Comparison of mirror functors of elliptic curves via LG/CY correspondence
CCOMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIALG/CY CORRESPONDENCE
SANGWOOK LEE
Abstract.
Polishchuk-Zaslow explained the homological mirror symmetry between Fukayacategory of symplectic torus and the derived category of coherent sheaves of elliptic curvesvia Lagrangian torus fibration. Recently, Cho-Hong-Lau found another proof of homologicalmirror symmetry using localized mirror functor, whose target category is given by graded matrixfactorizations. We find an explicit relation between these two approaches.
Contents
1. Introduction 12. Fukaya categories 32.1. Filtered A ∞ -categories 32.2. Triangulated A ∞ -categories 52.3. Fukaya category on surfaces 52.4. Derived Fukaya categories 83. Graded matrix factorizations 84. Orlov’s LG/CY correspondence 95. Polishchuk-Zaslow mirror symmetry 116. CY-LG mirror symmmetry(graded localized mirror functors) 136.1. Ungraded localized mirror functor 136.2. Graded localized mirror functor 157. Main theorem 167.1. Computations via G ◦ F LM L gr ◦ S S i for any i ∈ Z Introduction
Homological Mirror Symmetry(HMS) conjecture by Kontsevich has been a powerful moti-vation in recent developments of geometry and physics. Inspired by string theory, Kontsevichconjectured the equivalence of the derived Fukaya category of a Calabi-Yau manifold X and thederived category of coherent sheaves of the other Calabi-Yau manifold ˇ X , which is called themirror of X .The elliptic curve case was studied by Polishchuk-Zaslow[PZ]. Then Seidel[Sei3] proved theconjecture for the quartic surface. Also Abouzaid-Smith[AS] proved homological mirror symme-try for higher-dimensional(in particular 4-dimensional) tori. Many more important works hasfollowed, but we will not mention them further.On the other hand, inspired by the work of Seidel on genus two curve [Sei2], Cho-Hong-Lau[CHL1] developed, so called localized mirror functors formalism, and applied it to the studyof HMS for orbifold spheres. Their idea is to think of an immersed Lagrangian submanifold L in a r X i v : . [ m a t h . S G ] M a r SANGWOOK LEE a symplectic orbifold, and consider the Maurer-Cartan solutions of its A ∞ -algebra whose weakbounding cochains are given by immersed sectors. The superpotential which given by the count-ing of decorated polygons is a (quasi)homogeneous polynomial W . Then an explicit homologicalmirror functor is constructed by considering the (curved) Yoneda functor LM L ( · ) := CF ∗ ( L , · ),which gives an A ∞ -functor F u ( X ) → M F ( W ). Here F u means the subcategory whose objectsare unobstructed Lagrangians. In this correspondence the Floer complex CF ∗ ( L , L ) for an un-obstructed Lagrangian L , directly gives a matrix factorization of W . Taking twisted complexesand cohomologies on both sides, we get an exact functor.From now on we concentrate on the HMS of elliptic curves. Categorical mirror symmetry ofPolishchuk-Zaslow([PZ]) compares the derived category of coherent sheaves of an elliptic curve X and derived Fukaya category of a symplectic torus T . This foundational work gave a first non-trivial example of homological mirror symmetry. Roughly, they matched line bundles of degree d on X to the lines of slope d in X both of which may come with additional data (tensoring withhigher dimensional bundles and flat connections on bundles respectively). Intersections betweenlines translates to theta functions and the Floer product corresponds to theta identities.In Cho-Hong-Lau [CHL1], one first considers the the symplectic torus T (with Z / Z / P , , . The immersed Lagrangian in this orbifold (called SeidelLagrangian) defines the Landau-Ginzburg mirror (Λ , W ) with an A ∞ -functor from Fukayacategory of P , , to the dg-category of matrix factorizations. To recover the mirror symmetryof the original symplectic torus T , one need to take the Z -graded version of this functor fromgraded Fukaya category of T to the graded matrix factorization category of M F Z ( W ) ([CHL2])Hence, we have two different kinds of homological mirror symmetries of elliptic curves. Theyhave B -model categories as a derived category and a category of matrix factorizations, respec-tively. Indeed, by Orlov’s theorem[Or], these two categories are equivalent. Namely, if W is a(quasi-)homogeneous polynomial which defines a smooth projective CY hypersurface X , then D b Coh ( X ) (cid:39) HM F Z ( W ). This equivalence is called a Landau-Ginzburg/Calabi-Yau (LG/CYfor short) correspondence .So far we have different exact functors between triangulated categories, and we can ask howthey are related to each other. We find an explicit relation as follows.
Theorem.
We have a commutative diagram of exact functors D π F u ( T ) S i (cid:47) (cid:47) Φ (cid:15) (cid:15) D π F u ( T ) LM L (cid:15) (cid:15) D b Coh ( X ) G i (cid:47) (cid:47) HM F Z ( W ) . where Φ is the mirror functor of [PZ, AS] and (1.1) S i = [ − j ] ◦ τ d ◦ t (0 , / ◦ Ç − i + 2 1 å where j = (cid:98)− i (cid:99) , d = − i − j , τ is the rotation by − π/ and t (0 , / is the parallel transport by (0 , / . Remark 1.1.
The definition of LM L involves a choice of a character γ : Z / → U (1) . Varyingthe choice of γ , the functors S i may also vary. Here we have fixed one choice. Namely, two homological mirror symmetry are equivalent after certain geometric transforma-tion S i (rotation and translation) and shifts.We remark that in [CHL2], non-commutative homological mirror symmetry of elliptic curvehas been discussed (whose mirror is given by non-commutative Landau-Ginzburg model, which OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 3 is a choice of central element W in Sklyanin algebra). The relation between commutative andnon-commutative mirror functors is not known, and we hope to apply the method of this paperto compare commutative and non-commutative mirror functors in the future.Let us comment on the proof of the theorem. Recall that Orlov’s argument is based on thefact that D b Coh ( X ) and HM F Z ( W ) are Verdier quotients of D b (gr − A ). Instead of consideringquotients of D b (gr − A ) itself, consider quotients of its subcategory as π i : D b (gr − A ≥ i ) (cid:44) → D b (gr − A ) → D b (gr − A ) /D b (tors − A ) (cid:39) D b Coh ( X ) ,q i : D b (gr − A ≥ i ) (cid:44) → D b (gr − A ) → D b (gr − A ) / Perf − A (cid:39) HM F Z ( W ) . Then Orlov constructed adjoint functors of above ones: R ω i : D b Coh ( X ) → D b (gr − A ≥ i ) ,ν i : HM F Z ( W ) → D b (gr − A ≥ i ) . Then he proves that π i ◦ ν i : HM F Z ( W ) ∼ −→ D b Coh ( X ) and G i := q i ◦ R ω i : D b Coh ( X ) ∼ −→ HM F Z ( W ) if W defines a CY variety.The most nontrivial part for the proof is to compute images of R ω i which is a right derivedfunctor. It is not enough to know only cohomologies of the images, but we need to know themas genuine R Hom-complexes precisely, because we need to compare morphisms between them,not just objects themselves. The scheme for the computation is to use the Gorenstein propertyof the ring
R/W , because it can be used to make many terms in the cohomology long exactsequence vanish, so that the object which we suspect to be an image of G i is in the subcategorywhich is a component of the semiorthogonal decomposition, hence it is indeed an image of G i .We also remark that when we compare morphisms of matrix factorizations we do not have tocompute all entries, but it is sufficient to compare constant entries which in fact determine themorphism completely. This observation considerably reduces the counting of holomorphic strips.The organization of the paper is as follows. In Section 2 we recall basic ingredients of Fukayacategories. In Section 3 we introduce the notion of graded matrix factorizations and relate themwith a quotient of a derived category. Then we relate derived categories and matrix factorizationsby recalling Orlov’s LG/CY correspondence. In following two sections we introduce two differentkinds of mirror symmetry of elliptic curves. Finally in Section 7 we prove our main theorem. Acknowledgements.
The author thanks Cheol-hyun Cho for the encouragement and a lotof helpful suggestions. He also thanks Hansol Hong and Siu-Cheong Lau for generously sharingtheir ideas and results. He is grateful to Yong-Geun Oh for his interests in this problem anda number of useful comments. He thanks Dohyeong Kim and Dong Uk Lee for letting him tocare about crucial issues about elliptic curves. He is grateful to the Center for Geometry andPhysics(IBS) for its hospitality and support when he worked on this paper as a postdoctoralresearch fellow of the center. This work was supported by IBS-R003-D1.2.
Fukaya categories
We recall the definitions and relevant theorems of A ∞ -categories and Lagrangian Floer theorymainly to set the notations (we refer readers to [FOOO], [Aur] for example).2.1. Filtered A ∞ -categories.Definition 2.1. The
Novikov field is Λ := (cid:110) (cid:88) i ≥ a i T λ i | a i ∈ C , λ i ∈ R , λ i → ∞ as i → ∞ (cid:111) . SANGWOOK LEE
A filtration F • Λ of Λ is given by F λ Λ := (cid:110) (cid:88) i ≥ a i T λ i | λ i ≥ λ for all i (cid:111) ⊂ Λ , F + Λ := (cid:110) (cid:88) i ≥ a i T λ i | λ i > i (cid:111) . The
Novikov ring Λ is defined as Λ := F Λ. Definition 2.2. A filtered A ∞ -category C over Λ consists of a class of objects Ob ( C ) and theset of morphisms hom C ( A, B ) for a pair of objects A, B of C with the following conditions: (1) hom ( A, B ) is a filtered Z -graded Λ -vector space for any A, B ∈ Ob ( C ) , (2) for k ≥ there are multilinear maps of degree 1 m k : hom ( A , A )[1] ⊗ hom ( A , A )[1] ⊗ · · · ⊗ hom ( A k − , A k )[1] → hom ( A , A k )[1] such that they preserve the filtration and satisfy the A ∞ -relation (cid:88) k + k = n +1 k (cid:88) i =1 ( − (cid:15) m k ( x , ..., x i − , m k ( x i , ..., x i + k − ) , x i + k , ..., x n ) = 0 where (cid:15) = (cid:80) i − j =1 ( | x j | + 1) . Here, m means that for each object A we have m A ∈ hom ( A, A )[1] = hom ( A, A ) . If m (cid:54) = 0 , C is called a curved A ∞ -category. Otherwise, C is called strict. If there is only one object, then C is called an A ∞ -algebra. If only m and m are nonzero, then C is called a dg category.In this paper, every A ∞ -category is filtered over Λ. A ∞ -categories are generalizations of dgcategories where composition of morphisms may be associative only up to homotopy.To understand the meaning of the A ∞ -relation with possibly nonzero m , we write down therelation for the simplest case. For x ∈ hom ( A, B ),(2.1) m ( x ) + m ( m A , x ) + ( − | x | +1 m ( x, m B ) = 0 . Hence if m (cid:54) = 0, m may not be a differential (i.e. m = 0). Definition 2.3.
For an object A in an A ∞ -category, e A ∈ hom ( A, A ) is called a unit if itsatisfies (1) m ( e A , x ) = x, m ( y, e A ) = ( − | y | y for any x ∈ hom ( A, B ) , y ∈ hom ( B, A ) , (2) m k +1 ( x , ..., e A , ..., x k ) = 0 for any k (cid:54) = 1 . we recall the deformation theory of A ∞ -category. Definition 2.4.
An element b ∈ F + hom ( A, A ) is called a weak bounding cochain of A if it isa solution of the weak Maurer-Cartan equation (2.2) m ( e b ) := m A + m ( b ) + m ( b, b ) + · · · = P O ( A, b ) · e A for some P O ( A, b ) ∈ Λ . If such a solution exists, then A is called weakly unobstructed . If thereexists a solution b such that P O ( A, b ) = 0 , then b is called a bounding cochain and A is called unobstructed . P O ( A, b ) is called the Landau-Ginzburg superpotential of b . We denote M weak ( A ) be the set of weak bounding cochains of A . Then P O ( A, · ) is a functionon M weak ( A ). We also define M λweak ( A ) := { b ∈ M weak ( A ) | P O ( A, b ) = λ } . Following Proposition 1.20 of [Fu], given an A ∞ -category C , under the assumption M λweak ( A )is nonempty for some objects, we define a new A ∞ -category C λ as Ob ( C λ ) = (cid:91) A ∈ Ob ( C ) { A } × M λweak ( A ) , OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 5 hom C λ (( A , b ) , ( A , b )) = hom C ( A , A )with the following A ∞ -structure maps m b ,...,b k k : hom C λ (( A , b ) , ( A , b )) ⊗· · ·⊗ hom C λ (( A k − , b k − ) , (( A k , b k )) → hom C λ (( A , b ) , ( A k , b k )) ,m b ,...,b k k ( x , ..., x k ) := (cid:88) l ,...,l k m k + l + ··· + l k ( b l , x , b l , ..., b l k − k − , x k , b l k k )where x i ∈ hom C λ (( A i , b i ) , ( A i +1 , b i +1 )) . A ∞ -relation is induced by the weak Maurer-Cartanequation (2.2). Theorem 2.5.
Let ( A , b ) , ( A , b ) ∈ Ob ( C λ ) . Then ( m b ,b ) = 0 . Proof.
Let x ∈ hom C λ (( A , b ) , ( A , b )) . Then the A ∞ -equation is( m b ,b ) + m ( m ( e b ) , x ) + ( − | x | +1 m ( x, m ( e b )) = 0 . By m ( e b ) = λ · e A , m ( e b ) = λ · e A and by definition of units, m ( m ( e b ) , x ) + ( − | x | +1 m ( x, m ( e b )) = 0 , so ( m b ,b ) = 0 . (cid:3) So, under the existence of the weak Maurer-Cartan solutions, we get strict A ∞ -categories byrestricting to objects sharing certain value of the Landau-Ginzburg(LG for short) superpotential. Definition 2.6.
Let C and C (cid:48) be A ∞ -categories. An A ∞ -functor between C and C (cid:48) is a collection F = {F i } i ≥ consisting of • F : Ob ( C ) → Ob ( C (cid:48) ) , • F k : hom C ( A , A )[1] ⊗ · · · ⊗ hom C ( A k − , A k )[1] → hom C (cid:48) ( F ( A ) , F ( A k ))[1] of degree 0which are subject to the following condition: (cid:80) i,j ( − | x | (cid:48) + ··· + | x i − | (cid:48) F i − j + k ( x , ..., x i − , m C j − i +1 ( x i , ..., x j ) , x j +1 , ..., x k )= (cid:80) l m C (cid:48) l +1 ( F i − ( x , ..., x i ) , F i − i ( x i +1 , ..., x i ) , ..., F k − i l ( x i l +1 , ..., x k )) . Triangulated A ∞ -categories. By [Sei1], we know that any A ∞ -category C admits acohomologically fully faithful functor into another A ∞ -category which is called a triangulatedenvelope of C , in which we have exact triangles and shift functors. We take the most commonconstruction of triangulated envelope given by so-called twisted complexes. Since we do not usenon-trivial twisted complex in this paper, we omit its precise definition(and refer readers to[Sei1]) and just give a short summary: given an A ∞ -category C we add formal shifts and formaldirect sums, and equip an object E = N (cid:77) i =1 E i [ k i ] with a strictly lower triangular map δ : E → E such that (cid:88) k ≥ m k ( δ, ..., δ ) = 0 . Then the pair (
E, δ ) is called a twisted complex . Morphisms amongthem and A ∞ -structure maps are defined in the most canonical way, and denote the resulting A ∞ -category by T w ( C ) . Fukaya category on surfaces.
We will use a version of Fukaya category of surface M described in [Sei2] with a different set of conventions(as used in [CHL1]). We recall relevant ingre-dients for readers convenience. Roughly, Fukaya category of a symplectic manifold M (denotedby F u ( M )) is an A ∞ -category whose objects are Lagrangian submanifolds with additional dataand with morphisms given by Floer complexes. For simplicity, assume that L and L (cid:48) are orientedspin Lagrangian submanifolds which intersect transversely. Then the Floer complex CF ( L, L (cid:48) )is a vector space over Λ whose basis elements are intersections of L and L (cid:48) . Each intersection SANGWOOK LEE
Figure 1.
The left picture is a path from T p L with phase 0 to T p L with phase π . In this case deg( p ) = 1 . The right one is a path from T p L with phase 0 to T p L with phase − π , and deg( p ) = 0 . has an associated index (or parity in Z / p ∈ L ∩ L (cid:48) , choosea smooth path of oriented Lagrangian subspaces λ p ,p ( t ) for 0 ≤ t ≤ p , λ p ,p (0) = T p L and λ p ,p (1) = T p L (cid:48) . Then concatenate the positive definite path γ from T p L (cid:48) to T p L , whichdoes not depend on the orientation of Lagrangians. The homotopy class of the loop γ ◦ λ p ,p inLagrangian Grassmannian from T p L to itself gives a winding number, which is called the degreeof λ p ,p . Here, a positive definite path from T p L to T p L is defined by identifying L ∼ = R n and L ∼ = i R n at p = (0 , , ...,
0) and taking the path exp ( πit ) · R n for 0 ≤ t ≤ / λ p ,p ( t ), and also ifwe consider a Calabi-Yau manifold M and its graded Lagrangians, then there is a canonicalLagrangian path between T p L and T p L (cid:48) , in the sense that the path should preserve phases.Recall that an oriented Lagrangian submanifold in a CY manifold ( M, ω,
Ω) is graded if there isa function θ L : L → R such thatΩ( X ( p ) ∧ · · · ∧ X n ( p )) | Ω( X ( p ) ∧ · · · ∧ X n ( p )) | = e iθ L ( p ) for any positively oriented wedge of vector fields X ∧ · · · ∧ X n of L . θ L is called the phasefunction of L . If θ L is constant, then L is called special Lagrangian . Hence in the graded casethe degree of each intersection point is well-defined in Z .Floer differential m : CF ( L, L (cid:48) ) → CF ( L, L (cid:48) ) is defined as m ( p ) := (cid:88) q ∈ L ∩ L (cid:48) ind([ u ])=1 M ( p, q ; [ u ])) T ω ( u ) q where u is a J -holomorphic strip u : R × [0 , → M with u ( s, ∈ L, u ( s, ∈ L (cid:48) , lim s →−∞ u ( s, t ) = p, lim s →∞ u ( s, t ) = q. And ω ( u ) is the symplectic area of u . The index of the strip u is defined bythe Maslov index. Higher A ∞ -operations on morphisms m k : CF ( L , L ) ⊗· · ·⊗ CF ( L k − , L k ) → CF ( L , L k ) is defined by counting J -holomorphic polygons.Let p i ∈ CF ( L i − , L i ) and q ∈ CF ( L , L k ) . We define a moduli space M ( p , ..., p k ; q ) of J -holomorphic polygons whose domains are D minus k + 1 boundary points cyclically or-dered by z , ..., z k , z , arcs between p i and p i +1 are mapped inside L i (and inside L k between p k and q ), and the images near those punctures are asymptotically p , ..., p k , q , respectively. OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 7
Figure 2.
Holomorphic polygon u contributing to m k ( p , ..., p k ).Let M ( p , ..., p k ; q ; β ) be a subset of M ( p , ..., p k ; q ) which consists of holomorphic polygons ofhomotopy class β . Then the dimension of the moduli space is given bydim M ( p , ..., p k ; q ; β ) = k − β ) . The index of u ∈ M ( p , ...p k ; q ) is also given by the Maslov index. Fix a trivialization of u ∗ T M so that we get paths of Lagrangian subspaces l , l , ..., l k on L , L , ..., L k respectively. Then westart from T p L , at corners p i concatenate negative definite paths, move along l i until we arriveat T q L k . At q concatenate the positive definite path and move along l to arrive at T p L again.The index of u is defined by the Maslov index of the loop described above, and it depends onlyon the homotopy class of u . Now we define m k : CF ( L , L ) ⊗ · · · ⊗ CF ( L k − , L k ) → CF ( L , L k )by m k ( p , ..., p k ) := (cid:88) pi ∈ Li − ∩ Li,q ∈ L ∩ Lk ind[ u ]=2 − k M ( p , ..., p k ; q ; [ u ])) T ω ( u ) q. Recall that in graded case we can define degrees of Lagrangian intersections in Z . Then if aholomorphic polygon u has corners p ∈ L ∩ L , ..., p k ∈ L k − ∩ L k , q ∈ L ∩ L k , thenind( u ) = deg( q ) − deg( p ) − · · · − deg( p k )where q ∈ CF ( L , L k ).Now let us consider the case of surfaces. The precise construction of Fukaya category is moreinvolved since one has to deal with non-transverse Hom spaces CF ( L, L ). In [Sei2], a Morsefunction on S has been chosen so that the Hom space is generated by critical points. Werefer readers to [Sei2] for further discussions. Let us recall the definition of orientation for thecounting of polygons from [Sei2]. Let u ∈ M ( p , ..., p k ; q ) whose boundary lies on Lagrangiansubmanifolds as above. The sign of u is determined by the following steps. • If a Lagrangian is equipped with a nontrivial spin structure, put a point ◦ on it, onwhich the nontrivial spin bundle is twisted. • Disagreement of the orientation of ∂u on L is irrelevant. • If the orientation of ∂u on ˙ p i p i +1 does not agree with L i , the sign is affected by ( − | p i | . • If the orientation of ∂u on ˜ p k q does not agree with L k , the sign is affected by ( − | p k | + | q | . • Mutiply ( − l when ∂u passes through nontrivial spin points ◦ l times.The structure maps { m k } k ≥ define an A ∞ -structure, and the resulting A ∞ -category is called the Fukaya category of M and written as F u ( M ). In general Fukaya category may be obstructed, i.e. m is not zero, so CF ( L, L (cid:48) ) might not be a chain complex. But if we form an A ∞ -subcategory SANGWOOK LEE
F uk λ ( M ) of weakly unobstructed objects equipped with weak bounding cochains whose LGsuperpotentials have same value λ , then m on F uk λ ( M ) is a differential, and if ( L, b ) , ( L (cid:48) , b (cid:48) ) ∈ F uk λ ( M ), the cohomology of ( CF (( L, b ) , ( L (cid:48) , b (cid:48) )) , m ) is called the Floer cohomology of the pair((
L, b ) , ( L (cid:48) , b (cid:48) )) , denoted by HF (( L, b ) , ( L (cid:48) , b (cid:48) )) . In particular,
F u ( M ) is an A ∞ -subcategory of F u ( M ) of unobstructed objects. We remark another important fact that CF ( L, L (cid:48) ) is homotopyequivalent to CF ( L, φ ( L (cid:48) )) if φ is a Hamiltonian diffeomorphism. In particular, if L = L (cid:48) , then CF ∗ ( L, L ) ∼ = CF ∗ ( L, φ ( L )) for any Hamiltonian diffeomorphism φ , and HF ∗ ( L, L ) ∼ = H ∗ ( L, Λ).By definition, weak bounding cochains of L are in CF ( L, L ) . Derived Fukaya categories.
Since
F u ( M ) is an A ∞ -category, we also have its triangu-lated envelope T w ( F u ( M )) by adding twisted complexes of Lagrangians, and taking its coho-mology category, we get the derived Fukaya category
DF u ( M ) . Taking split-closure, we get thesplit-closed derived Fukaya category D π F u ( M ) . We will be mainly concerned with direct sums of Lagrangian submanifolds with new kindsof bounding cochains which occur by intersections between direct summands. First we clar-ify the meaning of direct sums and shifts in Fukaya categories. A direct sum of Lagrangiansubmanifolds is just the union of them. It can be also considered to be an immersed La-grangian. Given an object A in a triangulated A ∞ -category, A [1] is featured by the property hom i ( A [1] , B ) ∼ = hom i − ( A, B ), hom i ( B, A [1]) ∼ = hom i +1 ( B, A ). Hence, by definition of thedegree of morphisms(or intersections) between Lagrangian submanifolds, in non-graded case [1]is just reversing of the orientation. In 1-dimensional graded case in which we are interested, itcorresponds to the change of phase by − π .If L = L ⊕ · · · ⊕ L n , then CF ∗ ( L, L ) ∼ = (cid:77) ≤ i,j ≤ n CF ∗ ( L i , L j ) . Then there is another kind ofdegree 1 cochains which are given by CF ( L i , L j ) with i (cid:54) = j , in addition to those of degree1 cochains of a single embedded Lagrangian submanifold. In particular if L is 1-dimensionaland all L i are transverse to each other without triple(or more multiple) intersections, then someintersections among them are degree one cochains, and furthermore they can contribute to bea part of weak bounding cochains. We will encounter such an example later, namely (lifts of)Seidel Lagrangian on T . 3. Graded matrix factorizations
Let R = Λ[ x , ..., x n ] be a graded ring with deg( x i ) = d i . Definition 3.1.
Let W ∈ R be a (quasi)homogeneous polynomial of degree d . M F Z ( W ) is a dgcategory whose object ( P, d P ) is represented as a pair of graded morphisms p : P → P and p : P → P ( d ) , where P and P are graded free R -modules and p ( d ) ◦ p = W · id : P → P ( d ) ,p ◦ p = W · id : P → P ( d ) . Equivalently, an object described above is also expressed as a quasi-periodic infinite sequence K · : · · · (cid:47) (cid:47) K i k i (cid:47) (cid:47) K i +1 k i +1 (cid:47) (cid:47) K i +2 (cid:47) (cid:47) · · · where K i +1 = P ( i · d ) , K i = P ( i · d ) , k i = p ( i · d ) , k i +1 = p ( i · d ) . Then hom jMF Z ( W ) ( K · , L · ) := (cid:110) f · : K · → L · + j , graded | f i +2 = f i ( d ) (cid:111) and d : hom jMF Z ( W ) ( K · , L · ) → hom j +1 MF Z ( W ) ( K · , L · ) is defined by ( d ( f · )) i := l i + j ◦ f i + ( − j f i +1 ◦ k i where l n : L n → L n +1 . Compositions are defined as usual.
OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 9
Observe that given a matrix factorization K · it is natural to define the shift K · [1] such that K [1] i = K i +1 and k [1] i = − k i +1 . It is clear that K · [2] = K · ( d ) . Definition 3.2.
Given a dg category C , its cohomology category H ( C ) is defined by the sameobjects as those of C , and morphism spaces as 0th cohomologies of d : hom j → hom j +1 . Proposition 3.3.
The cohomology category H ( M F Z ( W )) is a triangulated category with exacttriangles K · f (cid:47) (cid:47) L · (cid:47) (cid:47) C · ( f ) (cid:47) (cid:47) K · [1] where the mapping cone of f is defined as C · ( f ) : · · · (cid:47) (cid:47) L i ⊕ K i +1 c i (cid:47) (cid:47) L i +1 ⊕ K i +2 c i +1 (cid:47) (cid:47) L i +2 ⊕ K i +3 (cid:47) (cid:47) · · · such that c i = Ç l i f i +1 − k i +1 å . The proof is standard, as in the case of homotopy categories over abelian categories.
Definition 3.4.
We write
HM F Z ( W ) := H ( M F Z ( W )) and call it the category of gradedmatrix factorizations of W . Remark 3.5.
The category
HM F Z ( W − λ ) is nontrivial(i.e. it contains nonzero objects) onlywhen λ is a critical value. Since we only deal with homogeneous polynomials, 0 is a criticalvalue(a degree 1 polynomial does not admit any nontrivial matrix factorization), so the categorywe are interested in this paper is nontrivial. Let A = R/W . Since W is homogeneous, A is a graded ring. Then the category gr − A of finitely generated graded A -modules is an abelian category. We define Perf − A as the fullsubcategory of chain complexes of A -modules, which are quasi-isomorphic to bounded complexesof projectives. Then Perf − A is a thick subcategory of D b (gr − A ) . We recall a useful lemma.
Lemma 3.6.
HM F Z ( W ) (cid:39) D grsg ( A ) , where D grsg ( A ) := D b (gr − A ) / Perf − A. We omit the proof but explain its origin. Since A is singular, the minimal A -free resolutionof an object in D b (gr − A ) need not terminate, but it eventually become quasi-(2-)periodic byEisenbud’s theorem [Eis]. If we replace free A -modules in the resolution by free R -modulesof same ranks and consider differentials as morphisms of R -modules, then the asymptotic 2-periodic part indeed becomes a matrix factorization of W . If two objects in D b (gr − A ) havefree resolutions which are asymptotically the same, then they define equivalent object in D grsg ( A )by definition, or equivalently, they give the same matrix factorization. Finally, we remark thatgiven a matrix factorization K · we take Cok( k − : K − → K ) to obtain an object of D grsg ( A ) . Orlov’s LG/CY correspondence
Let X = Proj( A ) where A = R/W as above, i.e. R = Λ[ x , ..., x n ] and W is a homogeneouspolynomial. In this section we recall the correspondence between HM F Z ( W ) and D b Coh ( X )in [Or]. Remark 4.1. A is a Gorenstein algebra, i.e. it has finite injective dimension n and if D ( k ) := R Hom A ( k , A ) where k ∼ = A/ ( x , ..., x n ) , (observe that k ∼ = Λ as a vector space. Nevertheless, wedistinguish the notation k from Λ because we want to emphasize that it is an A -module) thenit is isomorphic to k ( a )[ − n ] for some integer a which is called the Gorenstein parameter . Thishomological condition on A enables us to construct various derived functors (in later sections)between bounded derived categories. Also, if W defines a Calabi-Yau variety, then a = 0 . Theseproperties will be crucially used in Section 7. The idea of LG/CY correspondence comes from the fact that two categories are both Verdierquotients of D b (gr − A ). Let tors − A be the subcategory of gr − A of torsion modules, i.e. A -modules which are finite dimensional over A ∼ = Λ . By Serre’s theorem, D b Coh ( X ) (cid:39) D b (qgr − A ) := D b (gr − A ) /D b (tors − A ), whereas HM F Z ( W ) (cid:39) D b (gr − A ) / Perf − A just as shown above.If the quotient functors π : D b (gr − A ) → D b (qgr − A ) and q : D b (gr − A ) → D grsg ( A ) haveadjoint functors, then we can lift objects and morphisms in a quotient category to those in D b (gr − A ), and project them to the other quotient, and then we would obtain functors betweentwo quotient categories. Unfortunately, q does not admit any adjoint functor while π admitsa right adjoint, but if we consider restrictions π i : D b (gr − A ≥ i ) (cid:44) → D b (gr − A ) → D b (qgr − A )and q i : D b (gr − A ≥ i ) (cid:44) → D b (gr − A ) → D grsg ( A ), where gr − A ≥ i consists of modules M such that M p = 0 for p < i , then π i still admits a right adjoint, and q i has a left adjoint. Now we describethe adjoint of π i . Lemma 4.2.
Define R ω i : D b (qgr − A ) → D b (gr − A ≥ i ) by R ω i ( M ) := ∞ (cid:77) k = i R Hom D b (qgr − A ) ( πA, M ( k )) . Then R ω i is fully faithful and R ω i is the right adjoint to π i . Moreover, all cohomologies R j ω i ( M ) are contained in tors − A for j > . We use the same notation for the functor R ω i : D b Coh ( X ) → D b (gr − A ≥ i ), R ω i ( E ) := ∞ (cid:77) k = i R Hom D b Coh ( X ) ( O X , E ( k )) . Note that R ω i is well-defined from the Gorenstein condition. Definition 4.3.
Let C be a triangulated category. C = (cid:104)A , B(cid:105) is called a semiorthogonal decom-position of C if (1) A and B are full triangulated subcategories. (2) For any object in C ∈ Ob ( C ) , there is an exact triangle A [1] (cid:47) (cid:47) B (cid:126) (cid:126) C (cid:95) (cid:95) for some A ∈ Ob ( A ) and B ∈ Ob ( B ) . In this case, we call A and B as orthogonalprojections of C onto A and B respectively. (3) Hom C ( B, A ) = 0 for any B ∈ Ob ( B ) , A ∈ Ob ( A ) . Lemma 4.4.
Let S ≥ i be the triangulated subcategory generated by k ( e ) with e ≤ − i , and P ≥ i be the triangulated subcategory generated by A ( e ) with e ≤ − i . Then for any i ∈ Z we havesemiorthogonal decompositions D b (gr − A ≥ i ) = (cid:104)D i , S ≥ i (cid:105) = (cid:104)P ≥ i , T i (cid:105) where D i is equivalent to D b (qgr − A ) under the functor R ω i and T i is equivalent to D grsg ( A ) . Remark 4.5.
Given an object in D b (qgr − A ) , we can directly construct an object in D b (gr − A ≥ i ) because the right adjoint functor is explicitly given as R ω i . On the other hand, it is more dif-ficult to obtain an object in D b (gr − A ≥ i ) from an object in D grsg ( A ) , because the left adjointfunctor is not explicitly given. In the construction of the latter semiorthogonal decomposition of OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 11 D b (gr − A ≥ i ) , P ≥ i is first shown to be left admissible, and then T i is just given by the left orthog-onal of P ≥ i . One can prove that T i ∼ = D b (gr − A ≥ i ) / P ≥ i is equivalent to D grsg ( A ) by consideringsemiorthogonal decompositions of D b (gr − A ) and D b (grproj − A ) . Now we introduce Orlov’s main theorem.
Theorem 4.6 ([Or]) . Let a be the Gorenstein parameter of A . Then D grsg ( A ) and D b (qgr − A ) are related as following: (1) if a > , for each i ∈ Z there are fully faithful functors Φ i : D grsg ( A ) → D b (qgr − A ) andsemiorthogonal decomposition D b (qgr − A ) = (cid:104) πA ( − i − a + 1) , ..., πA ( − i ) , Φ i D grsg ( A ) (cid:105) where π : D b (gr − A ) → D b (qgr − A ) is the natural projection and Φ i : D grsg ( A ) (cid:39) T i (cid:44) → D b (gr − A ) π → D b (qgr − A ) . (2) if a < , for each i ∈ Z there are fully faithful functors G i : D b (qgr − A ) → D grsg ( A ) andsemiorthogonal decomposition D grsg ( A ) = (cid:104) q k ( − i ) , ..., q k ( − i + a + 1) , G i D b (qgr − A ) (cid:105) where q : D b (gr − A ) → D grsg ( A ) is the natural projection and G i : D b (qgr − A ) (cid:39) D i − a (cid:44) → D b (gr − A ) q → D grsg ( A ) . (3) if a = 0 , D grsg ( A ) and D b (qgr − A ) are equivalent via Φ i and G i . If X = Proj( A ) is a Calabi-Yau variety, i.e. a is 0, then D b (qgr − A )( ∼ = D b Coh ( X )) and D grsg ( A )( ∼ = HM F Z ( W )) are equivalent. In particular, in this case T i = D i in D b (gr − A ≥ i ).The equivalence for the Calabi-Yau case is illustrated as follows. Suppose that we aregiven an object E in D b Coh ( X ). Then we get a complex R ω i ( E ) ∈ D b (gr − A ≥ i ) . We com-pute its minimal free resolution over A such that the differentials give a matrix factoriza-tion of W . The other direction, namely from marix factorizations to coherent sheaves, isnot as straightforward as before, because we do not have an explicit form of the functorfrom HM F Z ( W )( (cid:39) D grsg ( A )) to D b (gr − A ≥ i ). So, given a matrix factorization K · , we take M = Cok( k − ) ∈ D b (gr − A ) = (cid:104) D b (gr − A ≥ i ) , P
Let T be a symplectic torus given by C / ( Z ⊕ e πi/ Z ). Let α be its symplectic area, and q := T α in Λ . Any unobstructed Lagrangian circle L is described by a straight line ( c,
0) + t −→ v , t ∈ R , where −→ v = a + e πi/ b with a, b ∈ Z , and c ∈ [0 , L = L ( a,b ) ,c ,and if c = 0 we omit it.Then the categorical mirror symmetry of elliptic curves proved by [PZ] is given by the followingtheorem: Theorem 5.1 ([PZ]) . There is an equivalence of categories F : D b Coh ( X ) (cid:39) D π F u ( E ) , Figure 3.
Mirror correspondence by Polishchuk-ZaslowΦ( O X ( np )) = L (1 , − n ) , Φ( O p ) = L (0 , − , where X is a mirror elliptic curve and p is a pointin X . The idea of the construction of F in [PZ] comes from theta identities. Indeed, theta functionscan be understood as morphisms among line bundles on elliptic curves. Here we briefly illustratetheir idea. They observe that theta identities given by compositions of morphisms between linebundles(i.e. products of theta functions) also naturally arise in the compositions of morphismsof the Fukaya category. To be more precise, first define a function θ [ c ] on Λ ∗ , θ [ c ]( w ) := (cid:88) m ∈ Z q ( m + c ) / w m + c . Then there is a degree 1 line bundle L on the elliptic curve such that its space of global sections isgenerated by θ [0]( w ), and L n has global sections generated by θ [ a/n ]( w n ) where a ∈ { , , ..., n − } . The addition formula of theta functions is given by θ [0]( w ) · θ [0]( w ) = θ [0](1) θ [0]( w ) + θ [1 / θ [1 / w ) . On the other hand, let { p } = L (1 , ∩ L (1 , − = L (1 , − ∩ L (1 , − and { q , q } = L (1 , ∩ L (1 , − where p and q are origin. p is a morphism from L (1 , to L (1 , − , or a morphism from L (1 , − to L (1 , − . Then m ( p, p ) = aq + bq is a morphism from L (1 , to L (1 , − . a and b are given by counts of holomorphic triangles whosevertices are p , p , q and p , p , q , respectively. The count is arranged with respect to the area,and it is easy to see that a = 2( q (1 · + q (2 · + q (3 · + · · · ) = 2( q + q + q + · · · ) = (cid:88) m ∈ Z q m ,b = 2( q ( · + q ( · + · · · ) = 2( q / + q / + q / + · · · ) = (cid:88) m ∈ Z q ( m +1 / , so a = θ [0](1) and b = θ [1 / . It implies that the mirrors of morphisms p , q and q amongLagrangian submanifolds are precisely theta functions which are natural basis of global sectionsof corresponding line bundles, or equivalently morphisms among them.In [AS] the mirror of T is given by the Tate curve, while in [CHL1] the mirror is X =Proj( R/W ) where R = Λ[ x, y, z ], W = φ ( x + y + z ) + ψxyz for some φ, ψ ∈ Λ which will bedefined later.We describe Polishchuk-Zaslow mirror correspondence between T and X . Again, we take O X as the mirror of L (1 , . Then the mirror of L (1 , − is a line bundle of degree 3 whose global OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 13 sections are generated by θ := θ [0]( w ), θ := θ [1 / w ) and θ := θ [2 / w ) . They aremirrors of Lagrangian intersections of L (1 , and L (1 , − . On the other hand, considering X asan abstract elliptic curve, they are used to define an embedding of X into Λ P . Since we let X as a projective cubic, the embedding is a priori given, so the mirrors of Lagrangian intersectionsare x , y and z which form a basis of global sections of O X (1) . (From now on we write O for O X if there is no confusion.) Therefore, we construct the mirror functor F : D π F u ( T ) → D b Coh ( X ) by F ( L (1 , ) := O , F ( L (1 , − ) := O (1) and the morphisms (0 , , (1 / , , (2 / , ∈ Hom D π F u ( T ) ( L (1 , , L (1 , − ) are mapped to y , x and z respectively. Remark 5.2.
As a divisor, from O Λ P (1) = ( x = 0) = ( y = 0) = ( z = 0) , letting ζ := e πi/ , O X (1) ∼ [1 : − − ζ : 0] + [1 : − ζ : 0] ∼ [0 : 1 : −
1] + [0 : 1 : − ζ ] + [0 : 1 : − ζ ] ∼ [1 : 0 : −
1] + [1 : 0 : − ζ ] + [1 : 0 : − ζ ] and these nine points [1 : − ζ k : 0] , [0 : 1 : − ζ k ] , [1 : 0 : − ζ k ] ( k = 0 , , are called inflectionpoints of X . Then for any inflection point p , O X (3 p ) ∼ = O X (1) . Now, we need an argument of Abouzaid-Smith [AS] to make the above functor exact. Insteadof constructing the functor explicitly on arbitrary objects and morphisms, they compare A ∞ -subcategories of F u ( E ) and D b ∞ Coh ( X ) which consist of split-generators. Let Γ A ⊂
F u ( E )be an A ∞ -subcategory of Lagrangians L (1 ,n ) for n ∈ Z , and Γ A ∨ ⊂ D b ∞ Coh ( X ) be a subcategoryconsisting of O ( np ). They use Polishchuk’s theorem on the A ∞ -structure of Γ A ∨ : it is uniquelydetermined by its cohomology category and its lack of formality. By [PZ], H (Γ A ) (cid:39) H (Γ A ∨ ) . Then they prove that Γ A is also nonformal, so that there is an A ∞ -quasiequivalence betweenΓ A and Γ A ∨ . Since the equivalence is between subcategories which consist of split-generators,it extends to an A ∞ -quasiequivalence between whole A ∞ -categories, and taking its cohomologywe get an exact equivalence between triangulated categories.6. CY-LG mirror symmmetry(graded localized mirror functors)
We review the construction of the localized mirror functor due to [CHL1] for T equippedwith Z / dz be the holomorphic volume form on T , where z is the complex coordinate of C .If we consider grading structure on E , then the above Z / Z / → Auteq ( D π F u ( T )) / Z where Z is generated by theshift functor [1].Let ξ = e πi/ , L be an oriented Lagrangian given by the line ξ + √− t with orientationupward and phase π , i.e. Ω( V ) | Ω( V ) | = e i · π for any nonzero positively oriented vector field V of L .We construct an immersed Lagrangian L := L ∪ τ ( L ) ∪ τ ( L )and equip L , τ ( L ) and τ ( L ) with nontrivial spin structures. We define the phase of τ ( L ) as π/ − π/
3, and that of τ ( L ) as π/ − π/
3, i.e. τ is the rotation by − π/
3. For simplicity,we call L the (lift of) Seidel Lagrangian.6.1. Ungraded localized mirror functor.
In this subsection we forget grading structuresand just consider the Z / e , e and e ofMorse complexes of L , τ ( L ) and τ ( L ) representing fundamental classes, and e := e + e + e . Then e is the unit of L , and we also have the following important theorem. Figure 4. Z / L on E and self-intersection points of L . A parallelogramwhose sides are solid lines is a fundamental domain of T and the union of threedotted lines is L . e , e and e represent fundamental classes of L , τ ( L ) and τ ( L ) respectively, and e = e + e + e is the unit of L . Theorem 6.1 ([CHL1]) . Let X , Y and Z be immersed generators of ¯ L ⊂ T / ( Z / , and X i , Y i and Z i for i = 1 , , be their liftings(see Figure 4). Then b = x ( X + X + X ) + y ( Y + Y + Y ) + z ( Z + Z + Z ) for any x, y, z ∈ Λ is a weak bounding cochain of L . Furthermore, P O ( L , b ) = φ ( x − y + z ) − ψxyz where φ and ψ are power series of q α = T ∆ xyz , such that ∆ xyz is the area of minimal xyz triangle and φ ( q α ) = ∞ (cid:88) k =0 ( − k +1 (2 k + 1) q (6 k +3) α ,ψ ( q α ) = − q α + ∞ (cid:88) k =1 ( − k +1 ((6 k + 1) q (6 k +1) α − (6 k − q (6 k − α ) . Let W := P O ( L , b ) . Suppose that L (cid:48) is an unobstructed Lagrangian submanifold. CF (( L , b ) , L (cid:48) )is generated by even and odd intersections(for example see Figure 5). If we compute m on it,i.e. count strips between even and odd generators, then m = W · id n × n where n is the numberof odd(or even) generators, namely ( CF (( L , b ) , L (cid:48) ) , m ) is a matrix factorization of W of rank n . More precisely, we have Theorem 6.2 ([CHL1]) . Let C be an A ∞ -category and ( A, b ) be a weakly unobstructed objectwith weak bounding cochain b . Let C be the A ∞ -subcategory of unobstructed objects. Define acollection of maps {LM ( A,b ) ∗ } from C such that • LM ( A,b )0 sends an object B to the matrix factorization ( hom (( A, b ) , B ) , m ) . • LM ( A,b )1 ( x ) is defined as ( − ∗ m b, ( · , x ) : ( hom (( A, b ) , B ) , m ) → ( hom (( A, b ) , B ) , m ) OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 15 where x ∈ hom ( B , B ) . • LM ( A,b ) k ( x , ..., x k ) is defined by F ( A,b ) k ( x , ..., x k )( y ) = ( − ∗ m b, ,..., k +1 ( y, x , ..., x k ) . Then {LM ( A,b ) ∗ } : C → M F ( P O ( A, b )) is an A ∞ -functor where M F ( P O ( A, b )) is the differen-tial Z / -graded category of matrix factorizations of P O ( A, b ) . We call the above functor {LM ( A,b ) ∗ } the (non-graded) localized mirror functor at ( A, b ).6.2.
Graded localized mirror functor.
Now we explain the construction of [CHL2] in thecase of cyclic group action (while the construction there of is for any finite group action). We alsorefer to the Chapter 5 of [CHL1] for more details. Let (
M, ω,
Ω) be a Calabi-Yau manifold whereΩ is the holomorphic volume form, and suppose Z /d Z acts on M . Consider d -grading on animmersed Lagrangian, i.e. define a phase function θ /dL : L → R such that Ω ⊗ d ( T p L ) = e πiθ /dL ( p ) . If there is such a phase function, then L is called d -graded. Furthermore if it can be equippedwith a constant phase function, then it is called special Lagrangian with respect to the d -grading. Proposition 6.3 ([CHL2]) . On the full A ∞ -subcategory of d -graded Lagrangians, the A ∞ -multiplication has degree − k under d -grading. Under d -grading, the degree of an intersection p between d -graded Lagrangians is defined as(6.1) deg /d ( p ) := 1 d ( θ /dL ( p ) − θ /dL ( p ) + θ d ( ˘ L L ) | p ) ∈ d Z where p is considered as a morphism L → L and πθ d ( ˘ L L ) | p is the phase angle of the positivedefinite path from T p L to T p L measured by Ω ⊗ d ( p ) . We go back to our example T on which Z / L can be madeinto a special Lagrangian with respect to -grading, defining the phase θ / L = − , for example.Then each immersed point which is considered to be a morphism τ ( L ) → τ ( L ), τ ( L ) → L or L → τ ( L )(i.e. an odd-degree morphism in Z / / b = x (cid:80) i =1 X i + y (cid:80) i =1 Y i + z (cid:80) i =1 Z i of degree 1, let deg / ( x ) = deg / ( y ) =deg / ( z ) = 2 /
3. Suppose that an unobstructed(or -graded) Lagrangian L (cid:48) is given. Themorphisms L → L (cid:48) are equipped with -gradings, and since b has degree 1, W = P O ( L , b ) is ofdegree 2. CF (( L , b ) , L (cid:48) ) is equipped with m of degree 1 and m = W · id n × n for some n .Now fix a labelling on components of L by elements of Z / L = L and according to the Z / L − = τ ( L ) and L − = τ ( L )(here 0, − − Z / Z / → U (1), − j (cid:55)→ e − πi · j/ . Take an unobstructed Lagrangian L (cid:48) again, and define a pair Ä (cid:77) p i ∈ CF (( L ,b ) ,L ) A g i [deg p i ] , m ( L ,b ) ,L ä ,p i ∈ L g i ∩ L (cid:48) and deg p i is the usual Z -grading by Ω.We explain the expressions above. Given a polynomial ring R and a (quasi-)homogeneouspolynomial W of degree d , we define a category T w Z ( R W Z /d ) which consists of objects aspairs ( (cid:76) g i ∈ Z /d A g i [ σ i ] , δ ). Hom( A g , A h ) is a vector space consisting of f which is required tosatisfy (cid:103) deg f := deg /d f + ( α h − α g ) ∈ Z . Here we fix a character g (cid:55)→ e πiα g . We give a Z -grading on Hom spaces by (cid:103) deg. δ is a degree 1endomorphism of (cid:76) g i ∈ Z / A g i [ σ i ] where the degree of a morphism A g [ σ ] → A h [ σ (cid:48) ] is shifted by σ (cid:48) − σ from the degree of the morphism A g → A h . The upshot is the following: Theorem 6.4 ([CT]) . T w Z ( R W Z /d ) is equivalent to the category of graded matrix factoriza-tions of W by the correspondence: Ä (cid:77) i A g i [ k i ] , δ ä (cid:55)→ ( · · · → E → E → · · · ) where E = (cid:77) k i :even R Ä − k i d − d α g i ä ,E = (cid:77) k i :odd R Ä (1 − k i ) d − d α g i ä and the structure maps p i : E i → E i +1 are given by the corresponding matrix defined by δ . Proposition 6.5 ([CHL2]) . The above pair Ä (cid:77) p i ∈ CF (( L ,b ) ,L ) A g i [deg p i ] , m ( L ,b ) ,L ä is an object of T w Z ( R W Z / where W = P O ( L , b ) ∈ R = Λ[ x, y, z ] , i.e. it gives a graded matrix factorizationof W with the usual grading on R , i.e. deg( x ) = deg( y ) = deg( z ) = 1 . Furthermore, thecollection of maps {F L ,b ∗ } becomes an A ∞ -functor F u Z ( T ) → T w Z ( R W Z / . The above A ∞ -functor in the proposition is called the graded localized mirror functor anddenoted by LM L ,bgr . We often omit b from the notation. We remark that we followed theconvention of [CHL2] which is different from that of [CT]. Also we mention that in [CHL2]Proposition 6.5 is proved in much more general setting, as noncommutative matrix factorizations.7. Main theorem
Now we are ready to state and prove our main theorem. Recall the notation G i = q i ◦ R ω i : D b ( qgr − A ) → D grsg ( A ) from Section 4. Theorem 7.1.
Suppose that S = t (0 , / ◦ Ç å be a symplectomorphism of E , where t (0 , / is the translation by (0 , / . Let S : F u ( T ) → F u ( T ) be an autoequivalence inducedby S . Then we can construct an equivalence ‹ F : D π F u ( T ) → D b Coh ( X ) which equals toPolishchuk-Zaslow’s functor on a split-generating subcategory A ⊂ D π F u ( T ) such that G ◦ ‹ F can be identified by LM L gr ◦ S . This section is devoted to the proof of this theorem. Pick two Lagrangians L := L (1 , and L := L (1 , − in D π F u ( E ), both equipped with nontrivial spin structures. Let A be the A ∞ -subcategory consisting of L and L . Let F be the Polishchuk-Zaslow’s mirror functor. Then L and L are mapped to O and O (1) via F respectively. Observe that the Lagrangians whichwe picked generate D π F u ( E ) and their images via F also generate D b Coh ( X ).7.1. Computations via G ◦ F . We need to compute image objects of O , O (1) and morphismsbetween them via G . R ω ( O ) = ∞ (cid:77) k =0 R Hom D b Coh ( X ) ( O , O ( k )) ∈ D b (gr − A ≥ ) (cid:44) → D b (gr − A ) , R ω ( O (1)) = ∞ (cid:77) k =0 R Hom D b Coh ( X ) ( O , O (1)( k )) ∈ D b (gr − A ≥ ) (cid:44) → D b (gr − A ) . OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 17
We recall a useful lemma of homological algebra.
Lemma 7.2.
For a chain complex C · , if H i ( C · ) ∼ = M and H j ( C · ) = 0 for j (cid:54) = i , then C · isquasi-isomorphic to M [ − i ] . A simple computation of Ext groups gives R ω ( O (1)) ∼ = A (1) ≥ , and R ω ( O (1)) = 0 . So bythe lemma we have a quasi-isomorphism of complexes R ω ( O (1)) (cid:39) A (1) ≥ (the right hand side is considered to be a complex concentrated in degree 0). Its minimal A -freeresolution gives rise to the corresponding matrix factorization via G .It is also easy to see that R ω ( O ) ∼ = A and R ω ( O ) ∼ = Λ. The complex E · := R ω ( O ) itselfcan be explicitly obtained E · by the following lemma. Lemma 7.3.
The complex E · fits into an exact triangle A → E · → k [ − → A [1] in D b (gr − A ) , where k [ − → A [1] is a nonzero morphism.Proof of the lemma. We follow the method of [Asp]. Recall that A is a Gorenstein algebrasatisfying R Hom A ( k , A ) = k [ − . Hence,Ext A ( k , A ) ∼ = Λ , or Hom D b (gr − A ) ( k , A [2]) = Hom D b (gr − A ) ( k [ − , A [1]) ∼ = Λand Ext iA ( k , A ) = 0 if i (cid:54) = 2.Pick any nonzero morphism f : k [ − → A [1] and let C be its cocone, i.e.(7.1) A (cid:47) (cid:47) C (cid:47) (cid:47) k [ − f (cid:47) (cid:47) A [1]is an exact triangle.Then applying Hom( · , A ( r )) for r ≤ D b (gr − A ) gives a long exact sequence · · · (cid:47) (cid:47) Hom( k [ − , A ( r )) (cid:47) (cid:47) Hom(
C, A ( r )) (cid:47) (cid:47) Hom(
A, A ( r )) (cid:47) (cid:47) Hom( k [ − , A ( r )) (cid:47) (cid:47) Hom( C [ − , A ( r )) (cid:47) (cid:47) Hom( A [ − , A ( r )) (cid:47) (cid:47) · · · . Let r = 0. Then a part of the above sequence is given by0 (cid:47) (cid:47) Hom(
C, A ) (cid:47) (cid:47) Λ f [ − ∗ (cid:47) (cid:47) Λ (cid:47) (cid:47) Hom( C [ − , A ) (cid:47) (cid:47) Hom( A [ − , A ) = 0 . Since f is nonzero, f [ − ∗ is an injective linear map from Λ to itself. So it is also surjective, andHom( C, A ) = Hom( C [ − , A ) = 0. If i (cid:54) = 0 , −
1, then the exact sequence0 = Hom( k [ i − , A ) (cid:47) (cid:47) Hom( C [ i ] , A ) (cid:47) (cid:47) Hom( A [ i ] , A ) = 0gives Hom( C [ i ] , A ) = 0 . Hence Hom( C [ i ] , A ) = 0 for all i ∈ Z . Now let r < . Then Hom( k [ i ] , A ( r )) =0 for all i ∈ Z by Gorenstein condition. Clearly Hom( A [ i ] , A ( r )) = 0 for any i ∈ Z and r < C [ i ] , A ( r )) = 0 for any i ∈ Z , r ≤ . By the semiorthogonal decomposition D b (gr − A ≥ ) = (cid:104)P ≥ , T (cid:105) , since Hom( C, P ) = 0 for all P ∈ P ≥ , C is in T . Finally, via π : D b (gr − A ) → D b (qgr − A ), by (7.1), πC is equivalent to πA which correspondsto O ∈ D b Coh ( X ). Since R ω : D b Coh ( X ) → D (= T ) is an equivalence, C is isomorphic to E · = R ω ( O ) . (cid:3) By the lemma, E · and the mapping cone of g : k [ − → A are quasi-isomorphic as chaincomplexes. Since k ∼ = A/ ( x, y, z ) and ( x, y, z ) is a regular sequence of R = Λ[ x, y, z ], we followthe algorithm in [Dyc] to take the free resolution of k . It is given by a double complex... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) A ( − (cid:47) (cid:47) (cid:15) (cid:15) A ( − (cid:47) (cid:47) (cid:15) (cid:15) A ( − (cid:15) (cid:15) A ( − (cid:47) (cid:47) A ( − (cid:47) (cid:47) ( w x w y w z ) (cid:15) (cid:15) A ( − (cid:47) (cid:47) Ç wzα − wyα − wzα wxαwyα − wxα å (cid:15) (cid:15) A ( − (cid:16) w x w y w z (cid:17) (cid:15) (cid:15) A ( − Ä xyz ä (cid:47) (cid:47) A ( − (cid:16) − αz αyαz − αx − αy αx (cid:17) (cid:47) (cid:47) A ( − x y z ) (cid:47) (cid:47) A (cid:47) (cid:47) α = ∞ (cid:88) k =0 Ä ( − k T et (1+6 k ) + ( − k +1 T et (5+6 k ) ä , W = xw x + yw y + zw z with w x = x ∞ (cid:88) k =0 ( − k +1 (2 k + 1) q (6 k +3) α + yz ∞ (cid:88) k =1 ( − k +1 (2 kq (6 k +1) α − kq (6 k − α ) ,w y = y ∞ (cid:88) k =0 ( − k (2 k + 1) q (6 k +3) α + zx ∞ (cid:88) k =1 ( − k (2 kq (6 k +1) α − kq (6 k − α ) ,w z = z ∞ (cid:88) k =0 ( − k +1 (2 k + 1) q (6 k +3) α + xy ∞ (cid:88) k =1 ( − k +1 ((2 k + 1) q (6 k +1) α − (2 k − q (6 k − α ) − xyq α . Here et ( a ) means the area of an equilateral triangle of face length a (we let the length of theminimal triangle be 1). α equals to γ which is in Definition 7.8 of [CHL1]. We choose thisspecific free resolution to make the comparison of objects more easily.The maps which are not specified are just copies of written ones. Observe that each rowis given by the usual Koszul complex of ( x, y, z ), and the vertical maps are needed to capturerelations caused by W . It is clear that the complex is eventually quasi-2-periodic and gives amatrix factorization of W . Now the free resolution of k [ −
2] is written as · · · (cid:47) (cid:47) A ( − ⊕ A ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) A ( − ⊕ A ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 19 (cid:16) wx − αz αywy αz − αxwz − αy αx (cid:17) (cid:47) (cid:47) A ( − x y z ) (cid:47) (cid:47) A (cid:47) (cid:47) . k [ − → A is given by φ : A ( − ⊕ A ( − → A such that φ ◦ Ü x y zx w z α − w y α y − w z α w x α z w y α − w x α ê = 0and it is easy to see that φ = (cid:0) w x w y w z (cid:1) gives a nontrivial morphism. It is nothing butthe first row of the consecutive differential map of the resolution. Any other row also defines achain map, but then it becomes homotopically trivial.So the mapping cone C ( φ ), which is isomorphic to R ω ( O ), is given by · · · Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) A ⊕ A ( − x y z ) (cid:47) (cid:47) A (cid:47) (cid:47) . R ω ( O (1)) (cid:39) A (1) ≥ . Since A (1) ≥ is generated by x, y and z , the resolution starts from · · · (cid:47) (cid:47) F − (cid:47) (cid:47) A x y z ) (cid:47) (cid:47) A (1) ≥ (cid:47) (cid:47) A (1) ≥ : · · · (cid:47) (cid:47) A ( − ⊕ A ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − (cid:16) wx − αz αywy αz − αxwz − αy αx (cid:17) (cid:47) (cid:47) A (cid:47) (cid:47) . Now we are ready to compare morphisms, HF k ( L i , L j ) and Hom HMF Z ( W ) ( M i , M j [ k ]) , i, j =0 or 1 and k = 0 or 1. Here M i and M j are matrix factorizations corresponding to L i and L j viaabove correspondence. HF ( L , L ) and HF ( L , L ) are generated by identity morphisms, andany functor preserves identities, so we do not need any computation for degree 0 endomorphisms.Recall that three intersections of L (1 , and L (1 , − , which are basis of HF ( L , L ), correspondto O x −→ O (1), O y −→ O (1) and O z −→ O (1) via F . We need to know how they correspondto morphisms between matrix factorizations via G . We compute the example O x −→ O (1). f := R ω ( x ) : R ω ( O ) → R ω ( O (1)) is a map · · · Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − f − (cid:15) (cid:15) Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) A ⊕ A ( − f (cid:15) (cid:15) ( x y z ) (cid:47) (cid:47) A (cid:15) (cid:15) (cid:47) (cid:47) · · · Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − (cid:16) wx − αz αywy αz − αxwz − αy αx (cid:17) (cid:47) (cid:47) A (cid:47) (cid:47) (cid:47) (cid:47) which induces x : A → A (1) ≥ , namely the map x between R ω ( O )( ∼ = A ) and R ω ( O (1))( ∼ = A (1) ≥ ). It turns out that maps between the 0th cohomologies completely determine the mapsbetween genuine complexes, becausedim Λ Hom D b (gr − A ) ( R ω ( O ) , R ω ( O (1))) = dim Λ Hom D b Coh ( X ) ( O , O (1))= 3= dim Λ Hom gr − A ( A, A (1) ≥ )(the first identity comes from the fact that R ω is fully faithful). Therefore, instead of tryingto compute the morphism f completely, we just try to describe data of f which are sufficient todetermine it.From O x −→ O (1), the induced morphism of complexes is determined by H ( f ), and it induces A x −→ A (1) ≥ if f = Ñ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ é . Similarly, if we start from O y −→ O (1) or O z −→ O (1), then the induced maps are g = Ñ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ é or h = Ñ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ é respectively.We also need to examine the correspondence of higher degree morphisms, i.e. we com-pare HF ( L i , L j ) ∼ = HF ( L i , L j [1]) and Hom HMF Z ( W ) ( M i , M j [1]) . The Serre duality for X gives Ext ( O (1) , O ) ∼ = Ext ( O , O (1)) ∗ and Ext ( O , O (1)) ∼ = Ext ( O (1) , O ) ∗ = 0 . We de-scribe the morphism O (1) x ∗ −→ O [1] as a morphism between R ω ( O (1)) and R ω ( O [1]) , where x ∗ ∈ Ext ( O (1) , O ) is the dual of O x −→ O (1). g := R ω ( x ∗ ) : R ω ( O (1)) → R ω ( O [1]) is a map · · · (cid:47) (cid:47) A ( − ⊕ A ( − f (cid:48)− (cid:15) (cid:15) Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) A ( − ⊕ A ( − f (cid:48)− (cid:15) (cid:15) (cid:16) wx − αz αywy αz − αxwz − αy αx (cid:17) (cid:47) (cid:47) A f (cid:48) (cid:15) (cid:15) (cid:47) (cid:47) · · · (cid:47) (cid:47) A ( − ⊕ A ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) A ⊕ A ( − x y z ) (cid:47) (cid:47) A (cid:47) (cid:47) . (7.2)It is also determined by the map of the 0th cohomologies by dimension arguments as above.Recall that R ω ( O (1)) ∼ = A (1) ≥ and it is generated by x , y and z which are identifiedas morphisms from O to O (1). On the other hand, R ω ( O [1]) ∼ = A/ ( x, y, z ) ∼ = Λ, andHom gr − A ( A (1) ≥ , A/ ( x, y, z )) has basis { φ x , φ y , φ z } where φ ( x ) is defined by φ x ( x ) = 1 , φ x ( y ) = φ x ( z ) = 0 ,φ y and φ z are defined similarly. Hence x ∗ corresponds to φ x , which is induced by g = (1 0 0) : A → A . Similarly y ∗ corresponds to (0 1 0) : A → A , and z ∗ to (0 0 1) : A → A. As before, they completely determine g − , g − , · · · , so give rise to a morphism of matrixfactorizations. Finally, under the duality Ext ( O , O ) ∼ = Ext ( O , O ) ∗ and Ext ( O (1) , O (1)) ∼ =Ext ( O (1) , O (1)) ∗ , x ∗ ◦ x = y ∗ ◦ y = z ∗ ◦ z = (id O ) ∗ ∈ Ext ( O , O ) , OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 21
Figure 5. (Hamiltonian perturbations of) the images of L (1 , and L (1 , − under S . Black dots are even intersections and red dots are odd intersections. x ◦ x ∗ = y ◦ y ∗ = z ◦ z ∗ = (id O (1) ) ∗ ∈ Ext ( O (1) , O (1))and they are mirrors of the basis of HF ( L , L ) and HF ( L , L ) respectively. There are cor-responding morphisms of matrix factorizations given by compositions of maps computed above.Finally, replacing all free A -modules by R -modules, i.e. considering above chain complexes over A as matrix factorizations of W , and extending it (quasi)2-periodically, we get objects andmorphisms in HM F Z ( W ) . Computations via LM L gr ◦S . Via S , L is mapped to L which is a branch of L and L ismapped to τ ( L ) which is another branch of L . Write L (cid:48) := S ( L ) = L and L (cid:48) := S ( L ) = τ ( L ) . Corresponding graded matrix factorizations M = LM L gr ( L (cid:48) ) and M := LM L gr ( L (cid:48) ) are givenby counting strips between even and odd intersections from L to L (cid:48) i for i = 0 ,
1. As in [CHL1],to see the picture more intuitively, we take a small Hamiltonian perturbation φ t of L (cid:48) i , constructFloer complexes ( CF ( L , φ t ( L (cid:48) i )) , m t ) and take the limit t → m : CF ( L , L (cid:48) i ) → CF ( L , L (cid:48) i ) . According to the definition of the graded localized mirror functor,we fix a character − j (cid:55)→ e πi · ( − j ) , i.e. α − j = − j , where the components are labelled by L = L ,τ ( L ) = L − , τ ( L ) = L − . For L = S ( L (1 , ), deg( a ) = 0, deg( a i ) = 2, deg( b ) = 1 = deg( b i )for i = 1 , ,
3. By Theorem 6.4, the 0th part M of the corresponding matrix factorization is R (0) ⊕ R ( − where R (0) comes from a and R ( − from a i for i = 1 , , M which is at the 1st position of M is R ⊕ R (1) and the basis is { b , b , b , b } . Itis also straightforward to see that M and M are R (1) ⊕ R and R (1) ⊕ R (2) respectively, andtheir basis are similarly given by { a (cid:48) , a (cid:48) , a (cid:48) , a (cid:48) } and { b (cid:48) , b (cid:48) , b (cid:48) , b (cid:48) } . Under these basis choices,by computations given in Chapter 7 of [CHL1] M is as follows: · · · (cid:47) (cid:47) R ( − ⊕ R ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ( − ⊕ R ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R ⊕ R ( − (cid:47) (cid:47) · · · and M is given by: Figure 6. CF ( φ t ( L (cid:48) ) , φ t ( L (cid:48) )) generated by x , y and z , which map b i (cid:55)→ b (cid:48) (yellow strips) and a (cid:55)→ a (cid:48) i (gray strips) for i = 1 , , . · · · (cid:47) (cid:47) R ( − ⊕ R ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ( − ⊕ R ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R (1) ⊕ R (cid:47) (cid:47) · · · . So we observe that G ◦ F and LM L gr ◦ S are identical on objects L and L .As noticed above, morphisms between M [ i ] and M [ j ] ( i, j = 0 ,
1) are determined by constantentry parts, so we do not compute all entries of corresponding morphisms. An intersection(0 , ∈ CF ( L , L ) is mapped to (0 , / y ∈ CF ( φ t ( L (cid:48) ) , φ t ( L (cid:48) )) as inFigure 6. Similarly, S (1 / ,
0) = x and S (2 / ,
0) = z .As t →
0, strips in Figure 6 collapse to strips of area zero, and they contribute with positivesigns by the criterion discussed in section 2.3. It is clear that there are no more strips which con-tribute to morphisms b i (cid:55)→ b (cid:48) and a (cid:55)→ a (cid:48) i . For example, the morphism of matrix factorizationsgiven by x is given as follows: · · · (cid:47) (cid:47) R ( − ⊕ R ( − p − (cid:15) (cid:15) Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ( − ⊕ R ( − p − (cid:15) (cid:15) Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R ⊕ R ( − p (cid:15) (cid:15) (cid:47) (cid:47) · · ·· · · (cid:47) (cid:47) R ( − ⊕ R ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ( − ⊕ R ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R (1) ⊕ R (cid:47) (cid:47) · · · OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 23
Figure 7. CF ( φ t ( L (cid:48) ) , φ t ( L (cid:48) [1])) generated by x ∗ , y ∗ and z ∗ which are equal to x , y and z as intersection points. They map a (cid:48) (cid:55)→ b i (yellow strips), a (cid:48) i (cid:55)→ b (graystrips) for i = 1 , , p i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê and p i − = Ü ∗ − ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ê . Similarly, the morphism induced by y is given by q i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê , q i − = Ü ∗ − ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ê and z gives the morphism r i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê , r i − = Ü ∗ − ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ê . Therefore, LM L gr ◦ S and G ◦ F are identical on morphisms L (1 , → L (1 , − . We can also obtain morphisms of graded matrix factorizations M → M [1] from morphisms L (cid:48) → L (cid:48) [1] in the Fukaya category. As already commented, it suffices to calculate constantentries. They are given by holomorphic strips in Figure 7. Again in this case there are no morestrips which map a (cid:48) (cid:55)→ b i and a (cid:48) i (cid:55)→ b . It is also clear that they collapse to area zero and havepositive signs.Hence, the morphism of graded matrix factorizations induced by x ∗ is given by · · · (cid:47) (cid:47) R ( − ⊕ R ( − p (cid:48)− (cid:15) (cid:15) Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ( − ⊕ R ( − p (cid:48)− (cid:15) (cid:15) Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R (1) ⊕ R p (cid:48) (cid:15) (cid:15) (cid:47) (cid:47) · · ·· · · (cid:47) (cid:47) R ( − ⊕ R ( − Å wx wy wzwx − αz αywy αz − αxwz − αy αx ã (cid:47) (cid:47) R ⊕ R ( − Ñ x y zx wzα − wyαy − wzα wxαz wyα − wxα é (cid:47) (cid:47) R ⊕ R (1) (cid:47) (cid:47) · · · such that p (cid:48) i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê . Then(7.3) Ü x y zx w z α − w y α y − w z α w x α z w y α − w x α ê ◦ p (cid:48) i − = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê ◦ Ü w x w y w z w x − αz αyw y αz − αxw z − αy αx ê . On the other hand, recalling (7.2), the morphism { f (cid:48) j } j ∈ Z was induced by the lifting of the map(1 0 0) : R → R, namely (cid:0) x y z (cid:1) ◦ f (cid:48)− = (1 0 0) ◦ Ñ w x − αz αyw y αz − αxw z − αy αx é ,f (cid:48)− is defined as the successive lifting of f (cid:48)− and then the lifting becomes 2-periodic. It is clearthat p (cid:48)− is also realized by the lift of (1 0 0) : R → R , i.e. (cid:0) x y z (cid:1) ◦ p (cid:48)− = (1 0 0) ◦ Ñ w x − αz αyw y αz − αxw z − αy αx é , so { p (cid:48) j } j ∈ Z is the same morphism as that induced in (7.2). Similarly, y ∗ induces a morphism q (cid:48) i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê and z ∗ induces r (cid:48) i = Ü ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ê and they also induce same morphisms as those coming from G ◦ F . Finally we construct ‹ F . By [CHL1], the functor LM L gr ◦S is an A ∞ -quasiequivalence between F u ( E ) and M F Z ( W ). By the result of [CT], the functor G also extends to an A ∞ -equivalencebetween D b ∞ Coh ( X ) and M F Z ( W ). Hence we define an A ∞ -functor ‹ F : F u ( E ) → D b ∞ Coh ( X )by ‹ F := G − ◦ LM L gr ◦ S , so that it is an A ∞ -quasiequivalence which realizes the Polishchuk-Zaslow’s mirror functor F on A . The cohomology functor of ‹ F gives an exact functor whichgives the CY-CY homological mirror symmetry of T of [AS]. OMPARISON OF MIRROR FUNCTORS OF ELLIPTIC CURVES VIA LG/CY CORRESPONDENCE 25
Description of S i for any i ∈ Z . By definition of R ω i , it is easy to observe the following: R ω i ( O ( − i )) ∼ = R ω ( O )( − i ) , R ω i ( O ( − i + 1)) ∼ = R ω ( O (1))( − i ) . By F , L (1 , i ) corresponds to O ( − i ), so via G i ◦F , L (1 , i ) corresponds to the matrix factorization M ( − i ) and L (1 , i − corresponds to M ( − i ). On the other hand, if LM L gr ( L (cid:48) ) = M (cid:48) , then LM L gr ( τ − ( L (cid:48) )) = M (cid:48) ( −
1) by definition of LM L gr . Recall that L = S ( L (1 , ) is mapped to M via LM L gr . τ − is the rotation by 2 π , which corresponds to [ −
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