90) / n -equivariant structure to this matrix factorization ( P • , d • ).Let us explain the equivariant structure of ∆ W following Section 2.5 of [26]. Supposea finite group H acts on R = k [ x , ··· , x n ], hence H also acts diagonally on R e . ∆ W canbe modified to produce a H -equivariant matrix factorization ∆ HW as follows. Define H -action on θ , ··· , θ n to be the same as that on x , ··· , x n which induces H -action on { ∂ θ i }.This defines an action on the module part of ∆ W . For δ = s + s , note that the expression s = (cid:80) i ( y i − x i ) ∂ θ i is H -invariant. Taking H -averaging of s = (cid:80) j ( ∆ j W ) θ j , we obtain anew s , which defines an H -equivariant modification ∆ HW . Observe that if we forget the H -action, then it is isomorphic to ∆ W .Now ∆ HW can be used to produce an H × H -equivariant matrix factorization (analo-gous to diagonal {( x , y ) | x = y } versus orbifold diagonal (cid:83) h ∈ H {( x , y ) | hx = y }). Theorem 2.11. [26]
Let ∆ W be a kernel for Id MF ( W ) and ∆ HW = ( R e [ θ , ··· , θ n ], d ( x , y )) beits H-equivariant modification. Then the following is a kernel for Id MF HA ∞ ( W ) ; ∆ H × HW : = (cid:77) h ∈ H (cid:161) R e [ θ , ··· , θ n ], d ( hx , y ) (cid:162) which is an H × H-equivariant matrix factorization of W ( y ) − W ( x ) . Notation 2.12.
For an R -module M , Let r v be an element in M with r ∈ R . We distin-guish the action on the generator v by ρ as follows: h · ( r v ) = ( h · r ) ρ ( h ) v .We explain H × H -equivariant structure of ∆ H × HW by describing the action of h × h .Denote each summand of ∆ H × HW by ∆ hW : = (cid:161) R e [ θ , ··· , θ n ], d ( hx , y ) (cid:162) . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 9
For an element r ( x , y ) v ∈ ∆ hW ,( h × h ) · ( r ( x , y ) v ) : = r ( h x , h y ) ρ ( h ) v ∈ ∆ h hh − W . (2.6)Then the H × H -equivariance of ∆ H × HW is verified as follows:( h × h ) · (cid:161) d ( hx , y )( r ( x , y ) v ) (cid:162) = ( h hh − × · ( h × h ) · ( h − × · (cid:161) d ( hx , y )( r ( x , y ) v ) (cid:162) = ( h hh − × · ( h × h ) · (cid:161) d ( x , y )( r ( h − x , y ) v ) (cid:162) = ( h hh − × · (cid:179) d ( x , y ) (cid:161) ( h × h ) · ( r ( h − x , y ) v ) (cid:162)(cid:180) = ( h hh − × · (cid:179) d ( x , y ) (cid:161) ( r ( h h − x , h y ) ρ ( h ) v ) (cid:162)(cid:180) = d ( h hh − x , y ) (cid:161) r ( h x , h y ) ρ ( h ) v (cid:162) = d ( h hh − x , y ) (cid:161) ( h × h ) · ( r ( x , y ) v ) (cid:162) .Note that we essentially used H -equivariance of d ( x , y ) for the third equality. Remark . The description of a kernel for Id MF H ( W ) does not depend on the abovespecific construction of ∆ HW . The proof of the Theorem 2.11 is based on facts that ∆ HW is H -equivariant, and that the cokernel of ∆ HW is isomorphic to R e /( y − x , ··· , y n − x n ) asa maximal Cohen-Macaulay module over R e /( W ( y ) − W ( x )). The latter fact implies thatthe cokernel of ∆ H × HW is isomorphic to (cid:76) h ∈ H R e /( y − h · x , ··· , y n − h · x n ) as a MCM mod-ule. Any matrix factorization P which is also H -equivariant and isomorphic to ∆ W as anonequivariant matrix factorization can be used to construct a kernel for Id MF H ( W ) inthe same way as in Theorem 2.11. We will appeal to this point later when we encountera matrix factorization of W ( y ) − W ( x ) from Floer theory.3. P RELIMINARIES ON ORBIFOLD J ACOBIAN ALGEBRAS
Consider a finite abelian group H acting on the commutative algebra R = k [ x , ··· , x n ].Let W be a H -invariant element in R . Then the triple ( R , W , H ) is called an orbifoldLandau-Ginzburg model, and we are interested in closed string theory of it as a B -model. It is given by an orbifold Jacobian algebra or equivalently defined as Hochschildcohomology of G -equivariant matrix factorization category of W .Let us recall the definition of orbifold Jacobian algebra following [32]. In fact it hassimple description as a (cid:90) /2-graded module, but its product structure is quite non-trivialand interesting. Recall that Jacobian algebra of W is k -algebra defined byJac( W ) = k [ x , ··· , x n ]( ∂ W ∂ x , ··· , ∂ W ∂ x n ) .For mirror symmetry applications, we take k = Λ . We remark that R = Λ [ x , ··· , x n ] is aTate algebra and hence any ideal is closed.We assume H acts on R diagonally. Namely, for h ∈ H , h · ( x , ··· , x n ) = ( h x , ··· , h n x n )for h i ∈ k ∗ . We set I h : = { i | h i = I h = { i | h i (cid:54)= d h = | I h | .Hence d h is the codimension of fixed subspace by h , Fix( h ) ⊂ k n .Denote by W h the restriction of W to Fix( h ). Definition 3.1. [7, 32] The twisted Jacobian algebra of ( R , W , H ) is a (cid:90) /2-graded algebraJac (cid:48) ( R , W , H ) = (cid:77) h ∈ H Jac( W h ) · ξ h where ξ h is a formal generator of degree d h mod 2. Its product structure is defined asfollows. For g , h ∈ H and φ g ∈ Jac( W g ), φ h ∈ Jac( W h ), we consider their restriction toFix( g h ) denoted as φ ghg , φ ghh and define φ g ξ g • φ h ξ h : = φ ghg · φ ghh · σ g , h · ξ gh where the definition of σ g , h ∈ Jac( W gh ) will be explained in Definition 3.4. We have σ g , h = d g + d h − d gh ∉ (cid:90) (3.1)which implies that the twisted Jacobian algebra Jac (cid:48) ( R , W , H ) is (cid:90) /2-graded. Definition 3.2.
Define the H -action on { ξ g | g ∈ G } by h · ξ g : = (cid:89) i ∈ I h h − i ξ g where h = ( h , ··· , h n ). The orbifold Jacobian algebra Jac( R , W , H ) is the H -invariant partof Jac (cid:48) ( R , W , H ). Theorem 3.3 ([32]) . Jac( R , W , H ) is a (cid:90) /2 -graded commutative algebra. Let us explain the definition of the products between ξ h following [32]. The construc-tion is based on an explicit relation between bar resolution and Koszul resolution andwe refer readers to the references for the details. Consider a morphism ∆ : R ⊗ ⊗ k [ θ , ··· , θ n ] → R ⊗ ⊗ k [ θ , ··· , θ n ] ⊗ , f ( x , y ) · p ( θ , ··· , θ n ) (cid:55)→ f ( x , z ) · p ( θ ⊗ + ⊗ θ , ··· , θ n ⊗ + ⊗ θ n )and ∆ : = e H W · ∆ , where H W is an element in R ⊗ ⊗ k [ θ , ··· , θ n ] ⊗ defined by H W ( x , y , z ) : = (cid:88) ≤ j ≤ i ≤ n ∇ y → ( y , z ) j ∇ x → ( x , y ) i ( W ) θ i ⊗ θ j .Identifying R ⊗ [ θ , ··· , θ n ] ⊗ R R ⊗ [ θ , ··· , θ n ] ∼= R ⊗ ⊗ k [ θ , ··· , θ n ] ⊗ ∼= via f ( x , y ) p ( θ ) ⊗ f ( y , z ) p ( θ ) ↔ f ( x , y ) f ( y , z ) ⊗ p ( θ ) ⊗ p ( θ ). Definition 3.4. [32] The structure coefficient σ g , h ∈ Jac( W | Fix( gh ) ) of the product be-tween g - and h -sectors is given by the coefficient of ∂ θ Igh : = (cid:81) i ∈ I gh ∂ θ i in the followingexpression1 d g , h ! Υ (cid:161) ( (cid:98) H W ( x , g · x , x ) (cid:99) gh +(cid:98) H W , g ( x ) (cid:99) gh ⊗ + ⊗(cid:98) H W , h ( g · x ) (cid:99) gh ) d g , h ⊗ ∂ θ Ig ⊗ ∂ θ Ih (cid:162) . (3.2)We explain various notations in (3.2). • H W , g : = (cid:88) i , j ∈ I g , j < i − g j ∇ x → ( x , x g ) j ∇ x → ( x , g · x ) i ( W ) θ j θ i ∈ R [ θ , ··· , θ n ] where x gi = x i if i ∈ I g and x gi = i ∈ I g . The operations ∇ x → ( x , g · x ) i and ∇ x → ( x , x g ) i are com-puted by ∇ x → ( x , y ) i followed by substitutions y = g · x and y = x g , respectively. • (cid:98) f (cid:99) g : = [ f | Fix( g ) ] ∈ Jac( W | Fix( g ) ). • d g , h : = d g + d h − d gh . We define σ g , h = d g , h is not an integer. • The map Υ is defined as Υ : R [ θ , ··· , θ n ] ⊗ ⊗ R [ ∂ θ , ··· , ∂ θ n ] ⊗ → R [ ∂ θ , ··· , ∂ θ n ], p ( θ ) ⊗ p ( θ ) ⊗ q ( ∂ θ ) ⊗ q ( ∂ θ ) (cid:55)→ ( − | q || p | p ( q ) · p ( q ),where p i ( q i ) means the action of p i ( θ ) on q i ( ∂ θ ) on Cl n /Cl n · 〈 θ , ··· , θ n 〉 via(2.2). ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 11
Then the product structure on Jac (cid:48) ( R , W , H ) is defined as follows: (cid:98) f (cid:99) g · ξ g • (cid:98) f (cid:99) h · ξ h : = (cid:98) f f (cid:99) gh · σ g , h ξ gh .We end this section by mentioning the following important theorem. Theorem 3.5 ([32]) . Orbifold Jacobian algebra is isomorphic to Hochschild cohomologyalgebra of H-equivariant matrix factorization category.
Jac( R , W , H ) ∼= H ∗ ( MF H ( W ), MF H ( W )).4. MC FORMALISM OF A ∞ - ALGEBRA A AND A NEW A ∞ - ALGEBRA B Let A : = ( V ,{ m k }) is a unital gapped filtered A ∞ -algebra over Λ . In this section, weuse a Maurer-Cartan formalism of A [17] to give a definition of a new A ∞ -algebra B . If A is weakly unobstructed with a potential function W , m -cohomology of B is shownto be isomorphic to Jacobian ring of W .4.1. Maurer-Cartan setup.
We briefly recall Maurer-Cartan formalism of A ∞ -algebrafrom [17] to set the notations. Here V be a (cid:90) /2-graded module over Λ equipped with Λ -multi-linear A ∞ -operation m k : V ⊗ k → V of degree 2 − k . We assume that it is gappedfiltered (which means m k = (cid:80) β ∈ G m k , β for a monoid G ⊂ (cid:82) ≥ such that { x ∈ G | x ≤ N } isfinite for any N ∈ (cid:78) ), and V ⊗ Λ Λ defines the A ∞ -algebra. We denote by a unit of A ∞ -algebra. For an element b ∈ F + V with positive valuation, weak Maurer-Cartan equationis given by m + m ( b ) + m ( b , b ) + ··· = W ( b ) ,The sum in the left hand side converges in T -adic sense and it is sometimes written as m ( e b ). Given b , ··· , b k , we can deform an A ∞ -operation m k to m b , ··· , b k k ( w , ··· , w k ) : = (cid:88) l , ··· , l k ≥ m k + l +···+ l k ( b l , w , b l , ··· , b l k − k − , w k , b l k k ). (4.1)For the case b = ··· = b k = b , we write m bk : = m b , ··· , bk . Maurer-Cartan equation can berestated as m b = W ( b ) . If b satisfies MC equation, m b defines a chain complex, andthis was used to define Floer cohomology of a Lagrangian L with m (cid:54)= A ∞ -algebra with formal parameters of deformation as follows.Take e , ··· , e n ∈ V . Assume that any b : = (cid:80) x i e i ∈ Λ + 〈 e , ··· , e n 〉 satisfies the weak MCequation m ( e b ) = W ( b ) · . Here, x i ’s are (formal) dual variables to e i ’s and may be con-sidered as coordinate functions of Maurer-Cartan solution space.In applications, one may take either { x i } or { e x i } as local mirror coordinates. In mir-ror symmetry, the former corresponds to immersed Lagrangians, and the latter for La-grangian torus. In this paper, we will develop the theory for the variables { x i }. Thetheory for exponentiated variables can be constructed analogously following the local-ized mirror construction of [10] of Lagrangian torus. We refer readers to Section 3.3 andLemma 4.2 of [13] to see the summary and basic constructions for the case of exponen-tiated variables.4.2. Jac( W ) from MC theory. Now, we set k = Λ and still denote by R = Λ [ x , ··· , x n ] thecompletion of the polynomial ring with respect to the valuation ν of Λ .For a gapped filtered A ∞ -algebra A = ( V ,{ m k }), we assume that at least its coho-mology is finite dimensional: If Λ -module V itself is not finitely generated, we takeits canonical model following [24] and [19]. We choose a finite dimensional subspace H ⊂ V , π : V → H , i : H → V as well as the contraction homotopy Q such that id − π = − ( m Q + Qm ). Then, it is well known that we can transfer the A ∞ -structure on H andfind A ∞ -quasi-isomorphisms between them in a combinatorial way. Definition 4.1.
We take a tensor product V ⊗ Λ R . By linearly extending A ∞ -operationover formal variables x , ··· , x n (with b = (cid:80) ni = x i X i ), { m bk } defines an A ∞ -algebra on V ⊗ Λ R . We denote B : = ( V ⊗ Λ R ,{ m bk }).If b satisfies the Maurer-Cartan equation m b = W ( b ) · , and ( B , m b ) defines a chaincomplex, and m b defines a product on its cohomology Remark . Cohomology of A ∞ -algebras A and B are quite different. In mirror sym-metry applications, A is A ∞ -algebra for a Lagrangian L (open string theory), but associ-ated m b cohomology B will be Jacobian ring of the potential W L (closed string theory). Remark . The process of taking canonical model, and tensoring R can be taken at thesame time (which was used in Section 4.2 [1]). Namely, we may carry out the same con-struction of canonical model for H ⊗ Λ [ x , ··· , x n ] ⊂ V ⊗ Λ [ x , ··· , x n ] and transfer A ∞ -structure { m bk } on the latter to the former. We can construct A ∞ -quasi-isomorphismsbetween them as before.We find a sufficient condition that cohomology of B becomes Jacobian ring. We firstdefine e I , e bI for a subset I ⊂ {1, ··· , n }, which are successive m or m b products. For any I = { i , ··· , i k } with i < i < ··· < i k , we set e (cid:59) = e b (cid:59) = and denote e I = m ( e i , m ( e i , m ( ··· , e i k ) ··· )) (4.2) e bI = m b ( e i , m b ( e i , m b ( ··· , e i k ) ··· )). (4.3) Assumption 4.4. R -module V ⊗ Λ R is generated by { e bI } I ⊂ {1, ··· n } .We compare the generation of e I and e bI . From the gapped filtered condition (withthe valuation ν on Λ -modules) we obtain the following lemma. Lemma 4.5.
Suppose Λ -module V ⊗ Λ Λ is generated by { e I } I ⊂ {1, ··· n } and that ν ( m b ( v , w ) − m ( v , w )) > (cid:178) for some positive (cid:178) > for any v , w. Then V ⊗ Λ R satisfies the Assumption 4.4.
Examples satisfying Assumption 4.4.Lemma 4.6.
Any Lagrangian torus L (at critical points of the potential W L ) satisfies theAssumption 4.4.Proof. We will show that e I generate V = H ∗ ( L , Λ ) and use it to show that e bI gener-ate V ⊗ Λ R . Let L be a Lagrangian torus, with exponential coordinates z i = e x i onMC space. Assume that the potential function W L ( z ) has a critical point at (1, ··· ,1)(which can be achieved for any critical point by change of coordinates). It is well-known that in this case L with b = L (see [10] for example). For degree one generators e , ··· , e n of singular cohomology, m products generate the whole cohomology classes, andtherefore so do their m products (because of the filtration). We argue that their m b product generate H ∗ ( L ; Λ ) ⊗ Λ [ x , ··· , x n ]. To see this, first we may take a canonicalmodel m cank of A ∞ -algebra. We may further assume that classical part m cank ,0 = k ≥ L is a torus. Here we write m k = m k ,0 + m k , + where m k ,0 is the classical ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 13 A ∞ -structure on L , and m k , + are defined using non-constant holomorphic discs. Then, m b ( v , w ) = (cid:80) m k ,0 ( e b , v , e b , w , e b ) = m ( v , w ). Therefore, e bI − e I has strictly positiveenergy. We apply the previous lemma to obtain the claim. (cid:3) Lemma 4.7.
The A ∞ -algebra of Seidel Lagrangian [28] in (cid:80) a , b , c satisfies the above as-sumption.Proof. We check the assumptions of Lemma 4.5 in this case. Recall that we consideran immersed Lagrangian (cid:76) (called Seidel Lagrangian) in the orbi-sphere (cid:80) a , b , c . First, m products for Seidel Lagrangian are computed in [28]: Recall that C F ( (cid:76) , (cid:76) ) is gen-erated by six immersed sectors X , X , X , ¯ X , ¯ X , ¯ X as well as two Morse generators e , p . We have m ( X i , X i + ) = T c ¯ X i + = − m ( X i + , X i ) for the area c of minimal triangle.Also, m ( X i , ¯ X i ) = p = − m ( ¯ X i , X i ), and e is the unit for m multiplication. Other m products are shown to vanish by grading considerations (it is shown for (5,5,5) but itis straightforward to generalize it for general ( a , b , c )). Therefore { e I } generate V . Notethat only non-trivial holomorphic curve contribution comes from minimal triangle en-closed by Seidel Lagrangian. It is immediate that any other holomorphic polygons hasstrictly larger area. Therefore, we have ν ( m b ( v , w ) − m ( v , w )) > (cid:178) for any v , w for some (cid:178) < c . This proves the lemma. (cid:3) Let us further assume that W has isolated singularity at 0. i.e. ( ∂ x W , ··· , ∂ x n W ) de-fines a regular sequence and an associated Koszul complex ( K • , s ) from (2.3). Proposition 4.8.
Suppose an A ∞ -algebra A satisfies the Assumption 4.4. The chain com-plex ( V ⊗ R , m b ) is isomorphic to the Koszul complex ( K • , s ) for the regular sequence ( ∂ x W , ··· , ∂ x n W ) . More precisely, a chain map Ψ : ( V ⊗ Λ R , m b ) → ( K • , s ) defined by Ψ ( e bI ) = θ i ··· θ i k gives an isomorphism. Moreover m b -cohomology of B with the product given by m b , isisomorphic to Jac( W ) as an algebra.Remark . If W does not have isolated singularity, Ψ defines an isomorphism to Koszulcohomology. But this may not be an algebra isomorphism. Proof.
By taking ∂∂ x i of the equation m ( e b ) = W ( b ) · , we obtain m b ( e i ) = ∂ x i W · Recall that m ( , (cid:63) ) = ( − | (cid:63) | m ( (cid:63) , ) = (cid:63) . Then the claim follows from Leibniz rule for m b , m b . For example, consider the identity m b ( e i i ··· i k ) = − m b (cid:161) m b ( e i ), m b ( e i , m b ( ··· , e i k ) ··· ) (cid:162) − m b (cid:161) e i , m b ( m b ( e i , m b ( ··· , e i k ) ··· )) (cid:162) where the first term equals ( − ∂ x i W ) m b ( e i , m b ( ··· , e i k ) ··· ). In this way, we can provethe claim inductively.For the products, we show that the induced product structure on cohomology arethe same. Recall that the homology of the Koszul complex, Jac( W ) is concentrated atthe 0-th wedge product, and the product structure of Jacobian ring is induced from thatof R . Correspondingly, m b -cohomology comes from R · e (cid:59) = R · . Then m b is just givenby the products of the coefficients in R . This proves the proposition. (cid:3) Thus, B provides a chain complex model for Jacobian ring of W . In Section 8, wewill define a Kodaira-Spencer ring homomorphism from quantum cohomology to m b cohomology of B .5. K ERNEL FROM MC FORMALISM OF F LOER THEORY
In this section, we will propose a new assumption, the Assumption 5.1. This iden-tifies Floer theoretic construction of matrix factorization and the kernel for matrix fac-torization category. Therefore we call the former Floer theoretic Kernel. We show thatmonotone Lagrangian torus as well as Seidel Lagrangian in the orbisphere (cid:80) a , b , c satisfythe Assumption 5.1.Let us first construct Floer Kernel. For an A ∞ -algebra A = ( V ,{ m k }) we consider twoset of Maurer-Cartan elements which are the same except the naming of variables. b ( x ) = n (cid:88) i = x i e i , b ( y ) = n (cid:88) i = y i e i .We denote by R = Λ [ x , ··· , x n ], R (cid:48) = Λ [ y , ··· , y n ] and R e = R ˆ ⊗ Λ R (cid:48) = Λ [ x , ··· , x n , y , ··· , y n ]Let us also write W ( y ) = W ( b ( y )), W ( x ) = W ( b ( x )). Then, we have degree one opera-tion m b ( x ), b ( y )1 (defined in (4.1)) on (cid:90) /2-graded finitely generated R e -module V ⊗ Λ R e satisfying (cid:161) m b ( x ), b ( y )1 (cid:162) = ( W ( y ) − W ( x )) · id.Let us call this matrix factorization a Floer kernel ∆ FL . Assumption 5.1.
We assume that (cid:161) V ⊗ Λ R e [ n ], m b ( x ), b ( y )1 (cid:162) ∼= (cid:161) R e [ θ , ··· , θ n ], (cid:88) i ( y i − x i ) ∂ θ i + (cid:88) i ∇ x → ( x , y ) i W · θ i (cid:162) .Namely, Floer kernel ∆ FL is quasi-isomorphic to a Koszul matrix factorization ∆ W of W ( y ) − W ( x ) (see (2.5)) up to shift of (cid:90) /2 grading by [ n ]. Remark . Here is an equivalent version without the shift [ n ]. (cid:161) V ⊗ Λ R e , m b ( x ), b ( y )1 (cid:162) ∼= (cid:161) R e [ ∂ θ , ··· , ∂ θ n ], (cid:88) i ( y i − x i ) ∂ θ i + (cid:88) i ∇ x → ( x , y ) i W · θ i (cid:162) . (5.1)We use this in Section 9 for comparison of orbifold Jacobian algebra and Floer theoryLet us give another conjectural explanation of the above assumption. If a symplecticmanifold M is mirror to W , one may expect that the product symplectic manifold M − × M is mirror to − W ( x ) + W ( y ). The diagonal Lagrangian ∆ M ⊂ M − × M gives identifyfunctor on Fukaya category in the language of Lagrangian correspondence. Hence it isnatural to expect that ∆ M and ∆ W should be mirror to each other under homologicalmirror symmetry.Given a localized mirror functor F (cid:76) : Fukaya( M ) → MF ( W ), there should be a local-ized mirror functor for the product F (cid:76) × (cid:76) that send the diagonal ∆ M in Fukaya categoryto the diagonal ∆ W in matrix factorization category. Now, let us explain how it is relatedto the Assumption 5.1. The functor F (cid:76) × (cid:76) sends ∆ M to the following decorated Floercomplex (which becomes a matrix factorization) F (cid:76) × (cid:76) ( ∆ M ) = (cid:161) C F ( ∆ , (cid:76) × (cid:76) ) ⊗ R e , m b ( x ) ⊗ + ⊗ b ( y )1 (cid:162) . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 15
We conjecture that this is isomorphic to matrix factorization (cid:161)
C F ( (cid:76) , (cid:76) ) ⊗ R e , − m b ( x ), b ( y )1 (cid:162) .We hope to explore this in more detail elsewhere.Therefore, we expect this assumption 5.1 to hold in general but we are only able tocheck this for two set of examples in the rest of this section.5.1. Monotone Lagrangian torus satisfy Assumption 5.1.Proposition 5.3.
Fukaya A ∞ -algebra for a monotone Lagrangian torus satisfies Assump-tion 5.1.Proof. We prove this lemma following Theorem 9.1 [9]. Let us recall that A ∞ -algebra formonotone Lagrangian torus L has the following structure. First, without the holomor-phic disc contribution, the classical cochain algebra is quasi-isomorphic to the exterioralgebra ∧ • W for W = H ( L , Λ ) = Λ 〈 e , ··· , e n 〉 .Therefore we can transfer its A ∞ -structure to ( (cid:86) • W ,{ m k }) such that m k = (cid:88) µ ∈ (cid:90) , µ ≥ T µ m k , µ . m = m k ,0 = k (cid:54)=
2, and ( − | u | m ( u , u ) for u , u ∈ (cid:86) • W corresponds to theexterior algebra structure. Here m k , µ records the quantum contribution from Maslovindex µ holomorphic discs, and monotone condition guarantees that µ > A ∞ -algebra is only (cid:90) /2-graded, but in terms of the degree of exterior algebra, m k , µ has degree 2 − k − µ . For example, we have m = m + Tm + T m + ··· where m µ : (cid:94) • W → (cid:94) •+ − µ W .Hence, wedge grading is an enhancement of (cid:90) /2-grading. Also, from weak Maurer-Cartan equation, the potential W ( b ) comes from Maslov index two disc contribution(namely, { m k ,2 })Let us assume n is even so that the shift [ n ] is trivial. The case of odd n is similar andomitted. We set m b ( x ), b ( y )1, µ ( w ) : = (cid:88) m k + + l , µ (cid:161) b ( x ), ··· , b ( x ) (cid:124) (cid:123)(cid:122) (cid:125) k , w , b ( y ), ··· , b ( y ) (cid:124) (cid:123)(cid:122) (cid:125) l (cid:162) .Then, degree of m b ( x ), b ( y )1, µ ( w ) is the same as deg( w ) + − µ , and we have m b ( x ), b ( y )1,0 ( e I ) = m ( b ( x ), e I ) + m ( e I , b ( y )) = (cid:88) i ( y i − x i ) e i ∧ e I .Therefore, m b ( x ), b ( y )1,0 is given by wedge operation (cid:80) i ( y i − x i ) e i ∧ · .Now, consider the shift [ n ] in (cid:90) /2-grading of (cid:161) ∧ • W ⊗ Λ R e [ n ], m b ( x ), b ( y )1,0 (cid:162) , which canbe realized by the suitable dual complex. Namely, we consider the same complex (cid:161) ∧ • W ⊗ Λ R e , d b ( x ), b ( y )1,0 (cid:162) (5.2)but we replace wedge operation of m b ( x ), b ( y )1,0 by contraction operation and denote it by d b ( x ), b ( y )1,0 = (cid:88) i ( y i − x i ) ι e i . We denote by d k , µ the operations corresponding to m k , µ . In this way, (5.2) defines aKoszul complex for the regular sequence ( y − x , ··· , y n − x n ). Let I be the ideal of R e generated by this regular sequence.Recall that for a matrix factorization ( P • , δ ) of W , the corresponding sheaf (via Orlovequivalence) in the singularity category is given by the cokernel of δ : P odd → P even .Polishchuk-Vaintrob computes this cokernel of Koszul matrix factorization using a spec-tral sequence in Proposition 2.3.1 [26]. We will use this spectral sequence following [10]to prove our claim. The bi-complex L • , • is concentrated on two diagonals i + j = i + j = − L − i , i = (cid:94) i W , L − i , i − = (cid:94) i − W with the differentials d b ( x ), b ( y )1, µ : L − i , i − → L − ( i − + µ ), i − + µ .Consider the spectral sequence of this bi-complex coming from horizontal filtration. Itis shown in [26] that this spectral sequence has an E -page (after taking homology with d b ( x ), b ( y )1,0 ) with E = R e / I , E i , j = i (cid:54)= − j , and E − i , i becomes zero in the singularitycategory for i (cid:54)=
0. Hence, the cokernel is isomorphic to R e / I . Note that higher Maslovindex contribution d b ( x ), b ( y )1, µ with µ ≥ E -page is only non-trivialon the diagonal i + j =
0. Therefore our spectral sequence degenerates at E -page alsoand the cokernel for the matrix factorization (cid:161) ∧ • W ⊗ Λ R e [ n ], m b ( x ), b ( y )1 (cid:162) is isomorphicto R / I in the singularity category. Hence it is isomorphic to the desired Koszul matrixfactorization. (cid:3) In fact, we can strengthen Theorem 9.1 [9] if we apply the similar argument as in theabove proof and we write the result for reader’s convenience.
Proposition 5.4.
For a monotone Lagrangian torus L with a potential function W , thelocalized mirror functor of L sends L to the Koszul matrix factorization of W obtained byclassical and Maslov index two contributions
Orbi-spheres (cid:80) a , b , c satisfy Assumption 5.1. The Seidel Lagrangian (cid:76) in the orbi-sphere (cid:80) a , b , c with bounding cochain b = x X + x X + x X for three degree one im-mersed sectors X , X , X are shown to be weakly unobstructed and localized mirrorfunctor with reference (cid:76) provides homological mirror symmetry in elliptic and hyper-bolic cases [9]. Recall that we equip (cid:76) a complex line bundle with holonomy ( −
1) whichis uniformly distributed (as in [12]). We prove
Proposition 5.5.
Fukaya A ∞ -algebra for the Seidel Lagrangian (cid:76) in (cid:80) a , b , c , satisfies As-sumption 5.1.Proof. We will show that the matrix factorization
C F (cid:161) ( (cid:76) , b ( x )),( (cid:76) , b ( y )) (cid:162) is a Koszul ma-trix factorization for ( y − x , (cid:126) w ) for some (cid:126) w = ( w , ··· , w n ) (see (2.3)). As mentioned inthe proof of Lemma 4.7, C F ( (cid:76) , (cid:76) ) has 8 generators and its m product structure maybe identified with an exterior algebra (cid:86) • 〈 X , X , X 〉 by setting T c ¯ X i + = X i ∧ X i + for i = T c p = X ∧ X ∧ X . It turns out we need a quantum correction tothis identification to match with a Koszul matrix factorization. Proposition 5.6.
There exist γ e ∈ Λ [ x , x , x , y , y , y ] (to be defined in (5.5) ) such thatin terms of a basis { p new , X , X , X , e , ¯ X new , ¯ X new , ¯ X new }, (5.3) ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 17 with ¯ X i new : = γ e ¯ X i and p new : = γ e p, the map m b ( x ), b ( y )1 on C F (cid:161) ( (cid:76) , b ( x )),( (cid:76) , b ( y )) (cid:162) can bewritten as the following linear transformation y − x y − x y − x y − x c c y − x − c c y − x − c − c f f f f y − x − ( y − x ) 0 0 0 0 f − ( y − x ) 0 y − x f y − x − ( y − x ) 0 0 0 0 0 .(5.4) Remark . Recall that in [9], the matrix factorization of W ( y ) that is mirror to (cid:76) wasshown to be Koszul. We also needed quantum change of variables for C F (cid:161) (cid:76) ,( (cid:76) , b ( y )) (cid:162) with γ playing the role of γ e here. In fact, we have γ e | x = x = x = = γ . Corollary 5.8.
We have c = f , c = f , c = f . Therefore, C F (cid:161) ( (cid:76) , b ( x )),( (cid:76) , b ( y )) (cid:162) defines a Koszul matrix factorization for ( y − x , (cid:126) w ) withw i = f i = c i + i + for i = .Proof. Let us first prove the corollary. Using the fact that (5.4) defines a matrix factoriz-tion for W ( y ) − W ( x ), we obtain that W ( y ) − W ( x ) = (cid:88) i = ( y i − x i ) f i = (cid:88) i = ( y i − x i ) c i + i + and − f c + f c = f c − f c = − f c + f c = (cid:3) Proof.
Let us prove Proposition 5.6. By unital property of A ∞ -algebra m b ( x ), b ( y )1 ( e ) = m ( e , b ( y )) + m ( b ( x ), e ) = b ( y ) − b ( x )and this explains the case of input e . The map from ¯ X i to p has only constant disc (withMorse trajectory) contribution, and we omit the details.For the rest of the proof, reflection symmetry along the equator of (cid:80) a , b , c plays themain role. Recall that (cid:76) is preserved by reflection and e , p are switched to each other.The coefficients from X i to e is the same as the coefficient from p to ¯ X i using reflectionargument in (iv) of Theorem [9]. Furthermore, we have the following skew-symmetryfrom reflection. Lemma 5.9.
Let c i j be the coeffcient of X j in m b ( x ), b ( y )1 ( γ e ¯ X i ) . We have c i j = − c ji .Proof. We check the sign following Lemma 7.4 [9]. Let P be a polygon contributing tothe coeffcient c i j , and then its reflection image P op contribute to the coefficient c ji (seeFigure 1). As observed in Lemma 7.4 [9], ∂ P and ∂ P op evenly covers (cid:76) , say l times. Since (cid:76) consists of 6 minimal edges, we may assume that ∂ P and ∂ P op each covers 3 l minimaledges.For polygon P , denote by a (resp. a ) the number of corners that lies betweenthe output corner to the input corner when we walk along ∂ P counter-clockwise (resp.clockwise) way. From the combinatorial sign convention, it is easy to see that the sign F IGURE
1. Reflection symmetry of holomorphic polygons.difference of A ∞ -operation for P and P op differ by ( − a + a + : If (cid:76) is oriented as in theFigure, then P carries signs ( − a + and P op carries signs ( − a . Extra ( −
1) factor for P comes from the output degree | X j | .One can also observe that for the holomorphic polygon P , the length of minimaledge between corners of same parity is odd, and the corners of different parity is even.Therefore, we can see that the parity of the number of edges of P is given by ( a − + ( a − = a + a . Hence, a + a ≡ l modulo 2.The additional sign difference from the non-trivial spin structure of (cid:76) for P and P op isgiven by ( − l = ( − a + a . Combining these two contributions ( − a + a + × ( − a + a ,we obtain the result. (cid:3) The following lemma can be proved in a similar way and we omit the proof.
Lemma 5.10.
Denote by h i j ∈ R e the coefficient of ¯ X j in m b ( x ), b ( y )1 ( X i ) as h i j . Then,h i j = − h ji . Now, let us explain the definition of γ e . The coefficient of p in (cid:161) m b ( x ), b ( y )1 (cid:162) ( X i ) shouldvanish (from A ∞ -identity) for i = y − x ) h + ( y − x ) h = y − x ) h + ( y − x ) h = y − x ) h + ( y − x ) h = γ e ∈ R e . h y − x = h y − x = h y − x = : γ e . (5.5)Here h is divisible by y − x by the above A ∞ equation. Therefore, the coefficient of¯ X new ( = γ e ¯ X ) in m b ( x ), b ( y )1 ( X ) is just ( y − x ) and so on. This proves the lemma. (cid:3) Thus, identifying the basis (5.3) with the basis of exterior algebra, we have shownthat the matrix factorization m b ( x ), b ( y )1 is a Koszul MF for ( y − x , (cid:126) f ). After shifting [3] in ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 19 (cid:90) /2-grading, we obtain a Koszul MF for ( (cid:126) f , y − x ), hence obtaining a Koszul resolutionof W ( y ) − W ( x ) for the diagonal ∆ . (cid:3) A priori, γ e ∈ R e and γ ∈ R are different. But for (cid:80) they are the same. Lemma 5.11.
For (cid:80) , γ e = γ is a scalar in Λ given by a modular form given in (10.3) .Proof. One can check that h i j is linear, hence γ e is scalar. Therefore it equals γ by theremark 5.7. (cid:3)
6. E
QUIVARIANT CONSTRUCTION AND A ∞ - ALGEBRA ( B (cid:111) (cid:98) G ) (cid:98) G In this section, we give an equivariant construction of Section 4. When a finite abeliangroup G acts on an A ∞ -category, we look at the quotient A ∞ -category and for a refer-ence object O , we construct a new A ∞ -algebra B by studying Maurer-Cartan equationin the quotient. Then we construct the theory for the original category by developing anequivariant construction for the dual group (cid:98) G -actionLet C be a filtered unital (cid:90) /2-graded A ∞ -category over Λ with a strict G -action. Thismeans that G acts on the set of objects Ob ( C ) and morphisms such that g : hom C ( O , O ) → hom C ( gO , gO ),and for composable morphisms w , ··· , w k , we have m k ( g w , ··· , g w k ) = g m k ( w , ··· , w k ).We introduce the following notation. Definition 6.1.
Define (cid:101) O : = (cid:76) g ∈ G gO for any object O of C , and hom C ( (cid:101) O , (cid:101) O ) : = (cid:77) g , g ∈ G hom C ( g O , g O ).This has an induced A ∞ -structure { m k } where m k is defined to be 0 if not composable.It has the strict diagonal G -action.One can define a (quotient) A ∞ -cateory C G as follows. Let us discuss this for the caseof single object for simplicity and let O ∈ Ob ( C ) be an object with g O (cid:54)= g O for g (cid:54)= g . Definition 6.2.
We define an A ∞ -algebra A (for the quotient object O ) on hom C G ( O , O ) : = (cid:77) g ∈ G hom C ( O , gO ).Let us denote an element v ∈ hom C ( O , gO ) as v g to keep track of the indices. An A ∞ -structure on hom C G ( O , O ) is defined as m k (( w ) g , ··· ,( w k ) g k ) = m k ( w , g · ( w ),( g g ) · ( w ), ··· ,( g , ··· , g k − ) · w k ) (6.1)One may check that the above m k operation is composable, and satisfies A ∞ -equations.Let (cid:98) G = Hom ( G , U (1)) be the character group of the finite abelian group G . We define (cid:98) G -action on quotient morphism spaces. Definition 6.3.
We define (cid:98) G -action on hom C G ( O , O ) by χ ( v g ) = χ ( g − ) v g , v g ∈ hom C ( O , gO ). (6.2) Remark . In [27], action was defined to be χ ( g ) v g . Our convention has the advantagethat the A ∞ -isomorphism in Lemma 6.7 preserves the eigen-spaces of G -action. Let us denote the action by ρ ( χ ). We may call this the first (cid:98) G -action, which we use todefine the following semi-direct product. Later, the second (cid:98) G -action will be defined inDefinition 6.16. Definition 6.5. [27] A semi-direct product A ∞ -algebra hom C G ( O , O ) (cid:111) (cid:98) G is an A ∞ -structure defined on hom C G ( O , O ) ⊗ Λ [ (cid:98) G ] with its A ∞ -operation is defined as m k ( w ⊗ χ , ··· , w k ⊗ χ k ): = m k (cid:161) ρ ( χ ··· χ k )( w ), ρ ( χ ··· χ k )( w ), ··· , ρ ( χ k )( w k − ), w k (cid:162) ⊗ χ ··· χ k . (6.3) Remark . This convention makes more sense after reversing the order of inputs of m k . This is analogous to the setup in Definition 2.2 that to make dg-algebra into an A ∞ -algebra, we take the opposite hom spaces.Seidel observed the following isomorphism. For later use, we describe an explicitisomorphism as follows (taking a sum over G -orbit twisted by a character χ ). Lemma 6.7.
We have an isomorphism of two A ∞ -algebras given by Φ . We set Φ k ≥ = and define Φ : hom C G ( O , O ) (cid:111) (cid:98) G → hom C ( (cid:101) O , (cid:101) O ) Φ : v ⊗ χ (cid:55)→ (cid:88) g ∈ G χ ( g − )( g · v ). Furthermore, Φ is G-equivariant map where G-action on the domain of Φ is defined byg · ( v ⊗ χ ) : = χ ( g )( v ⊗ χ ). Proof. G -equivariance is due to the following: Φ ( h · ( v ⊗ χ )) = Φ ( χ ( h ) v ⊗ χ ) = (cid:88) g ∈ G χ ( g − h ) g v = h · (cid:88) g ∈ G χ ( g − h ) h − g v = h · Φ ( v ⊗ χ ).To prove that it is an A ∞ -morphism, consider the following projection map for g ∈ G . π g : (cid:77) g , g ∈ G hom C ( g O , g O ) → (cid:77) h ∈ G hom C ( gO , g hO ).Then for v i ∈ hom C ( O , g i O ) and for g ∈ G , π g ◦ (cid:179) m k (cid:161) Φ ( v ⊗ χ ), ··· , Φ ( v k ⊗ χ k ) (cid:162)(cid:180) = m k (cid:161) χ ( g − ) g v , χ (( g g ) − ) g g v , ··· , χ k (( g g ··· g k − ) − ) g g ··· g k − v k (cid:162) = ( χ ··· χ k )( g − ) · ( χ ··· χ k )( g − ) · ( χ ··· χ k )( g − ) ··· χ k ( g − k − ) m k ( g v , g g v , ··· , g g ··· g k − v k ) = ( χ ··· χ k )( g − ) · ( χ ··· χ k )( g − ) · ( χ ··· χ k )( g − ) ··· χ k ( g − k − ) g · m k ( v , g v , ··· , g ··· g k − v k ) = ( χ ··· χ k )( g − ) · g · m k (cid:161) ( χ ··· χ k )( g − ) v , ··· , χ k ( g − k − ) v k − , v k (cid:162) = ( χ ··· χ k )( g − ) · g · m k ( ρ ( χ ··· χ k )( v ), ··· , ρ ( χ k )( v k − ), v k ) = π g ◦ (cid:161) Φ ( m k ( v ⊗ χ , ··· , v k ⊗ χ k )) (cid:162) .The map is clearly injective. For g v ∈ hom C ( gO , g hO ), we observe that Φ (cid:179) (cid:88) χ ∈ (cid:98) G χ ( g ) v ⊗ χ | (cid:98) G | (cid:180) = (cid:88) χ ∈ (cid:98) G (cid:88) g (cid:48) ∈ G χ ( g g (cid:48)− ) g (cid:48) v | (cid:98) G | . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 21
When g (cid:48) = g , the summand is g v . If g (cid:48) (cid:54)= g , then the summand is zero due to (cid:88) χ ∈ (cid:98) G χ ( g g (cid:48)− ) = Φ is also surjective. (cid:3) Since G is finite abelian, hom C ( (cid:101) O , (cid:101) O ) has eigenspace decomposition for the G -action. Corollary 6.8. Φ sends hom C G ( O , O ) ⊗ χ to a χ -eigenspace of hom C ( (cid:101) O , (cid:101) O ) . Bounding cochains.
We consider Maurer-Cartan theory for the semi-direct prod-uct, and the corresponding deformation of A ∞ -structure. The following observation iseasy but important for further development. It also appeared in Sheridan’s work [31]. Lemma 6.9.
If b satisfies weak Mauer-Cartan equation with potential W for the A ∞ -algebra hom C G ( O , O ) , then so does b ⊗ for the semi-direct product hom C G ( O , O ) (cid:111) (cid:98) G. In particular, we can define deformed A ∞ -maps { m b ⊗ } on hom C G ( O , O ) (cid:111) (cid:98) G . Notethat m b ⊗ preserves χ -eigenspace hom C G ( O , O ) ⊗ χ for any χ . Namely, for a ⊗ χ , m b ⊗ ( a ⊗ χ ) = ∞ (cid:88) k = m k + ( b ⊗ ··· , b ⊗ a ⊗ , b ⊗ ··· b ⊗ = ∞ (cid:88) k = m k + ( ρ ( χ )( b ), ··· , ρ ( χ )( b ), a , b , ··· , b ) ⊗ χ Here ρ ( χ )( b ) is a χ action defined in (6.2), and therefore acts only on X i ’s. We pretendthat χ − acts on variables x i ’s instead and not on X i ’s and make the following definition. Definition 6.10.
For b = (cid:80) i x i X i , we set b ( χ − ) : = (cid:88) i χ − ( x i ) X i We have b ( χ − ) = (cid:80) i x i χ ( X i ) = ρ ( χ ) b .Therefore, m b ⊗ ( a ⊗ χ ) for Floer theory uses contributions of J -holomorphic discswith the following inputs and write the output on the χ -sector. a outb b b b ( χ − ) b ( χ − ) b ( χ − )F IGURE
2. Geometric description of m b ( χ − ), b ( a ⊗ χ ) Lemma 6.11.
We have an isomorphism sending w ⊗ χ → w. (cid:161) hom C G ( O , O ) ⊗ χ , m b ⊗ (cid:162) ∼= (cid:161) hom C G ( O , O ), m b ( χ − ), b (cid:162) . In general, m b ⊗ k is given by m b ⊗ k ( a ⊗ χ , ··· , a k ⊗ χ k ) (6.4) = m ρ ( χ ··· χ k ) b , ρ ( χ ··· χ k ) b , ··· , ρ ( χ k ) b , bk (cid:161) ρ ( χ ··· χ k ) a , ρ ( χ ··· χ k ) a , ··· , a k (cid:162) ⊗ χ ··· χ k .In particular, m b product of χ and χ -eigenvectors goes to χ χ -eigenspace.Sometimes it is convenient to work on hom C ( (cid:101) O , (cid:101) O ). Definition 6.12.
For a bounding cochain b , we set (cid:101) b : = ( Φ ) ∗ ( b ) = Φ ( b )In particular, for b ( x ) = (cid:80) i x i X i ⊗
1, we get (cid:101) b ( x ) = (cid:88) i x i (cid:161) (cid:88) g ∈ G g ( X i ) (cid:162) As we assumed that g O (cid:54)= g O for g (cid:54)= g , G -action permutes the output of m k -operation in hom( (cid:101) O , (cid:101) O ) and one can observe that W ( (cid:101) b ) = W ( b ).We remark that in [9], the following localized mirror functor has been defined. Theorem 6.13 ([9]) . We have an A ∞ -functor F (cid:101) O (resp. F O ) which are cohomologicallyinjective on Hom ’s with (cid:101)
O (resp. O). C (cid:102) F (cid:101) O (cid:47) (cid:47) quotient by G (cid:15) (cid:15) MF A ∞ (cid:98) G ( W ) C G F O (cid:47) (cid:47) MF A ∞ ( W ) quotient by (cid:98) G (cid:79) (cid:79) A ∞ -algebra B (cid:111) (cid:98) G and its (cid:98) G -quotient. Recall that in non-equivariant case, we de-fined a new A ∞ -algebra B by tensoring R = Λ [ x , ··· , x n ] to an A ∞ -algebra A . Definition 6.14. An A ∞ -algebra B (cid:111) (cid:98) G is the data (cid:161) ( hom C G ( O , O ) (cid:111) (cid:98) G ) ⊗ R ,{ m b ⊗ k } (cid:162) where m k is the R -linear extension of m k on hom C G ( O , O ) (cid:111) (cid:98) G , and b ⊗ = (cid:80) x i X i ⊗ B (cid:111) (cid:98) G . Lemma 6.15.
Using Lemma 6.7 (tensoring R), we can identify the following two A ∞ -algebras via A ∞ -isomorphism Φ . (cid:161) B (cid:111) (cid:98) G ,{ m b ⊗ k } (cid:162) (cid:39) (cid:161) hom C ( (cid:101) O , (cid:101) O ) ⊗ R ,{ m (cid:101) bk } (cid:162) .We now define the second (cid:98) G -action on B (cid:111) (cid:98) G (which is strict). Definition 6.16.
First, for X i ∈ hom C ( O , g i O ), recall from (6.2) χ · X i = ρ ( χ )( X i ) = χ ( g − i ) X i .For the dual variable, we set χ · x i : = χ ( g i ) x i .Then we define the (cid:98) G -action on B (cid:111) (cid:98) G by χ · ( r ( x ) v ⊗ η ) : = r ( χ · x ) ρ ( χ ) v ⊗ η . Remark . On (cid:111) (cid:98) G part, (cid:98) G is supposed to act by conjugation in previous literatures,but since (cid:98) G is abelian, our action on (cid:98) G -part is trivial. ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 23
For this second (cid:98) G action, we have χ · ( b ⊗ = ( b ⊗ χ -action on x i and X i cancels each other. Remark . Two (cid:98) G -actions in Definition 6.3 and in Definition 6.16 are different. Thefirst (cid:98) G -action in Definition 6.3 only acts on hom ( O , O ) and is used to define semi-directproduct B (cid:111) (cid:98) G . The second actions in Definition 6.16 is an action on B (cid:111) (cid:98) G , and we willmostly use the second action from now on. Proposition 6.19.
The (cid:98)
G-action on B (cid:111) (cid:98) G is compatible with the A ∞ -structure { m b ⊗ k } .Proof. Let f i v i ⊗ χ i ∈ B (cid:111) (cid:98) G for i = ··· , k , where f i ∈ R and v i ∈ hom C ( O , g i O ). Let η ∈ (cid:98) G . Then, m k ( η · ( f v ⊗ χ ), ··· , η · ( f k v k ⊗ χ k )) = ( η · ( f ··· f k )) m k ( η ( g − ) v ⊗ χ , ··· , η ( g − k ) v k ⊗ χ k ) = ( η · ( f ··· f k )) η ( g − ··· g − k ) m k ( v ⊗ χ , ··· , v k ⊗ χ k ) = ( η · ( f ··· f k )) η · m k ( v ⊗ χ , ··· , v k ⊗ χ k ) = η · m k ( f v ⊗ χ , ··· , f k v k ⊗ χ k ).The third equality comes from the fact m k (cid:161) ( v ) g , ··· ,( v k ) g k (cid:162) ∈ hom C ( O , g ··· g k O )and the definition of (cid:98) G -action (6.2). Since b ⊗ (cid:98) G -action, m b ⊗ k is alsocompatible with the action. (cid:3) Corollary 6.20. (cid:98)
G-invariant part of B (cid:111) (cid:98) G, denoted as ( B (cid:111) (cid:98) G ) (cid:98) G , has an induced A ∞ -structure { m b ⊗ k } .
7. C
OHOMOLOGY ALGEBRA OF B (cid:111) (cid:98) G AND ORBIFOLD J ACOBIAN RING
From an A ∞ -algebra hom C G ( O , O ) with potential function W , we constructed an A ∞ -algebra B (cid:111) (cid:98) G and its (cid:98) G -quotient ( B (cid:111) (cid:98) G ) (cid:98) G in the previous section. We find therelation to the orbifold Jacobian ring of ( W , (cid:98) G ). Theorem 7.1.
Assume that W has an isolated singularity at the origin. Suppose anA ∞ -algebra hom C G ( O , O ) satisfies an Assumption 5.1. Then we have an algebra isomor-phism: H ∗ (cid:161) ( B (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg ∼= Jac( W , (cid:98) G ). Proof of the Theorem.
We will use the fact thatJac( W , (cid:98) G ) ∼= Hom MF (cid:98) G × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ).Let us briefly explain this. By Shklyarov, Jac( W , (cid:98) G ) is isomorphic to the Hochschild co-homology H ∗ ( MF (cid:98) G ( W ), MF (cid:98) G ( W )) which is (from Definition 2.5) hom RHom c ( MF (cid:98) G ( W ), MF (cid:98) G ( W )) (Id MF (cid:98) G ( W ) ,Id MF (cid:98) G ( W ) ).On the other hand, Polishchuk-Vaintrob [26] showed that R Hom c ( MF (cid:98) G ( W ), MF (cid:98) G ( W )) ∼= MF (cid:98) G × (cid:98) G ( W ( y ) − W ( x ))such that identity functor Id MF (cid:98) G ( W ) corresponds to a kernel ∆ (cid:98) G × (cid:98) GW .To prove the main theorem, we will show how to relate ∆ (cid:98) G × (cid:98) GW and B (cid:111) (cid:98) G using local-ized mirror functor of [9]. Let us make the following shorthand notation. O x : = (cid:161) O , b ( x ) (cid:162) , (cid:101) O y : = (cid:161) (cid:101) O , (cid:101) b ( y ) (cid:162) . Extend the (cid:98) G -action on R e by χ · y i : = χ ( g i ) y i when X i ∈ Hom( O , g i O ). Lemma 7.2.
With respect to the (cid:98)
G-action on hom C G ( O , O ) and R e ,hom C G ( O x , O y ) ⊗ R e , m b ( x ), b ( y )1 ) (7.1) is a (cid:98) G-equivariant matrix factorization of W ( y ) − W ( x ) .Proof of the Lemma. Let v ∈ hom C G ( O x , O y ) ⊗ R e . By Proposition 6.19, χ · m b ( x ), b ( y )1 ( v ) = m χ · b ( x ), χ · b ( y )1 ( χ · v ).Since χ · b ( x ) = b ( x ) and χ · b ( y ) = b ( y ), m b ( x ), b ( y )1 is (cid:98) G -equivariant. (cid:3) From Assumption 5.1 and by Theorem 2.11 (and appealing to Remark 2.13), we canobtain ∆ (cid:98) G × (cid:98) GW from the matrix factorization (7.1). ∆ (cid:98) G × (cid:98) GW : = (cid:77) χ ∈ (cid:98) G (cid:161) hom C G ( O χ · x , O y ) ⊗ R e , m b ( χ · x ), b ( y )1 (cid:162) .Using Lemma 6.11, we rewrite the kernel as ∆ (cid:98) G × (cid:98) GW = (cid:77) χ ∈ (cid:98) G (cid:161) ( hom C G ( O x , O y ) ⊗ χ ) ⊗ R e , m b ( x ) ⊗ b ( y ) ⊗ (cid:162) .Then the (cid:98) G × (cid:98) G -action on ∆ (cid:98) G × (cid:98) GW in (2.6) translates into( χ × χ ) · ( r ( x , y ) v ⊗ χ ) = r ( χ · x , χ · y ) ρ ( χ )( v ) ⊗ χ χχ − .Observe that if we restrict the action to the diagonal subgroup (cid:98) G , then it coincides withthe action in Definition 6.16.The following two propositions will prove the main theorem. Proposition 7.3.
There is an A ∞ -homomorphism between two A ∞ -algebras F : B (cid:111) (cid:98) G → hom MF A ∞ ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ). (7.2) Moreover, F is injective in cohomology. Proposition 7.4.
The cohomological image of B (cid:111) (cid:98) G under F are exactly morphismsthat are (1 × (cid:98) G ) -equivariant, hence given by Hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ). (7.3) The cohomological image of ( B (cid:111) (cid:98) G ) (cid:98) G are exactly (cid:98) G × (cid:98) G-equivariant morphisms given by
Hom MF A ∞ (cid:98) G × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ). Proof of Proposition 7.3.
We use the idea of localized mirror functor in [9] to define an A ∞ -functor F to matrix factorizations. Recall from [9] that given an A ∞ -algebra hom( (cid:101) O , (cid:101) O )with bounding cochains (cid:101) b ( y ) and potential function W ( y ), an A ∞ -functor F (cid:101) O y : C → MF ( W ( y ))(relative to ( (cid:101) O , (cid:101) b ( y ))) is defined by sending an object K of C to the matrix factorization (cid:161) hom( K , (cid:101) O ), − m (cid:101) b ( y )1 (cid:162) . Higher part of the functor is defined as F (cid:101) O y k ( p , ··· , p k ) = m k + ( p , ··· , p k , · ). ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 25
Lemma 7.19 of [9] states that this functor is cohomologically injective. This was shownby constructing an explicit right inverse using the unit of A ∞ -algebra. We remark thatwe use the sign convention of [12] by taking hom( · , (cid:101) O ) instead of hom( (cid:101) O , · ).For the proof, we will use the following variation of the above construction. Namely,we can apply the functor F (cid:101) O y to the same object (cid:101) O but equipped with a boundingcochain (cid:101) b ( x ). In this case, ( (cid:101) O , (cid:101) b ( x )) is mapped by F (cid:101) O y to (cid:161) hom C ( (cid:101) O x , (cid:101) O y ) ⊗ R e , − m (cid:101) b ( x ), (cid:101) b ( y )1 (cid:162) ,which is a matrix factorization of W ( y ) − W ( x ). Also, k -th part of the A ∞ -functor in thiscase is given as follows.( F (cid:101) O y ) k : (cid:161) hom C ( (cid:101) O x , (cid:101) O x ) ⊗ R e (cid:162) ⊗ k → hom MF A ∞ ( W ( y ) − W ( x )) (cid:161) ( hom C ( (cid:101) O x , (cid:101) O y ) ⊗ R e , − m (cid:101) b ( x ), (cid:101) b ( y )1 ),( hom C ( (cid:101) O x , (cid:101) O y ) ⊗ R e , − m (cid:101) b ( x ), (cid:101) b ( y )1 ) (cid:162) ,( p , ··· , p k ) (cid:55)→ m (cid:101) b ( x ), ··· , (cid:101) b ( x ), (cid:101) b ( y ) k + ( p , ··· , p k , • ).We can show that this defines an A ∞ -homomorphim by the same argument in [9](hence omit the proof). The injectivity can be also shown as in [9].We modify the sign for matrix factorization category using the following simple lemma Lemma 7.5.
For an A ∞ -algebra ( A , m , m , m ≥ = , there is an A ∞ -isomorphism ( A , m , m ) (cid:39) ( A , − m , m ), a (cid:55)→ ( − | a | a .Applying this to the object ( hom C ( (cid:101) O x , (cid:101) O y ) ⊗ R e , − m (cid:101) b ( x ), (cid:101) b ( y )1 ), we remove the negativesign. Using the isomorphism in Lemma 6.15, we have ∆ (cid:98) G × (cid:98) GW = (cid:161) ( hom C G ( O x , O y ) (cid:111) (cid:98) G ) ⊗ R e , m b ( x ) ⊗ b ( y ) ⊗ (cid:162) (cid:39) (cid:161) hom C ( (cid:101) O x , (cid:101) O y ) ⊗ R e , m (cid:101) b ( x ), (cid:101) b ( y )1 (cid:162) .Combining these isomorphisms, we get the desired A ∞ -morphism F : B (cid:111) (cid:98) G → hom MF A ∞ ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ). (cid:3) Now, it remains to verify the assertions about equivariance of morphisms.
Proof of Proposition 7.4.
We first prove that the image of F is included in hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ).Let f ( x ) p ⊗ η ∈ ( hom C G ( O x , O x ) ⊗ η ) ⊗ R ,and r ( x , y ) v ⊗ χ ∈ ( hom C G ( O x , O y ) ⊗ χ ) ⊗ R e .By definition of (cid:98) G × (cid:98) G -action,( χ × χ ) · (cid:161) m b ( x ) ⊗ b ( x ) ⊗ b ( y ) ⊗ ( f ( x ) p ⊗ η , r ( x , y ) v ⊗ χ ) (cid:162) (7.4) = ρ ( χ ) (cid:179) m ρ ( ηχ ) b ( χ · x ), ρ ( χ ) b ( χ · x ), b ( χ · y )2 (cid:161) ρ ( χ )( f ( χ · x ) p ), r ( χ · x , χ · y ) v (cid:162)(cid:180) ⊗ χ ηχχ − = m ρ ( χ ηχ ) b ( χ · x ), ρ ( χ χ ) b ( χ · x ), ρ ( χ ) b ( χ · y )2 (cid:161) ρ ( χ χ )( f ( χ · x ) p ), ρ ( χ )( r ( χ · x , χ · y ) v ) (cid:162) ⊗ χ ηχχ − = m ρ ( χ ηχχ − ) b ( x ), ρ ( χ χχ − ) b ( x ), b ( y )2 (cid:161) ρ ( χ χ )( f ( χ · x ) p ), ρ ( χ )( r ( χ · x , χ · y ) v ) (cid:162) ⊗ χ ηχχ − .On the other hand, m b ( x ) ⊗ b ( x ) ⊗ b ( y ) ⊗ (cid:161) f ( x ) p ⊗ η ,( χ × χ ) · ( r ( x , y ) v ⊗ χ ) (cid:162) (7.5) = m b ( x ) ⊗ b ( x ) ⊗ b ( y ) ⊗ (cid:161) f ( x ) p ⊗ η , r ( χ · x , χ · y ) ρ ( χ ) v ⊗ χ χχ − (cid:162) = m ρ ( ηχ χχ − ) b ( x ), ρ ( χ χχ − ) b ( x ), b ( y )2 (cid:161) ρ ( χ χχ − )( f ( x ) p ), r ( χ · x , χ · y ) ρ ( χ ) v (cid:162) ⊗ ηχ χχ − .If χ =
1, then (7.4)=(7.5), so the morphism F ( f ( x ) p ⊗ η ) = m b ( x ) ⊗ b ( x ) ⊗ b ( y ) ⊗ ( f ( x ) p ⊗ η , • )is 1 × (cid:98) G -equivariant for any f ( x ) p ⊗ η . F ( f ( x ) p ) ⊗ η is a (cid:98) G × (cid:98) G -equivariant morphism if and only if (7.4)=(7.5) for general χ × χ . It is equivalent to f ( χ · x ) p = ρ ( χ − )( f ( x ) p ),if and only if f ( x ) p is (cid:98) G -invariant.Since we have an injectivity result by Proposition 7.2, it suffices to show that anycohomology class of a 1 × (cid:98) G -invariant morphism is an image of F . Then it proves notonly the first but also the second statement, because any (cid:98) G × (cid:98) G -equivariant morphismis a 1 × (cid:98) G -equivariant morphism. For this, we show the following isomorphism of chaincomplexes ( hom C G ( O , O ) (cid:111) (cid:98) G ) ⊗ R (cid:39) hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ),which implies that H ∗ ( B (cid:111) (cid:98) G ) ∼= Hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ).Let φ ∈ hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ). Then the equivariance of φ gives us(1 × χ − ) · φ ( r v ⊗ χ ) = φ (cid:161) (1 × χ − ) · ( r v ⊗ χ ) (cid:162) = φ (cid:161) r ( x , χ − · y ) ρ ( χ − ) v ⊗ (cid:162) .We conclude that a 1 × (cid:98) G -equivariant morphism φ is completely determined by its re-striction to hom ( O x , O y ) ⊗
1. Therefore, hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ) (cid:39) hom R e (cid:161) ( hom C G ( O x , O y ) ⊗ ⊗ R e , (cid:77) χ ∈ (cid:98) G ( hom C G ( O x , O y ) ⊗ χ ) ⊗ R e (cid:162) (cid:39) hom R e (cid:161) hom C G ( O x , O y ) ⊗ R e , (cid:77) χ ∈ (cid:98) G hom C G ( O χ · x , O y ) ⊗ R e (cid:162) (cid:39) hom R e (cid:161) hom C G ( O x , O y ) ⊗ R e , (cid:77) χ ∈ (cid:98) G R e /( y − χ · x ) (cid:162) (cid:39) (cid:77) χ ∈ (cid:98) G ( hom C G ( O x , O χ · x ) ⊗ R ) ∨ .By Assumption 5.1, the differential of hom C G ( O x , O χ · x ) ⊗ R is a sum of Koszul differen-tials. By self-duality of Koszul complexes, we have( hom C G ( O x , O χ · x ) ⊗ R ) ∨ (cid:39) (cid:77) χ ∈ (cid:98) G hom C G ( O x , O χ · x ) ⊗ R .On each χ -summand, we take coordinate change x (cid:55)→ χ − · x . Then we have hom C G ( O x , O χ · x ) ⊗ R (cid:39) hom C G ( O χ − · x , O x ) ⊗ R , ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 27 hence hom MF A ∞ × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ) (cid:39) (cid:77) χ ∈ (cid:98) G hom C G ( O χ − · x , O x ) ⊗ R (cid:39) ( hom C G ( O , O ) (cid:111) (cid:98) G ) ⊗ R ,and we finish the proof of the Proposition. (cid:3) Let us finish the proof of the Theorem 7.1. By Proposition 7.4, we have H ∗ (( B (cid:111) (cid:98) G ) (cid:98) G ) ∼= Hom MF A ∞ (cid:98) G × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ).Modifying sign to the dg-setting, H ∗ (( B (cid:111) (cid:98) G ) (cid:98) G ) alg ∼= Hom MF (cid:98) G × (cid:98) G ( W ( y ) − W ( x )) ( ∆ (cid:98) G × (cid:98) GW , ∆ (cid:98) G × (cid:98) GW ) ∼= Jac( W , (cid:98) G ). (cid:3) It is natural to conjecture that the theorem holds before taking (cid:98) G -quotient. Conjecture 7.6.
With the same assumptions as in Theorem 7.1, we have H ∗ ( B (cid:111) (cid:98) G ) alg ∼= Jac (cid:48) ( W , (cid:98) G ).Although we have proved that left hand side equals (7.3), we do not know whetherthis is isomorphic to twisted Jacobian algebra defined by [7, 32].8. K ODAIRA -S PENCER MAP
In this section, we construct a general Kodaira-Spencer map from small quantum co-homology
Q H ∗ ( M ) to a cohomology of an A ∞ -algebra, which is isomorphic to Jacobianring of W for a non-equivariant case and orbifold Jacobian algebra of ( W , (cid:98) G ) for an equi-variant case. The construction should extend to the case of big quantum cohomologyusing bulk-deformations by following [18] but we omit it for simplicity.Recall that Fukaya-Oh-Ohta-Ono[18] defined a geometric map ks : Q H ∗ ( M ) → Jac( W )for general toric manifolds (with bulk-deformations) and showed that the map is a ringisomorphism. Such a construction was generalized to the case of orbi-sphere (cid:80) a , b , c [1]. The former construction uses T n -action in an essential way. For example, for theFukaya algebra A ( L ) of a Lagrangian torus L , it uses the observation that the outputof the closed-open map (using a J -holomorphic disc with one interior input and oneboundary output) C O : Q H ∗ ( M ) → H ∗ ( A ( L )) alg is always a multiple of the unit = [ L ] from T n -action. In this case, we can just readthe coefficient of the unit with boundary Maurer-Cartan deformation and this gives anelement of the algebra Jac( W ). The construction of [1] is similar, but uses (cid:90) /2-action(and low dimensionality of L ) instead of T n -action and show that C O always maps tothe multiple of .But in general, the image of the map C O is not expected to be a multiple of . Forexample, its virtual dimension is n + µ ( β ) − deg M ( A ) which is not necessarily n = dim( L ).Our simple but important idea is that we replace Jac( W ) by its associated Koszulcomplex. Also, we find a mild assumption 4.4 with which the A ∞ -algebra ( B , m b ) from L defines such a Koszul complex. In fact, C O naturally has an output in B (see Figure3 (a)) by decorating L with bounding cochain b . Here, we work in the setting of local-ized mirror in the sense that we only look at the part of the mirror given by the formalneighborhood of the reference Lagrangian (cid:76) . b b boutA (a) (b) (c) F IGURE
3. (a) Kodaira-Spencer map ks ( A ), (b) Σ , (c) Σ Theorem 8.1.
Let A ( L ) be Fukaya A ∞ -algebra of Lagrangian L, with bounding cochainb and potential function W . Denote by B ( L ) another A ∞ -algebra defined in Definition4.1. There is a Kodaira-Spencer map which is a ring homomorphism ks : Q H ∗ ( M , Λ ) → H ∗ ( B ( L )) alg . In particular for f , f ∈ Q H ∗ ( M ) , we have ks ( f ∗ Q f ) = ( − deg ks ( f ) m ( ks ( f ), ks ( f )). Under the Assumption 4.4, the target is isomorphic to
Jac( W L ) by Proposition 4.8. Most of the construction of Fukaya-Oh-Ohta-Ono[18], and its modifications [1] car-ries over to this setting and we give a sketch of proof later in the section (explainingwhich parts of [18] and [1] have to be modified).Let us explain an equivariant version of Kodaira-Spencer map. Let G be a finiteabelian group and M be a symplectic manifold with an effective G -action. Fukaya cate-gory of M can be given as a filtered (cid:90) /2-graded unital A ∞ -category with a strict G -action(using de Rham version of the work of Fukaya-Oh-Ohta-Ono but any other technicalsetting will be okay). Therefore, we can apply the construction of Section 5. Supposethe Lagrangian L ⊂ [ M / G ] satisfies the assumptions of Section 5 so that it has an A ∞ -algebra A ( L ) as well as the A ∞ -category (cid:102) A whose objects are | G | embedded lifts of L . b = (cid:80) x i X i satisfies a Maurer-Cartan equation for A ( L ). Theorem 8.2.
There is a Kodaira-Spencer map which is a ring homomorphism ks : Q H ∗ ( M , Λ ) → H ∗ (cid:161) ( B ( L ) (cid:111) (cid:98) G ) (cid:98) G (cid:162) alg . (8.1) Under the Assumption 5.1, the target of ks is isomorphic to the orbifold Jacobian algebra Jac( W L , (cid:98) G ) by Theorem 7.1.Remark . If we restrict the above ks map to the G -invariant part ( χ = H ∗ ([ M / G ])to H ∗ (cid:161) B ( L ) (cid:98) G , m b (cid:162) . This should be a part of orbifold version of ks : Q H ∗ or b ([ M / G ], Λ ) → H ∗ ( B ( L )) alg .We can define the map when the input is a fundamental class of a twisted sector fol-lowing [1], but there is a technical difficulty to define it in general cases. We will notconsider this map in this paper.Now, let us prove Theorem 8.1 and Theorem 8.2. ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 29
Proof of Theorem 8.1.
We will mainly follow the construction of [18], [1]. We con-struct a Kodaira-Spencer map using J -holomorphic discs as in Figure 3. As explainedbefore, the main difference in our case is that we do not need to make an output a mul-tiple of the fundamental class. This is why we can consider a general case. T n -action of[18] or (cid:90) /2-symmetry of [1] was used to achieve this. But we still need the Lagrangian L to be weakly unobstructed. After this adjustment, it is almost straightforward to adaptthe proof of [18], [1] to our setup, and we only give a brief sketch, and refer readers tothese references for the full construction. Also we are considering a general case where Q H ∗ ( M ) may have odd degree elements (like T ), our formula involves a sign. We ex-plain the sign computation at the end.For a moduli space of J -holomorphic discs M k + ( β ) for β ∈ H ( M , L ) and a cycle A ⊂ M , we define a Kodaira-Spencer map as follows. Let M k + ( β , A ) = M k + ( β ) × M A and consider their evaluation maps ev ki : M k + ( β , A ) → L at i -th marked point. Wedefine ks ( PD [ A ]) = ∞ (cid:88) k = ( ev k ) ! (( ev k ) ∗ b ∧ ··· ∧ ( ev kk ) ∗ b ) (8.2)From now on, we omit Poincaré dual from PD [ A ] and just write A . Gromov compact-ness shows that ks ( A ) ∈ B . We first check that this map is well-defined in cohomology. Lemma 8.4 (c.f. Proposition 2.4.15 [18]) . If A − B = ∂ R, then ks ( A ) − ks ( B ) = m b can H.Proof.
We define H : = ks ( R ), and consider its codimension one boundary contributions.One of the strata is ks ( ∂ R ) which corresponds to ks ( A ) − ks ( B ). The others come fromdisc bubbling. If the boundary marked point z and interior marked point z + are inthe same component, then the contribution from the bubble component is m b whichis a multiple of unit. Therefore such contribution vanishes by the usual argument. Theother case that z lies in a different disc component than z + , then it can be writtenas m b ( ks ( R )). We may take the projection to cohomology. Namely, we have a quasi-isomorphism from ( B ,{ m bk }) to B can ,{ m bk , can } and denote its linear component (partwith one input) by (cid:81) b . We have (cid:81) b ◦ m b = m b can (cid:81) b and hence we obtain the result. (cid:3) Now, let us show that ks is a ring homomorphism. The geometric idea behind this israther well-known. In the language of 2-dimensional open-closed topological confor-mal field theory, the closed-open map is a ring homomorphism. But the actual proofof this in [18] becomes rather technical. One of the issue there is that T n -equivarianceis essential to define ks -map, but the moduli space of J -holomorphic spheres does nothave T n -equivariant perturbation so they need to find a way to accommodate both. In[1], a simplified construction without such T n -equivariance is given. We follow closelythe construction in [1]. The main simplification is that we can use a uniform Kuran-ishi perturbation scheme, namely, a component-wise compatible continuous family ofmulti-section by Fukaya [16] for every moduli spaces involved.Consider the map forget : M maink + ( β , A ⊗ B ) → M main of forgetting the map and the boundary marked points except the first one. We are in-terested in forget − ( Σ ), forget − ( Σ ) where Σ is a stable disc with a sphere attached atthe interior, and Σ is a disc with two disc bubble each of which contains an interiormarked point (See Figure 3). The fiber product forget − ( Σ ) ( ev + , ev + ) × ( M × M ) ( f × f ) (8.3) For Σ = Σ , the above expression gives ks ( f ∗ Q f ), and for Σ = Σ , the above expressiongives ( − deg ks ( f ) m ( ks ( f ), ks ( f )). For an interval I ⊂ M main connecting Σ , Σ , (8.3)with Σ = I should give us the cobordism connecting these two operations. The restof the proof is the same as [1, Section 6.2] and [18] and we omit the details. But thesign ( − deg ks ( f ) did not appear previously due to even dimensionality of cohomologyclasses, and we will explain this later in the section.8.2. Proof of Theorem 8.2.
We label one of the embedded lift of L as (cid:101) L and label therest of the lift by (cid:101) L g = g · ( (cid:101) L ). Denote by (cid:101) L = (cid:76) g ∈ G (cid:101) L g . Th idea is to consider the modulispace of J -holomorphic stable polygon in M (similar to Figure 3) with interior insertionof A with boundary on (cid:101) L . We write it as M k + ( β , A ) for β ∈ H ( M , (cid:101) L ). From Lemma(6.15), we have a G -equivariant A ∞ -isomorphism between (cid:161) B (cid:111) (cid:98) G ,{ m b ⊗ k } (cid:162) (cid:39) (cid:161) C F ( (cid:101) L , (cid:101) L ) ⊗ Λ [ x , ··· , x n ],{ m (cid:101) bk } (cid:162) .Thus we can define ks following (8.2) as a map to the cohomology of the right hand side.We can prove that ks is well-defined in cohomology following Lemma 8.4.Now, G acts on the moduli space M k + ( β ) freely and we consider a G -equivariantKuranishi structure and perturbations. This defines a Fukaya A ∞ -algebra of Lagrangians C F ( (cid:101) L , (cid:101) L ) with a strict G -action on it, and its G -invariant part defines an A ∞ -algebra of L . Suppose that ambient cycle A is G -invariant. then moduli space for the Kodaira-Spencer map M k + ( β , A ) has a G -action on it, and the output of ks ( A ) become G -invariant as well. By Φ − , it corresponds to an element in the trivial sector B ⊗
1. Interms of Jac( W , (cid:98) G ), for a G -invariant cycle A , ks ( A ) should lie in the trivial sector ofJac( W , (cid:98) G ) which is Jac( W ) (cid:98) G . Since H ∗ ([ M / G ], Λ ) ∼= H ∗ ( M , Λ ) G (we consider Λ with (cid:67) -coefficient as usual) A may be considered as an element of H ∗ ([ M / G ]). Therefore, thismoduli space gives two associated maps, one from Q H ∗ ( M ) and one from H ∗ ([ M / G ]).Now, let us consider more interesting case that A is not G -invariant. Roughly ks ( A )will be in non-trivial sectors of Jac( W , (cid:98) G ). From finite abelian G -action, we can de-compose H ∗ ( M ) according to each character χ ∈ (cid:98) G . Namely if A is in χ -eigenspace, g · A = χ ( g ) A . By considering G -equivariant perturbations of the moduli space (cid:91) g ∈ G M ( g · β , g · A ),we observe that ks ( A ) lies in χ -eigenspace as well. Namely, we have Φ − ( ks ( A )) ∈ B ⊗ χ .Now, we consider the second (cid:98) G -action on B (see Definition 6.16). Lemma 8.5.
We claim that Φ − ( ks ( A )) ∈ ( B ⊗ χ ) (cid:98) G . Proof.
Recall that (cid:98) G -action acts on both Lagrangian Floer generators (by recording dif-ference of branches of (cid:101) L ) and on variables (from its action on the associated immersedsector at the quotient), but not on χ . Now, (cid:98) G -action on ks ( A ) come from two sources,one is the variables corresponding to b ’s, and the other is the output Floer generatorin C F ( (cid:101) L , (cid:101) L ). Given a J -holomorphic polygon u with k + (cid:101) L contributing to ks ( A ). Let us choose g , ··· , g k ∈ G such that j -thmarked point z j maps to the Lagrangian intersection L g i ∩ L g i + for i = ··· , k mod k .At j -th marked point, the next branch of lagrangian is obtained by the multiplication of g − i g i + ∈ G of the previous branch (in a counter-clockwise order), hence χ ∈ (cid:98) G action ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 31 on the dual variable will give χ ( g − i g i + ) for each i = ··· , k −
1. Hence χ -action onresulting variables on ks ( A ) is χ ( g − g ) ··· χ ( g − k − g k ) = χ ( g − g k ).On the other hand, the output of ks ( A ) lies in C F ( L g , L g k ) and by Definition 6.16, χ action on the output is given by the multiplication of χ (cid:161) ( g − g k ) − (cid:162) . Therefore, these twoactions cancel each other out and ks ( A ) is invariant under this additional (cid:98) G -action. (cid:3) Sign computation.
Let us explain the sign in the theorem 8.1. ks ( f ∗ Q f ) = ( − deg ks ( f ) m ( ks ( f ), ks ( f ))One can check deg L ( ks ( f )) = deg M ( f ). In [18], the above sign did not appear since non-trivial cohomology classes of toric manifolds (which are ( (cid:67) ∗ ) n -invariant) are of evendegree.Note that (8.3) can be identified with (by orientation formulae of [17, Chapter 8]) (cid:161) forget − ( Σ ) ev + × M f (cid:162) ev + × M f Let us consider the sign for forget − ( Σ ). We start by recalling the following orienta-tion formula for disk bubbling. Proposition 8.6. [17, Proposition 8.3.3]
We have an isomorphism ∂ M regm + ( β ) = (cid:91) β = β (cid:48) + β (cid:48)(cid:48) ( − ( m − m − + ( n + m ) M regm + ( β (cid:48) ) ev × ev M regm + ( β (cid:48)(cid:48) ) as oriented Kuranishi spaces where m = m + m − and n = dim( L ) . By applying the above formula twice, we identify a stratum in ∂ M regm + ( β ) (as ori-ented Kuranishi spaces) given by (cid:91) β = β + β + β (cid:161) M reg ( β ) ev × ev M reg ( β ) (cid:162) ev × ev M reg ( β )There is an analogous formula with interior marked points (cid:91) β = β + β + β (cid:161) M reg ( β ) ev × ev M reg ( β ) (cid:162) ev × ev M reg ( β )We take a fiber product (cid:161) • ev + × M f (cid:162) ev + × M f of the above from the right and after someyoga of moving around fiber products (which we leave as an exercise, but this does notbring any additional sign mainly because M is even dimensional), we get the following (cid:91) β (cid:161) M reg ( β ) ev × ev ks ( f ) (cid:162) ev × ev ks ( f )Fukaya-Oh-Ohta-Ono defined m -product to be the above with an additional sign.( − deg L ks ( f ) m ( ks ( f ), ks ( f ))Here deg L ks ( f ) = deg M f . Now we can argue as in [18] to find a cobordism between forget − ( Σ ) and forget − ( Σ ).
9. M
ATCHING ALGEBRAIC AND GEOMETRIC GENERATORS FOR ORBIFOLD J ACOBIANALGEBRA
We have shown in Theorem 7.1 under the Assumption 5.1 that we can use Floer the-ory to construct an A ∞ -algebra whose cohomology algebra is Jac( W , (cid:98) G ). But the proofthereof does not specify the precise isomorphism. In this section, we will describe moreprecise correspondence and identify geometric generators corresponding to the formalgenerators ξ h of orbifold Jacobian algebra in Definition 3.1.For this, let us recall briefly the construction of [32]. For a curved algebra R W , andits bimodule M (which means it is an R -bimodule whose left and right action by W agree), we have a mixed complex ( K ∗ ( R W , M )) whose total cohomology is Hochschildcohomology H ∗ ( R W , M ): To define this, we set K ∗ ( R ) = R e ⊗ k [ θ , ··· , θ n ] which is amixed complex with degree 1 differential δ Kos and degree ( −
1) differential δ curv given by δ Kos : = n (cid:88) i = ( x i − y i ) ∂ θ i , δ curv : = n (cid:88) i = ∇ x → ( x , y ) i ( W ) · θ i Also bimodule M may be considered as a R e -module with x i acting on the left and y i acting on the right. Then mixed complex ( K ∗ ( R W , M )) is defined as (cid:161) hom R e ( K −∗ ( R ), M ), ∂ Kos = δ ∨ Kos , ∂ curv = δ ∨ curv (cid:162) .This mixed complex can be identified (see (4.10) [32]) with the following (cid:161) M [ ∂ θ , ··· , ∂ θ n ], ∂ Kos : = n (cid:88) i = ( x i − y i ) ∂ θ i , ∂ curv : = − n (cid:88) i = ∇ x → ( x , y ) i ( W ) · θ i (cid:162) In the case of M = R (or R W ), left and right action agree and hence we may set x i = y i .Also, note that ∇ x → ( x , y ) i ( W ) | x = y = ∂ i W ( x ).Hence, K ∗ ( R W , R W ) computing Hochschild cohomology H ∗ ( R W , R W ) equals (cid:161) R [ ∂ θ , ··· , ∂ θ n ], ∂ Kos = ∂ curv : = − n (cid:88) i = ∂ x i W ( x ) · θ i (cid:162) whose cohomology is Jac( W ) if W has isolated singularities.For (cid:98) G -equivariant setting, consider the semi-direct product R W [ (cid:98) G ] = R W ⊗ k [ (cid:98) G ] withproduct structure ( r ⊗ g ) · ( r (cid:48) ⊗ g (cid:48) ) : = r ( g · r (cid:48) ) ⊗ g g (cid:48) .It is an R -bimodule. On R W ⊗ χ , x acts on the right by χ · x . Hence χ -sector of the complex K ∗ ( R W , R W [ (cid:98) G ]) can be identified with (cid:161) R [ ∂ θ , ··· , ∂ θ n ], δ Kos ( χ ) : = n (cid:88) i = ( x i − χ · x i ) ∂ θ i , δ curv ( χ ) : = − n (cid:88) i = ∇ x → ( x , y ) i ( W ) | y = χ · x · θ i (cid:162) (9.1)Let us consider the wedge degree of R [ ∂ θ , ··· , ∂ θ n ] (here ∂ θ i has degree one). Shklyarovshowed that the total cohomology of (9.1) is isomorphic to Jac( W | Fix( χ ) ) · ξ χ and 1 · ξ χ corresponds to a cohomology generator whose highest wedge degree term is (cid:81) i ∈ I χ ∂ θ i .More precisely, in [32], Shklyarov finds a quasi-isomorphic complex whose cohomol-ogy is generated by the above term only, and the quasi-isomorphism (to the originalcomplex) is given by exp tH W , χ . Since H W , χ action lowers the wedge degree, we obtainthe above claim. ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 33
On the other hand, we can compare (9.1) to the geometric setting via Assumption5.1. Given (cid:161) V ⊗ Λ R e , m b ( x ), b ( y )1 (cid:162) (which is a matrix factorization of W ( y ) − W ( x )), byrestricting it to ( χ − x , x ), on which we have a chain complex due to W ( x ) − W ( χ − · x ) = (cid:161) V ⊗ Λ R e , m b ( x ), b ( y )1 (cid:162) | ( χ − x , x ) (cid:39) (cid:161) R [ ∂ θ , ··· , ∂ θ n ], (cid:88) i ( x i − χ − · x i ) ∂ θ i + (cid:88) i ∇ x → ( x , y ) i W | ( χ − x , x ) · θ i (cid:162) .(9.2)By change of variables χ − x i → x i , we have an isomorphism (cid:161) V ⊗ Λ R , m b ( χ − · x ), b ( x )1 (cid:162) (cid:39) (cid:161) R [ ∂ θ , ··· , ∂ θ n ], (cid:88) i ( χ · x i − x i ) ∂ θ i + (cid:88) i ∇ x → ( x , y ) i W | y = χ · x · θ i (cid:162) .From these observations, we obtain the following. Proposition 9.1.
Given assumption 5.1, H ∗ ( B ⊗ χ , m b ⊗ ) is isomorphic as a module to χ -sector of Hochschild cohomology H ∗ ( R W , R W [ (cid:98) G ]) . In this isomorphism, ξ χ correspondsto a cohomology generator in ( B ⊗ χ , m b ⊗ ) whose highest wedge degree term (in the iso-morphism (9.2) ) is (cid:81) i ∈ I χ ( − ∂ θ i ) .Remark . The sign ( − | I χ | in the identification also appeared in the construction of(7.2). See Lemma 7.5. 10. O RBIFOLD J ACOBIAN ALGEBRA FOR T Take T which is obtained by identifying opposite sides of a regular hexagon. Thereis a (cid:90) /3-action given by rotation, and the quotient is an orbifold sphere (cid:80) . We canconsider an immersed Lagrangian (cid:76) ⊂ (cid:80) equipped with a non-trivial spin structure(for the symmetry, we follow [12] to equip (cid:76) with a unitary line bundle whose holonomyis ( −
1) which is equally distributed). This is the Seidel Lagrangian [28] (see also Efimov[20]). (cid:76) lifts to embedded circles in T , which are denoted by L , L g , L g where {1, g , g }denote elements of (cid:90) /3. See Figure 4.Define q : = T ω ( ∆ ) , where ω ( ∆ ) is the area of minimal triangle. In [9],[12], (cid:76) is shownto be weakly unobstructed and the potential W of (cid:76) is computed as follows. W = − φ ( q ) i ( x + x + x ) + ψ ( q ) i x x x where φ and ψ are elements of Novikov ring as follows φ ( q ) : = (cid:88) k ∈ (cid:90) ( − k + ( k +
12 ) q (6 k + , ψ ( q ) : = (cid:88) k ∈ (cid:90) ( − k + (6 k + q (6 k + . (10.1)Dual group (cid:98) G -action are given as follows. Defineˇ χ : = e π i /3 ,and denote elements of (cid:98) G by {1, χ , χ } where χ · x i = ˇ χ x i . We remark that the mirrorpotential given in [20] and [28] is just W alg = x n + x n + x n + x x x whereas the discpotentials in [9] and [11] are quantum corrected ones and they are infinite series forhyperbolic cases. Orbifold Jacobian algebra structure for Jac( W alg , (cid:90) / n ) has been com-puted in the appendix of [32].We are interested in the Kodaira-Spencer map from Q H ∗ ( T ) to Jac( W , (cid:90) /3) (includ-ing the quantum corrections). Define an A ∞ -algebra B : = C F (cid:161) ( (cid:76) , b ( x )),( (cid:76) , b ( x )) (cid:162) ⊗ Λ [ x , x , x ]. F IGURE
4. Dotted lines are embedded Lagrangians L , L g and L g . Ar-rows indicate how to move a point to one of the critical points of theMorse functions.Since the Seidel Lagrangian for (cid:80) = [ T / (cid:90) /3] satisfies Assumption 5.1, we have shownthe following sequence of isomorphisms (except the first map). Q H ∗ ( T ) ks → H ∗ (cid:161) ( B (cid:111) (cid:90) /3) (cid:90) /3 (cid:162) alg ∼= H ∗ ( MF (cid:90) /3 ( W ), MF (cid:90) /3 ( W )) ∼= Jac( W , (cid:90) /3). (10.2)We will show that ks is an isomorphism and furthermore we will find a precise corre-spondence as follows. First, let us introduce some notations for Q H ∗ ( T ) (which equals H ∗ ( T ) since there are no non-trivial holomorphic spheres). Let T be the fundamentalclass and pt T be the Poincaré dual of the point class in Q H ∗ ( T ). In addition, we de-note Poincaré duals of the meridian C h and longitude C v by [ C h ],[ C v ] respectively. Wewill take them as in the above Figure 4. Note that g · [ C h ] = [ C v ], g · [ C v ] = − [ C h ] − [ C v ].Therefore, [ C h ] − ˇ χ [ C v ],[ C v ] − ˇ χ [ C h ] are cohomology classes in χ and χ -sector of H ∗ ( T )respectively. Theorem 10.1.
Let the composition map be called ks , by abuse of notation. Then ks : Q H ∗ ( T ) → Jac( W , (cid:90) /3) in (10.2) is given by T (cid:55)→ T (cid:55)→ q ∂ W (cid:76) ∂ ql χ : = − i γ · [ C h ] − ˇ χ [ C v ]ˇ χ − (cid:55)→ ξ χ l χ : = − i γ · [ C v ] − ˇ χ [ C h ]ˇ χ − (cid:55)→ ξ χ . Here, γ is a modular form γ = (cid:88) k ∈ (cid:90) ( − k + i q (6 k + . (10.3) ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 35
Proof.
Let us first compute the products in Jac( W , (cid:90) /3). Lemma 10.2. In Jac( W , (cid:90) /3) , we have the following product ξ χ • ξ χ = σ χ , χ · where σ χ , χ can be computed (following the definition in [32] ) to be σ χ , χ = (27 φ − ψ ) i (1 − ˇ χ ) x x x ∈ Jac( W ). We have ξ χ • ξ χ = , ξ χ • ξ χ = from the degree condition (3.1) . We will prove this lemma in Appendix A.On the other hand, one can easily check that l χ ∪ l χ = l χ ∪ l χ =
0. Also, it is easy tocompute that l χ ∪ l χ = − γ · (cid:181) + ˇ χ − ˇ χ (cid:182) · pt T . (10.4)Now, we are ready to check that ks is a ring isomorphism. Observe that the Jacobianrelation identifies monomials x x x , x , x , x up to scaling and we can use it to showthat 124 q ∂ W ∂ q = i q (cid:161) − ψφ ∂φ∂ q + ∂ψ∂ q (cid:162) x x x ∈ Jac( W ).Also, we need the following identity of modular forms, which is proved in [13]. γ q (cid:179) − ψ ∂φ∂ q + φ ∂ψ∂ q (cid:180) = − φ (27 φ − ψ ). (10.5)By this identity, we can show the most nontrivial part ks ( l χ ∪ l χ ) = ks ( l χ ) • ks ( l χ )as follows: ks ( l χ ∪ l χ ) = − γ · (cid:181) + ˇ χ − ˇ χ (cid:182) ks (pt T ) = − γ · (cid:181) + ˇ χ − ˇ χ (cid:182) i q (cid:161) − ψφ ∂φ∂ q + ∂ψ∂ q (cid:162) x x x = (cid:181) + ˇ χ − ˇ χ (cid:182) i φ − ψ )24 x x x = i (27 φ − ψ )(1 − ˇ χ ) x x x = ξ χ • ξ χ .It finishes the proof of Theorem 10.1. (cid:3) Now, let us explain each map in (10.2) to justify our choice of l χ and l χ .First we will investigate images of ks in H ∗ (( B (cid:111) (cid:90) /3) (cid:90) /3 ). Recall that m b preserveseigenspaces of (cid:98) G -action. Let us first consider the trivial sector 1 ∈ (cid:98) G whose m b coho-mology is given by H ∗ (cid:161) ( B ⊗ (cid:90) /3 , m b (cid:162) ∼= H ∗ ( B , m b ) (cid:90) /3 ∼= Jac( W ) (cid:90) /3 Note that Jac( W ) is generated by 8 elements 1, x , x , x , x , x , x , x x x and (cid:90) /3 actson variables by multiplication of 3rd root of unity. Hence Jac( W ) (cid:90) /3 is generated by 1and x x x .In fact, we can choose a specific generator using the ks map. Since ks is a ring ho-momorphism, it sends the unit of quantum cohomology to the unit L ⊗
1. The class of x x x comes from ks map of the point class of T . In [1], they showed that ks map is anisomorphism for (cid:80) and computed ks (pt (cid:80) ). The same computation applies to ourcase and we have ks (pt T ) = q ∂ W ∂ q · ( L ⊗ T separated by the Seidel Lagrangian. Since each piece is set to haveequal area q , count of disks with a point insertion can be compared to the area of thecorresponding disc. This gives the above result.To find cohomology classes in χ , χ -sectors of Q H ∗ ( T ) that matches with ξ χ , ξ χ , westart with Kodaira-Spencer images of two circles [ C h ],[ C v ] generating Q H ( T ). Notethat C h intersect (cid:76) at 4 points and hence ks ([ C h ]) is given by Poincare duals of these 4points(see Figure 4). We wish to compare these points, but one should note that twodifferent points in (cid:76) are not m b , b cohomologous in general. Let us argue in terms ofsingular chains(which can be made into that of differential forms). For a line segment I connecting two points p , p (cid:48) in (cid:76) (in the domain of immersion and oriented as (cid:76) ), m b , b ( I )should have the classical term ∂ I = p (cid:48) − p as well as the constant disc contributionscoming from m ( b , I ), m ( I , b ). Since b comes from immersed sector, only one of thesetwo are composable, and the composable ones can be computed following the case ofthe unit identity m ( b ,1 (cid:76) ) = − b , m (1 (cid:76) , b ) = b .Using this idea, we compare each of 4 intersection points of C h ∩ (cid:76) , C v ∩ (cid:76) with chosenminimum points p , g · p , g · p of the Morse function on L , L g , L g respectively (seeFigure 4). From the Figure 4, it is not hard to obtain the following. Proposition 10.3.
We have ks ([ C h ]) = g · p − p − g · p + x ( g · X − X ) + x ( g · X − X ), ks ([ C v ]) = g · p − p − g · p + x ( X − g · X ) + x ( g · X − g · X ).Then, by Lemma 6.7, we have Φ − (cid:161) ks ([ C h ]) (cid:162) = ˇ χ p ⊗ χ + ˇ χ p ⊗ χ + (cid:161) ( ˇ χ − x X + ( ˇ χ − x X (cid:162) ⊗ χ + (cid:161) ( ˇ χ − x X + ( ˇ χ − x X (cid:162) ⊗ χ Φ − (cid:161) ks ([ C v ]) (cid:162) = ˇ χ p ⊗ χ + ˇ χ p ⊗ χ + (cid:161) (1 − ˇ χ ) x X + ( ˇ χ − ˇ χ ) x X (cid:162) ⊗ χ + (cid:161) (1 − ˇ χ ) x X + ( ˇ χ − ˇ χ ) x X (cid:162) ⊗ χ Φ − (cid:179) ks (cid:161) [ C h ] − ˇ χ [ C v ]ˇ χ − (cid:162)(cid:180) = (cid:161) p + ˇ χ ˇ χ − x X − χ − x X (cid:162) ⊗ χ (10.6) Φ − (cid:179) ks (cid:161) [ C v ] − ˇ χ [ C h ]ˇ χ − (cid:162)(cid:180) = (cid:161) p − ˇ χ ˇ χ − x X + ˇ χ ˇ χ − x X (cid:162) ⊗ χ . (10.7)Note that ([ C h ] − ˇ χ [ C v ]) and ([ C v ] − ˇ χ [ C h ]) are in fact χ and χ -eigenvectors for (cid:90) /3-action on H ( T ), and the above illustrates that ks preserves eigenspaces.Let us check that (10.6) and (10.7) are indeed nontrivial cocycles in B (cid:111) (cid:90) /3. FromLemma 6.11, we may consider the chain complex (cid:161) hom ( L , L ) ⊗ R , m b ( χ − ), b (cid:162) . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 37
By (5.4) with substitution ( x , y ) (cid:55)→ ( χ − · x , y ), we have m b ( χ − ), b ( e ) = (cid:88) ≤ i ≤ (1 − ˇ χ ) x i X i , m b ( χ − ), b ( ¯ X new ) = (1 − ˇ χ ) x p new − c X − c X , m b ( χ − ), b ( ¯ X new ) = (1 − ˇ χ ) x p new + c X − c X , m b ( χ − ), b ( ¯ X new ) = (1 − ˇ χ ) x p new + c X + c X .Thus, for any linear combination of { e , ¯ X new , ¯ X new , ¯ X new }, its image of m b ( χ − ), b cannotbe p + ˇ χ ˇ χ − x X − χ − x X , because the coefficient of p of the image is always a polyno-mial with zero constant term. In the same way, we can also prove that p − ˇ χ ˇ χ − x X + ˇ χ ˇ χ − x X is not an image of m b ( χ − ), b . This proves that (10.6) and (10.7) are nontrivialcocycles.Since Jac( W , (cid:98) G ) χ and Jac( W , (cid:98) G ) χ are 1-dimensional in this case, and same is true for χ and χ -eigenspaces of H ( T ). Therefore, we find that Kodaira-Spencer map sends χ and χ -sector of Q H ∗ ( T ) to the following spaces as a bijection. H ∗ ( B ⊗ χ , m b ⊗ ) ∼= Jac( W , (cid:98) G ) χ , H ∗ ( B ⊗ χ , m b ⊗ ) ∼= Jac( W , (cid:98) G ) χ .To find an exact generator that matches with ξ χ and ξ χ , we use Proposition 9.1.Namely, we look for cocycles in (cid:161) hom ( L , L ), m b ( χ − ), b (cid:162) and (cid:161) hom ( L , L ), m b ( χ − ), b (cid:162) respec-tively that has the highest wedge degree terms given by − p new = − γ p . (The sign appearsbecause | I χ | = | I χ | = − γ · [ C h ] − ˇ χ [ C v ]ˇ χ − − γ · [ C v ] − ˇ χ [ C h ]ˇ χ − ∈ H ∗ ( T )respectively. It concludes the explanation of the choice of l χ and l χ in Theorem 10.1. Remark . We can remove i = (cid:112)− H ∗ ( T ) alg (or alternatively Jac( W , (cid:90) /3) op ).A PPENDIX
A. A
LGEBRAIC COMPUTATION OF ORBIFOLD J ACOBIAN PRODUCT
In [32, Appendix A], Shklyarov computed orbifold Jacobian algebra structures for W : = x g + + x g + + x g + − x x x ,with the action by G : = {( ζ , ζ , ζ − ) ∈ ( (cid:67) ∗ ) | ζ g + = W , (cid:90) /3) for the quantum cor-rected potential W = − φ ( q ) i ( x + x + x ) + ψ ( q ) i x x x where (cid:90) /3 = {1, χ , χ } acts on Λ [ x , x , x ] by χ · x i = e π i /3 x i . Hence ˇ χ : = e π i /3 .To compute σ χ , χ (cid:48) , we first compute H W ( x , χ · x , x ) and H W , χ ( x ) for χ ∈ (cid:98) G . Recall that H W ( x , y , z ) = (cid:88) ≤ i ≤ j ≤ (cid:161) ∇ y → ( y , z ) i ∇ x → ( x , y ) j W (cid:162) θ j ⊗ θ i . The constant σ χ is easily computed as follows. One can check that d χ = σ χ = d χ ! Υ (cid:161) ( (cid:98) H W ( x , χ · x , x ) (cid:99) χ + (cid:98) H W ,1 ( x ) (cid:99) χ ⊗ + ⊗ (cid:98) H W , χ ( χ · x ) (cid:99) χ ) d χ ⊗ ∂ θ I ⊗ ∂ θ I χ (cid:162) = Υ (cid:161) (1 ⊗ ⊗ (1 ⊗ ∂ θ I χ ) (cid:162) = Υ (1 · ∂ θ I χ ) = Υ . We conclude that ξ is the multi-plicative identity of Jac (cid:48) ( W , (cid:90) /3).Now we compute σ χ , χ (cid:48) for both χ and χ (cid:48) are not 1. We have σ χ , χ = χ (cid:54)=
1, because d χ , χ = d χ + d χ − d χ = ∉ (cid:90) Let us compute σ χ , χ . From the definition of differences, we have (for j = (cid:161) ∇ y → ( y , z ) i ∇ x → ( x , y ) i W (cid:162) ( x , χ · x , x ) θ j ⊗ θ j = − φ i x j − ˇ χ θ j ⊗ θ j , (cid:88) ≤ i < j ≤ (cid:161) ∇ y → ( y , z ) i ∇ x → ( x , y ) j W (cid:162) ( x , χ · x , x ) θ j ⊗ θ i = ψ i x θ ⊗ θ + ˇ χψ i x θ ⊗ θ + ψ i x θ ⊗ θ .So we have H W ( x , χ · x , x ) = − φ i x − ˇ χ θ ⊗ θ − φ i x − ˇ χ θ ⊗ θ − φ i x − ˇ χ θ ⊗ θ + ψ i x θ ⊗ θ + ˇ χψ i x θ ⊗ θ + ψ i x θ ⊗ θ . (A.1)We also compute H W , χ ( x ) = − ˇ χψ i x − ˇ χ θ θ − ˇ χ ψ i x − ˇ χ θ θ , (A.2) H W , χ ( χ · x ) = − ψ i x − ˇ χ θ θ − ˇ χ ψ i x − ˇ χ θ θ , (A.3)Then σ χ , χ is the constant coefficient of the expression13! Υ (cid:161) ( (cid:98) H W ( x , χ · x , x ) (cid:99) + (cid:98) H W , χ ( x ) (cid:99) ⊗ + ⊗ (cid:98) H W , χ ( χ · x ) (cid:99) ) ⊗ ∂ θ ∂ θ ∂ θ ⊗ ∂ θ ∂ θ ∂ θ (cid:162) .If we denote coefficients of (A.1), (A.2) and (A.3) by H W ( x , χ · x , x ) = A θ ⊗ θ + A θ ⊗ θ + A θ ⊗ θ + A θ ⊗ θ + A θ ⊗ θ + A θ ⊗ θ and H W , χ ( x ) = B θ θ + B θ θ , H W , χ ( χ · x ) = C θ θ + C θ θ ,then by definition of Υ , we have σ χ , χ = (cid:98) A A A − A B C − A B C + A B C (cid:99) .Up to the relation 3 φ i x = φ i x = φ i x = ψ i x x x given by Jacobian ideal of W , we can check that σ χ , χ = (27 φ − ψ ) i (1 − ˇ χ ) x x x ∈ Jac( W ).Hence, we obtain ξ χ • ξ χ = (27 φ − ψ ) i (1 − ˇ χ ) x x x . ODAIRA-SPENCER MAP, LAGRANGIAN FLOER THEORY AND ORBIFOLD JACOBIAN ALGEBRAS 39 R EFERENCES[1] L. Amorim, C.-H. Cho, H. Hong and S.-C. Lau,
Big quantum cohomology of orbifold spheres ,arXiv:2002.11080.[2] M. Akaho and D. Joyce,
Immersed Lagrangian Floer theory , J. Differential Geom. 86 (2010), no. 3, 381-500.[3] D. Auroux,
Mirror symmetry and T-duality in the complement of an anticanonical divisor , J. Gökova Geom.Top. Vol 1 (2017) 51-91[4] V. Batyrev,
Quantum cohomology rings of toric manifolds,
Journées de Géométrie Algébrique d’Orsay, 1992,Astérisque 218, 9-34.[5] M. Ballard, D. Favero and L. Katzarkov,
A category of kernels for matrix factorizations and its implicationsfor Hodge theory , Publ. Math. Inst. Hautes Études Sci. 120 (2014), 1-111.[6] A. Basalaev and A. Takahashi,
Hochschild cohomology and orbifold Jacobian algebras associated to invert-ible polynomials , arXiv:1802.03912.[7] A. Basalaev, A. Takahashi and E. Werner,
Orbifold Jacobian algebras for invertible polynomials ,arXiv:1608.08962.[8] P. Candelas, X. de la Ossa, P. Green and L. Parkes,
A pair of Calabi-Yau manifolds as an exactly solublesuperconformal theory,
Nucl. Phys., B359 (1991), pp. 21-74. [9] C.-H. Cho, H. Hong and S.-C. Lau,
Localized mirror functor for Lagrangian immersions, and homologicalmirror symmetry for (cid:80) a , b , c , J. Differential Geom. 106 (2017), no. 1, 45-126.[10] C.-H. Cho, H. Hong and S.-C. Lau, Localized mirror functor constructed from a Lagrangian torus , J. Geom.Phys. 136 (2019), 284-320.[11] C.-H. Cho, H. Hong, S.-H. Kim and S.-C. Lau,
Lagrangian Floer potential of orbifold spheres , Adv. Math.306 (2017), 344-426.[12] C.-H. Cho, H. Hong and S.-C. Lau,
Noncommutative homological mirror functor , to appear at Memoirsof AMS. arXiv:1512.07128[13] C.-H. Cho, S. Lee and H.-S. Shin,
Pairings in mirror symmetry between a symplectic manifold and aLandau-Ginzburg B-model , Comm. Math. Phys. 375, 345-390 (2020).[14] C.-H. Cho and Y. -G. Oh,
Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toricmanifolds , Asian J. Math. 10 (2006), no. 4, 773-814.[15] T. Dyckerhoff,
Compact generators in categories of matrix factorizations , Duke Math. J. 159 (2011), no. 2,223-274.[16] K. Fukaya,
Cyclic symmetry and adic convergence in Lagrangian Floer theory , Kyoto J. Math. 50 (2010), no.3, 521-590.[17] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono,
Lagrangian intersection Floer theory: anomaly and obstruction.Part I , AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI.2009.[18] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono,
Lagrangian Floer theory and mirror symmetry on compact toricmanifolds , Astérisque No. 376 (2016), vi+340 pp.[19] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono,
Canonical models of filtered A ∞ -algebras and Morse complexes ,New perspectives and challenges in symplectic field theory, 201-227, CRM Proc. Lecture Notes, 49, Amer.Math. Soc., Providence, RI, 2009.[20] A. Efimov, Homological mirror symmetry for curves of higher genus , Adv. Math. 230 (2012), no. 2, 493-530. [21] A. Givental,
A mirror theorem for toric complete intersections,
Toplogical field theory, primitive forms and related topics (Kyoto, 1996), 141-175, Progr. Math. 160, Birkhäuser Boston, MA, 1998.[22] W. He, S. Li and Y. Li,
G-twisted braces and orbifold Landau-Ginzburg models , Comm. Math. Phys. 373,175-217 (2020).[23] H. Iritani,
An integral structure in quantum cohomology and mirror symmetry for toric orbifolds,
Adv.Math. 222 (2009), no. 3, 1016-1079.[24] M. Markl,
Transfering A ∞ (strongly homotopy associative) structures, Supplem. ai Rend. Circ. Matem.Palermo Ser. II 79(2006), 139-151.[25] A. Polishchuk and E. Zaslow,
Categorical mirror symmetry: the elliptic curve , Adv. Theor. Math. Phys.2:443-470, 1998.[26] A. Polishchuk and A. Vaintrob,
Chern characters and Hirzebruch-Riemann-Roch formula for matrix fac-torizations , Duke Math. J. 161 (2012), no. 10, 1863-1926.[27] P. Seidel,
Homological mirror symmetry for the quartic surface,
Memoirs of AMS, Vol 236, 1116 (2015). [28] P. Seidel,
Homological mirror symmetry for the genus two curve , J. Algebraic Geom. 20 (2011), no. 4, 727-769. [29] N. Sheridan,
On the homological mirror symmetry conjecture for pairs of pants , J. Differential Geom. Vol-ume 89, Number 2 (2011), 271-367.[30] N. Sheridan,
Homological mirror symmetry for Calabiâ ˘A¸SYau hypersurfaces in projective space , Invent.math. 199 (2015), 1-186.[31] N. Sheridan,
On the Fukaya category of a Fano hypersurface in projective space , Publications Mathéma-tiques de l’IHÉS, 124 (2016), no. 1, 165-317.[32] D. Shklyarov,
On Hochschild invariants of Landau-Ginzburg orbifolds , arXiv:1708.06030.[33] A. Strominger, S.-T. Yau and E. Zaslow,
Mirror symmetry is T-duality , Nucl. Phys. B 479 (1996), 243-259.[34] B. Toën,
The homotopy theory of dg-categories and derived Morita theory,
Invent. Math. 167 (2007), 615-667.[35] J. Tu,
Matrix factorizations via Koszul duality , Compos. Math. 150 (2014), no. 9, 1549-1578.C
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