Categorical primitive forms of Calabi-Yau A ∞ -categories with semi-simple cohomology
CCATEGORICAL PRIMITIVE FORMS OF CALABI–YAU A ∞ -CATEGORIES WITH SEMI-SIMPLE COHOMOLOGY LINO AMORIM AND JUNWU TU
Abstract.
We study categorical primitive forms for Calabi–Yau A ∞ cate-gories with semi-simple Hochschild cohomology. We classify these primitiveforms in terms of certain grading operators on the Hochschild homology. Weuse this result to prove that, if the Fukaya category Fuk ( M ) of a symplec-tic manifold M has semi-simple Hochschild cohomology, then its genus zeroGromov–Witten invariants may be recovered from the A ∞ -category Fuk ( M )together with the closed-open map. An immediate corollary of this is that inthe semi-simple case, homological mirror symmetry implies enumerative mirrorsymmetry. Introduction
Kontsevich’s proposal.
In his ICM talk [20], Kontsevich proposed thecelebrated homological mirror symmetry conjecture which predicts an equivalencebetween two A ∞ -categories, one constructed from symplectic geometry and theother from complex geometry. More precisely, there should exist a quasi-equivalence D b ( Fuk ( X )) ∼ = D b ( Coh ( Y )) , between the derived Fukaya category of a Calabi–Yau manifold X and (a dg-enhancement of) the derived category of coherent sheaves on a mirror dual Calabi–Yau manifold Y . In the same article, Kontsevich also suggested that by studyingvariational Hodge structures associated to these A ∞ -categories, homological mir-ror symmetry conjecture may be used to prove enumerative mirror symmetry . Byenumerative mirror symmetry, we mean an equality between the Gromov–Witteninvariants of X and some period integrals of a holomorphic volume form on Y , aswas first discovered by physicists [7]. Both versions of mirror symmetry have sincebeen extended to include manifolds which are not necessarily Calabi–Yau. For in-stance, the symplectic manifold X could be a toric symplectic manifold, while itsmirror dual is a Landau–Ginzburg model ( Y, W ) given by a space Y together with aholomorphic function W ∈ Γ( Y, O Y ). In this case, the category of coherent sheavesis replaced by the category of matrix factorizations MF ( Y, W ), and period integralsare replaced by Saito’s theory of primitive forms [28, 29] for the singularities of W .Genus zero Gromov–Witten invariants, period integrals and Saito’s theory ofprimitive forms all determine a Frobenius manifold. Therefore Kontsevich’s pro-posal can be realized by having a natural construction that associates a Frobeniusmanifold to an A ∞ -category, such that when applied to Fuk ( X ) it would reproducethe geometric Gromov–Witten invariants of X and when applied to MF ( Y, W ) itwould reproduce Saito’s invariants from primitive form theory.To carry out such construction one needs to restrict to saturated (meaningsmooth, proper and compactly generated) Calabi–Yau (CY) A ∞ -categories. Then a r X i v : . [ m a t h . S G ] S e p e proceed in two steps depicted in the following diagram:Saturated CY A ∞ -category VSHS FrobeniusmanifoldA BIn the middle box, VSHS stands for Variational Semi-infinite Hodge Structure, animportant notion introduced by Barannikov [3, 4], generalizing Saito’s frameworkof primitive forms [28, 29].Step A in the diagram is well understood thanks to the works of Getzler [17],Katzarkov–Kontsevich–Pantev [19], Kontsevich–Soibelman [21], Shklyarov [33, 34]and Sheridan [32]. In this step, one associates to a saturated CY A ∞ -category C , a VSHS V C on the negative cyclic homology HC −• ( C ) with C a versal defor-mation of C . For the deformation theory of C to be un-obstructed, one needs thenon-commutative Hodge-to-de-Rham spectral sequence of C to degenerate, whichaccording to a conjecture of Kontsevich–Soibelman is always the case for saturated A ∞ -categories.Step B depends on some choices and is less well understood in general. In the casewhen the VSHS is defined over a one dimensional base and is of Hodge-Tate type,Ganatra–Perutz–Sheridan [16] obtained a partial resolution of Step B. Remarkably,this is already enough to recover the predictions of [7] for quintic 3-folds.1.2.
The semi-simple case.
Here we want to understand the above construction,when one takes as input data, a saturated CY A ∞ -category whose Hochschild co-homology ring is semi-simple. This situation includes several interesting examples:categories of matrix factorizations MF ( Y, W ), when Y has Morse singularities; andFukaya categories of many Fano manifolds, including Fuk ( CP n ). These type of cat-egories differ significantly from Ganatra–Perutz–Sheridan’s setup. The base of theVSHS is not 1-dimensional, and even after restricting to a 1-dimensional base, it isnot of Hodge-Tate type. The key difference, which may look like a technicality atfirst glance, is that these categories, for example Fuk ( CP n ), are only Z / Z -graded,not Z -graded.In the semi-simple case, Step A is relatively easy since semi-simplicity impliesthat the Hochschild homology is purely graded (see Corollary 2.5), which in turnimplies degeneration of the Hodge-to-de-Rham spectral sequence by degree reasons.We first address Step B in a general setup. Let C be a saturated Z / Z -graded CY A ∞ -category and V C be its associated VSHS. It turns out that in order to obtaina Frobenius manifold from V C , an additional piece of data is needed: a choice of aprimitive element in V C satisfying a few properties detailed in Definition 4.1. Werefer to articles [29, 28] for the origin of primitive forms, and to the more recentwork [5] for its categorical analogue. Following [28], we call these primitive elements categorical primitive forms . The definition of categorical primitive forms is quiteinvolved. In particular, it is not easy to prove general existence results. We firstclassify categorical primitive forms in terms of some particular splittings of theHodge filtration. Theorem 1.1.
Let C be a saturated CY A ∞ -category. Assume that the Hodge-to-de-Rham spectral sequence for C degenerates. Then there is a naturally definedbijection between the following two sets: (1) Categorical primitive forms of V C . Good splittings of the Hodge filtration compatible with the CY structure.See Definition 3.7 for details.
When Hochschild cohomology ring is semi-simple, we can classify good splittings,and therefore categorical primitive forms, in terms of certain linear algebra data.
Theorem 1.2.
Let C be a saturated CY A ∞ -category, such that HH • ( C ) is semi-simple. Then the Hodge-to-de-Rham spectral sequence for C degenerates and thetwo sets in Theorem 1.1 are also naturally in bijection with the following set (3) Grading operators on HH • ( C ) . See Definition 3.9 for details. Hence we obtain for each grading operator µ on HH • ( C ), a categorical primitiveform ζ µ ∈ V C . Then the primitive form ζ µ defines a homogeneous Frobenius mani-fold M ζ µ on the formal moduli space Spec ( R ) parameterizing formal deformationsof the A ∞ -category C . In Section 5 we will describe this Frobenius manifold.In general, there exists no canonical choice of categorical primitive forms. How-ever, if HH • ( C ) is semi-simple, we prove that there exists a canonical categoricalprimitive form which corresponds to the zero grading operator on HH • ( C ) throughthe bijection in the above theorem. The corresponding Frobenius manifold M ζ isconstant (see discussion at the end of Section 4).1.3. Applications to Fukaya categories and mirror symmetry.
We may ap-ply Theorem 1.2 to the Fukaya category D b (cid:0) Fuk ( M ) (cid:1) with semi-simple Hochschildcohomology. In particular, this includes the projective spaces CP n , as well as generictoric Fano manifolds. We want to produce a Frobenius manifold M ζ µ isomorphicto the Frobenius manifold determined by the genus zero Gromov–Witten invariantsof M , known as the big quantum cohomology of M . To choose the grading µ thatdoes this, we need an extra piece of geometric information: the closed-open map CO : QH • ( M ) → HH • (cid:16) D b (cid:0) Fuk ( M ) (cid:1)(cid:17) . Under some conditions, detailed in Assumption 5.8, we can use CO and the dualitymap D : HH • (cid:16) D b (cid:0) Fuk ( M ) (cid:1)(cid:17) → HH • (cid:16) D b (cid:0) Fuk ( M ) (cid:1)(cid:17) (see Equation 3) , to pull-back the integral grading operator on QH • ( M ) to a grading operator de-noted by µ C O on the Hochschild homology HH • (cid:16) D b (cid:0) Fuk ( M ) (cid:1)(cid:17) . This corresponds,using Theorems 1.1 and 1.2, to a categorical primitive form denoted by ζ C O ∈V D b (cid:0) Fuk ( M ) (cid:1) . Theorem 1.3.
Let M be a symplectic manifold such that its Fukaya category D b (cid:0) Fuk ( M ) (cid:1) and the closed-open map C O satisfy Assumption 5.8. Assume further-more that HH • (cid:16) D b (cid:0) Fuk ( M ) (cid:1)(cid:17) is semi-simple. Then the big quantum cohomologyof M is isomorphic to the formal Frobenius manifold M ζ C O associated with thepair (cid:16) V D b (cid:0) Fuk ( M ) (cid:1) , ζ C O (cid:17) . In particular, the category D b (cid:0) Fuk ( M ) (cid:1) together with theclosed-open map CO determine the big quantum cohomology of M . As we will see in Section 5, compact toric manifolds for which the potentialfunction PO (in the notation of [14], sometimes also called Landau–Ginzburg po-tential) has only non-degenerate critical points are examples of symplectic manifolds or which the above theorem applies to. In the case when M is Fano, the poten-tial non-degenerate critical points for a generic choice of symplectic form (see [13,Proposition 8.8]).We would like to point out, that in our proof of Theorem 1.2 we calculate the R -matrix of the Frobenius manifold associated to the grading µ . Therefore, usingthe Givental group action (see [26]) and Teleman’s reconstruction theorem [36] weshould be able to recover the higher genus Gromov–Witten invariants from µ C O aswell. We will investigate this further in the future.An immediate corollary of the above theorem is a realization of Kontsevich’soriginal proposal that his homological mirror symmetry implies enumerative mir-ror symmetry in the case of semi-simple saturated CY A ∞ -categories. See Corol-lary 5.13 for a precise statement.1.4. Organization of the paper.
In Section 2 and Appendix A, we recall andprove some basic properties of Hochschild invariants of (cyclic) A ∞ -categories. InSection 3 we prove there exists a bijection between the sets (2) and (3) in The-orem 1.1. In Section 4 we prove the bijection between the sets (1) and (2), thusfinishing the proof of Theorem 1.1. In Section 5, we apply the results to Fukayacategories and prove Theorem 1.3.1.5. Conventions and Notations.
We shall work with Z / Z -graded A ∞ -algebras(or categories) over a field K of characteristic zero. If A is such an A ∞ -algebra (orcategory), denote by m k : A ⊗ k → A, ( k ≥ | − | and | − | (cid:48) to denote the degree in A ,or its shifted degree in A [1] respectively, i.e. | − | (cid:48) = | − | + 1 (mod 2).1.6. Acknowledgments.
L. A. would like to thank Cheol-Hyun Cho for usefulconversations about closed-open maps. J. T. is grateful to Andrei C˘ald˘araru and SiLi for useful discussions around the topic of categorical primitive forms. We wouldalso like to thank Nick Sheridan for very helpful discussions about sign conventionsfor the Mukai pairing.2.
Hochschild invariants of A ∞ -categories In this section we recall some basic facts about Hochschild invariants of A ∞ -algebras and categories, establishing the notation and conventions (mostly following[32]) and setting up the technical framework for the remainder of the paper.2.1. Hochschild (co)chain complexes.
All our A ∞ -categories will be Z / Z -graded, saturated and Calabi–Yau. Therefore we can assume, without loss ofgenerality, that they are smooth and after taking a minimal model have finitedimensional hom spaces, are strictly unital and have a cyclic pairing [21]. Moreoverwe will assume that any such category C has finitely many orthogonal objects, thatis the space of homomorphisms between distinct objects is trivial. Therefore we canconsider C as a direct sum (cid:76) i A i of Z / Z -graded, smooth and finite-dimensionalcyclic A ∞ -algebras of parity d ∈ Z / Z - here each A i is the endomorphism algebra ofeach object. By cyclic we mean there is a non-degenerate pairing (cid:104)− , −(cid:105) : A ⊗ i → K satisfying (cid:104) a, b (cid:105) = ( − | a | (cid:48) | b | (cid:48) (cid:104) b, a (cid:105) , (cid:104) m k ( a , . . . , a k − ) , a k (cid:105) = ( − ♥ (cid:104) m k ( a , . . . , a k ) , a (cid:105) , here ♥ = | a | (cid:48) ( | a | (cid:48) + . . . + | a k | (cid:48) ). Moreover, (cid:104) a, b (cid:105) = 0 if | a | + | b | (cid:54) = d .In addition we will allow C to be weakly curved, that is, each A ∞ algebra hasan m term of the form m ( A i ) = λ i i , where i is the unit and λ i is an elementof the ground field K .All the invariants we will consider, Hochschild (co)homology, negative cyclic ho-mology and periodic cyclic homology for C are obviously the direct sum of thecorresponding invariants for each A ∞ -algebra A i . Therefore, for notational sim-plicity, we will restrict ourselves to A ∞ -algebras. We work on the following setup:( † ) Let A be a A ∞ -algebra, with operations m = { m k } k ≥ . We assume A is Z/ Z -graded, strictly unital (unit denoted by ), smooth, finite-dimensional andcyclic of parity d ∈ Z / Z . We also assume that A is weakly curved: m = λ . Since we allow for curvature terms in our A ∞ -algebras, we must be careful whendefining the Hochschild (co)homology. Definition 2.1.
Let A be as in ( † ) . We define the Hochschild chain complex CC • ( A ) := (cid:77) k ≥ A ⊗ ( A/ K · ) ⊗ k , with the grading | α ⊗ α . . . α r | := | α | + | α | (cid:48) + . . . | α r | (cid:48) . The Hochschild cochaincomplex is defined as CC • ( A ) := Hom (cid:77) k ≥ ( A/ K · ) ⊗ k , A , where ϕ = { ϕ k } k ≥ is homogeneous of degree n if | ϕ k | = n − k . In both casesthe differentials are defined as in the uncurved case ( [32] ). More explicitly, in thehomology case the differential is given by b ( α | α . . .α r ) = (cid:88) ≤ i ≤ r ≤ j ≤ r − i +1 ( − (cid:63) α | α . . . m j ( α i , . . . , α i + j − ) . . . a r + (cid:88) ≤ i ≤ j ≤ r ( − @ m r − j + i +1 ( α j +1 , . . . , α r , α , α , . . . , α i ) | α i +1 . . . α j , where (cid:63) = | α ,i − | (cid:48) := | α | (cid:48) + . . . | α i − | (cid:48) and @ = | α ,j | (cid:48) | α j +1 ,r | (cid:48) . The complexes defined above are usually called the reduced
Hochschild (co)chaincomplexes. In the uncurved case, these are quasi-isomorphic to the usual ones[32], but when there is curvature this is not true and in this case, the unreducedcomplexes give trivial homologies (see [6]).All the constructions and operations in the Hochschild (co)chain complexes wewill consider in this paper behave in the reduced complexes exactly as they do inthe unreduced ones, assuming strict unitality (which we always do). Therefore thiswill not play a role in the remainder of the paper.
Remark 2.2.
From now on we will use the symbol @ for the sign obtained, fol-lowing the Koszul convention for the shifted degrees, from rotating the inputs of theexpression from their original order. .2. Homotopy Calculus structures.
The algebraic structures of the Hochschildpair ( CC • ( A ) , CC • ( A )) are extremely rich. Here we recall the ones which are mostrelevant to us: • A differential graded Lie algebra structure (DGLA) on CC • ( A )[1]. TheHochschild differential δ on the complex CC • ( A ) is given by bracketingwith the structure cochain, that is δ = [ m , − ]. • A cup product ∪ : CC • ( A ) ⊗ → CC • ( A ) which is associative and commu-tative on cohomology. It is defined as ϕ ∪ ψ := ( − | ϕ | (cid:48) m { ϕ, ψ } , where m {− , −} is Getzler brace operation (denoted by M in [32]).We need to introduce some more operators.– Given a Hochschild cochain ϕ we define a new cochain L ϕ by setting L ϕ ( α . . . α n ) := (cid:88) ( − (cid:63) α . . . ϕ ( α i +1 . . . ) . . . α n + (cid:88) ( − @ ϕ ( α j +1 . . . α . . . α i ) . . . α j where (cid:63) = | ϕ | (cid:48) ( | α | (cid:48) + . . . + | α i | (cid:48) ) and as before @ = ( | α | (cid:48) + . . . + | α j | (cid:48) )( | α j +1 | (cid:48) + . . . + | α n | (cid:48) ). Note that b = L m .– The Connes’ differential B ( α | α . . . | α r ) = (cid:88) ( − @ | α i , . . . , α r , α , α , . . . , α i − . – For a Hochschild cochain ϕ we set B { ϕ } ( α . . . α n ) := (cid:88) ( − (cid:63) | α j +1 . . . ϕ ( α i +1 . . . ) . . . α n , a . . . α j where (cid:63) = | ϕ | (cid:48) ( | α (cid:48) j +1 | + . . . + | α i | (cid:48) ) + @. This operation is denoted by B , in [32].– Given two Hochschild cochains ϕ, ψ we define: T ( ϕ, ψ )( α . . . α n ) := (cid:88) ( − (cid:63) ϕ ( α j +1 . . . ψ ( α k +1 . . . ) . . . α . . . ) α i +1 . . . α j where (cid:63) = | ψ | (cid:48) ( | α j +1 | (cid:48) + . . . + | α k | (cid:48) ) + @.Additionally we define b { ϕ } := T ( m , ϕ ).The Hochschild cochain complex CC • ( A ) acts on CC • ( A ) is two different ways: • The assignment φ (cid:55)→ L φ defines a differential graded Lie module structureon CC • ( A ) over the DGLA CC • ( A )[1]. • The assignment φ (cid:55)→ ϕ ∩ ( − ) := ( − | ϕ | b { ϕ } ( − ), called cap product, definesa left module structure, with respect to the cup product, on HH • ( A ), thatis ( ϕ ∪ ψ ) ∩ ( − ) = ϕ ∩ ( ψ ∩ ( − )) on homology.Next, note that B = bB + Bb = 0, therefore we can define the differential b + uB on CC • ( A )[[ u ]] (respectively CC • ( A )(( u ))), where u is a formal variable.The resulting homology HC −• ( A ) (respectively HP • ( A )) is called the negative cyclichomology of A (respectively periodic cyclic homology).We set the extended cap product action on the periodic cyclic chain complex CC • ( A )(( u )) by(1) ι { ϕ } := b { ϕ } + uB { ϕ } Proposition 2.3.
We have the following identities (1)
Cartan homotopy formula: [ b + uB, ι { ϕ } ] = − u · L ϕ − ι { [ m , ϕ ] } Daletskii–Gelfand–Tsygan homotopy formula: b { [ ϕ, ψ ] } = ( − | ϕ | (cid:48) [ L ϕ , b { ψ } ] − [ b, T ( ϕ, ψ )] + T ([ m , ϕ ] , ψ ) + ( − | ϕ | (cid:48) T ( ϕ, [ m , ψ ])The identity (1) is proved by Getzler [17]. The identity (2) is proved in [10], seealso [25].2.3. The duality isomorphism.
The cyclic pairing (cid:104)− , −(cid:105) on A has degree d ∈ Z / Z . Therefore it induces an isomorphism A → A ∨ [ d ] of A ∞ -bimodules, whichyields an isomorphism HH • ( A ) ∼ = HH • + d ( A ∨ ) . The right hand side is isomorphic to the shifted dual of the Hochschild homology HH •− d ( A ) ∨ .There is also a degree zero pairing naturally defined on Hochschild homology,known as the Mukai-pairing (cid:104)− , −(cid:105) Muk : HH • ( A ) ⊗ HH • ( A ) → K . Since A is smooth and compact, this pairing is non-degenerate by Shklyarov [33].Thus, it induces an isomorphism HH • ( A ) ∼ = HH −• ( A ) ∨ . Since we assume that A is finite-dimensional the Mukai pairing can be described at the chain-level (see [32,Proposition 5.22]): for α = α | α | . . . | α r and β = β | β | . . . | β s , we have(2) (cid:104) α, β (cid:105) Muk = (cid:88) tr (cid:2) c → ( − † m ∗ ( α j , .., α , .., m ∗ ( α i , .., c, β n , .., β , .. ) , β m , .. ) (cid:3) where tr stands for the trace of a linear map and † = 1 + | c || β | + ( | α j | (cid:48) + .. | α | (cid:48) + .. + | α i − | (cid:48) ) + @As before @ is the sign coming from rotating the α ’s and the β ’s.Putting together, we obtain a chain of isomorphisms:(3) D : HH • ( A ) ∼ = HH • + d ( A ∨ ) ∼ = HH •− d ( A ) ∨ ∼ = HH d −• ( A )Using the isomorphism to pull-back the Mukai-pairing yields a pairing which wedenote by D ∗ (cid:104)− , −(cid:105) Muk : HH • ( A ) ⊗ HH • ( A ) → K , defined as D ∗ (cid:104) ϕ, ψ (cid:105) Muk = ( − | ϕ | d (cid:104) D ( ϕ ) , D ( ψ ) (cid:105) Muk .We have the following folklore result.
Theorem 2.4.
Let A be a Z / Z -graded, smooth and finite-dimensional cyclic A ∞ -algebra. (a) The isomorphism D : HH • ( A ) → HH d −• ( A ) , is a map of HH • ( A ) -modules. (b) The triple (cid:0) HH • ( A ) , ∪ , D ∗ (cid:104)− , −(cid:105) Muk (cid:1) forms a Frobenius algebra.
We will provide a proof of this theorem in Appendix A.
Corollary 2.5. If HH • ( A ) is a semi-simple ring then HH odd ( A ) = 0 .Proof. By the above proposition, HH • ( A ) is a Frobenius algebra. This now followsfrom [18], we reproduce the argument here for the reader’s convenience. Supposethere is a class of odd degree ϕ (cid:54) = 0. Then ϕ ∪ ϕ = 0, by graded commutativity,and there is ψ such that D ∗ (cid:104) ϕ, ψ (cid:105) Muk = 1 by non-degeneracy of the pairing. Thisimplies D ∗ (cid:104) ϕ ∪ ψ, (cid:105) Muk = 1 and therefore ϕ ∪ ψ (cid:54) = 0. Since ( ϕ ∪ ψ ) = 0 we concludethat ϕ ∪ ψ is nilpotent. But this is a contradiction since there are no nilpotents ina semi-simple ring. (cid:3) .4. Pairing and u -connection. On HC −• ( A ) we can define a pairing and a con-nection. • The pairing, known as the higher residue pairing [35]:(4) (cid:104)− , −(cid:105) hres : HC −• ( A ) ⊗ HC −• ( A ) → K [[ u ]] , is obtained by extending the map defined by (2) sesquilinearly, that is (cid:104) uα, β (cid:105) hres = −(cid:104) α, uβ (cid:105) hres = u (cid:104) α, β (cid:105) Muk , for Hochschild chains α, β . • The meromorphic connection ∇ ddu : HC −• ( A ) → u − HC −• ( A ) is defined bythe formula (see [5][21][19][33]):(5) ∇ ddu := ddu + Γ2 u + ι { m (cid:48) } u . Where m (cid:48) is the cocycle in CC • ( A ) defined as m (cid:48) := (cid:81) k ≥ (2 − k ) m k and Γis the length operator Γ( a | . . . | a n ) = − n · a | . . . | a n . Proposition 2.6.
The higher residue pairing is parallel with respect to the u -connection, that is (6) ddu (cid:104) α, β (cid:105) hres = (cid:104)∇ ddu α, β (cid:105) hres − (cid:104) α, ∇ ddu β (cid:105) hres We will prove this proposition in Appendix B. Its dg-version was proved byShklyarov [35].3.
Splittings of the non-commutative Hodge filtration
Let A be as in ( † ). Assume that A has semi-simple Hochschild cohomology. Inthis section, we prove that A admits a canonical splitting of the Hodge filtrationin the sense of Definition 3.7. Furthermore, we exhibit a natural bijection betweenthe set of splittings with the set of grading operators on HH • ( A ).3.1. The semi-simple splitting.
We define the following cocycle in CC • ( A ): η := (cid:89) k ≥ (2 − k ) m k . One can easily verify that [ m , η ] = 0, hence it defines a class in the Hochschildcohomology [ η ] ∈ HH • ( A ). Lemma 3.1.
Assume that the Hochschild cohomology ring HH • ( A ) is semi-simple.Then we have [ η ] = 0 .Proof. Let [ e ] , . . . , [ e k ] be a basis of HH • ( A ), and let N be the maximum ofthe lengths of all the e i . By definition of the cap product, if α is a chain oflength n then η ∩ α is a chain of length less than or equal to n −
1. Therefore η ∪ N +1 ∩ e i = 0 for all i . Hence η ∪ N +1 ∩ ( − ) is the zero map on homology. Inparticular, 0 = η ∪ N +1 ∩ ω = η ∪ N +1 ∩ D ( ) = D ( η ∪ N +1 ), which gives η ∪ N +1 = 0,since D is an isomorphism. In other words, [ η ] is a nilpotent element, which mustnecessarily be zero in a semi-simple ring. (cid:3) By the previous lemma, there exists a cochain Q ∈ CC • ( A ) such that[ m , Q ] = η. We fix such a cochain Q in the following. Using Proposition2.3(1) we have b + uB, ι { Q } ] = − u · L Q − ι { η } . Moreover m (cid:48) = 2 λ + η since m = λ . Therefore we have a simplified formulaof the u -connection operator:(7) ∇ (cid:48) ddu = ddu + λ · id u + Γ − L Q u , using the fact that ι { } = id . Note that the two connections ∇ (cid:48) ddu and ∇ ddu differby [ b + uB,ι { Q } ]2 u , which implies that they induce the same map on homology. For thisreason, we shall not distinguish them, and slightly abuse the notation, using ∇ ddu for both operators.Next we will study the u − part of this connection. From now on we use thenotation: M := Γ − L Q . Lemma 3.2.
We have [ b, M ] = − b . Thus the operator M induces a map onHochschild homology, which we still denote by M : HH • ( A ) → HH • ( A ) . Proof.
Since b = L m , we have [ b, L Q ] = [ L m , L Q ] = L [ m ,Q ] = L η . On the otherhand, one checks that [ b, Γ] = L (cid:81) k ≥ (1 − k ) m k . The two identities imply that[ b, M ] = [ b, Γ − L Q ] = −L m = − b. (cid:3) Lemma 3.3.
The operator M = Γ − L Q as defined above is anti-symmetric withrespect to the Mukai pairing, i.e. (cid:104) M x, y (cid:105)
Muk + (cid:104) x, M y (cid:105) Muk = 0 , ∀ x, y ∈ HH • ( A ) . Proof.
For two chains x = a | a | · · · | a k and y = b | b | · · · | b l , it is clear we have (cid:104) Γ x, y (cid:105) Muk + (cid:104) x, Γ y (cid:105) Muk = ( − k − l ) (cid:104) x, y (cid:105) Muk . Next, we prove the following identity (cid:104)L Q x, y (cid:105) Muk + (cid:104) x, L Q y (cid:105) Muk = ( − k − l ) (cid:104) x, y (cid:105) Muk , which implies the lemma. We could prove this by showing that both sides differ byan explicit homotopy like we do in the proofs of Proposition A.1 or Proposition 2.6.Instead we will give a pictorial proof using the tree diagrams and sign conventionsas in Sheridan [32, Appendix C]. Indeed, the Mukai pairing can be graphicallywritten as (cid:104) x, y (cid:105) Muk = str (cid:0) x y (cid:1) here str denotes the super-trace. With this notation and using [ m , Q ] = η = (cid:81) k (2 − k ) m k , we have str (cid:0) L Q x y (cid:1) + str (cid:0) x L Q y (cid:1) = str (cid:0) L Q x y (cid:1) − str (cid:0) x y • Q (cid:1) + str (cid:0) x y • Q (cid:1) + str (cid:0) x L Q y (cid:1) = str (cid:0) x y • η (cid:1) − str (cid:0) x y • Q (cid:1) + str (cid:0) x y • Q (cid:1) + str (cid:0) x y • η (cid:1) = str (cid:0) x y • η (cid:1) + str (cid:0) x y • η (cid:1) = ( − k − l ) (cid:104) x, y (cid:105) Muk
The first equality uses the fact that str ( A ◦ B ) = ( − | A || B | str ( B ◦ A ), for homoge-neous maps A, B . The last equality uses the fact that if (2 − i ) m i and (2 − j ) m j are used at the two positions of η , then we must have that i + j = k + l + 4, whichimplies that the total sum has coefficient 4 − i − j = − k − l . (cid:3) Dually, on the Hochschild cochain complex, we define an operatorˇ M : CC • ( A ) → CC • ( A )by formula ˇ M ( − ) := [ Q, − ] − ˇΓ , ˇΓ( φ ) = k · ϕ, for a cochain ϕ ∈ Hom ( A ⊗ k , A ) , ∀ k ≥ . Analogously to Lemma 3.2, one verifies that [ δ, [ Q, − ]] = [ η, − ] and additionally[ δ, ˇΓ] = (cid:104)(cid:81) k ≥ (1 − k ) m k , − (cid:105) , which implies[ δ, ˇ M ] = δ. Therefore ˇ M induces a map on cohomologyˇ M : HH • ( A ) → HH • ( A ) . Lemma 3.4.
The operator ˇ M is a derivation of the Hochschild cohomology ring,i.e. we have ˇ M ( φ ∪ ψ ) = ˇ M ( φ ) ∪ ψ + φ ∪ ˇ M ( ψ ) . Furthermore, if the Hochschild cohomology ring HH • ( A ) is semi-simple, then ˇ M =0 . roof. To prove that ˇ M is a derivation, one first checks, by direct computation, thefollowing formula. Let ϕ and ψ be closed Hochschild cochains of even degree andlet Q be an odd cochain, we have the following:[ Q, m { ϕ, ψ } ] − m { [ Q, ϕ ] , ψ } − m { ϕ, [ Q, ψ ] } = − [ Q, m ] { ϕ, ψ } + [ m , Q { ϕ, ψ } ] . Using the fact that ϕ ∪ ψ = − m { ϕ, ψ } and [ Q, m ] = − η , this formula gives[ Q, ϕ ∪ ψ ] − [ Q, ϕ ] ∪ ψ − ϕ ∪ [ Q, ψ ] = η { ϕ, ψ } + [ m , Q { ϕ, ψ } ] . An easier computation, using the definition of ˇΓ, givesˇΓ( ϕ ∪ ψ ) − ˇΓ( ϕ ) ∪ ψ − ϕ ∪ ˇΓ( ψ ) = η { ϕ, ψ } . Subtracting the above two equations yields (on cohomology level) the intendedidentity(8) ˇ M ( φ ∪ ψ ) = ˇ M ( φ ) ∪ ψ + φ ∪ ˇ M ( ψ ) . To prove the second part, let ( e i ) i =1 ,..,n be a semi-simple basis of HH • ( A ).Applying ˇ M to the equality e i ∪ e i = e i and using (8) we obtain2 ˇ M ( e i ) ∪ e i = ˇ M ( e i ) . Here we used the fact that ∪ is graded commutative and the e i ’s have even degree.Now, using the notation ˇ M ( e i ) = (cid:80) j ˇ M ji e j , the above equation gives (cid:88) j ˇ M ji e j = (cid:88) j M ji e j ∪ e i = (cid:88) j δ ij ˇ M ji e i . Hence ˇ M ji = 0, if i (cid:54) = j and ˇ M ii = 2 ˇ M ii , which proves that ˇ M = 0. (cid:3) Lemma 3.5.
As operators on HH • ( A ) , the following identity holds: [ b { α } , M ] = b { ˇ M ( α ) } . Proof.
Applying Proposition 2.3(2) to ϕ = Q, ψ = α we get b { Q, α } = [ L Q , b { α } ] − [ b, T ( Q, α )] + T ( η, α ) , since [ m , Q ] = η and α is closed and even. Then using the definition of T it is easyto verify that b { ˇΓ α } = [Γ , b { α } ] + T ( η, α ) . Subtracting the above two equations yields b { ˇ M ( α ) } = [ − M, b { α } ] + [ b, T ( Q, α )] , which after passing to homology gives the desired identity in the lemma. (cid:3) Theorem 3.6.
Let A be as in ( † ) . Assume that Hochschild cohomology ring HH • ( A ) is semi-simple. Then the operator M : HH • ( A ) → HH • ( A ) is the zerooperator.Proof. By Lemma 3.4 and Lemma 3.5, we have[ b { α } , M ] = 0 , ∀ α ∈ HH • ( A ) , that is, the operators b { α } and M commute. As in the proof of Lemma 3.4, we let e , · · · , e n be a idempotent basis of HH • ( A ). Since the duality map D in Equation(3) is an isomorphism, D ( e ) , · · · , D ( e n ) form a basis of HH • ( A ). Moreover, byTheorem 2.4, this is an orthogonal basis with respect to the Mukai pairing. irst note that(9) b { e j } ( D ( e i )) = e j ∩ D ( e i ) = D ( e j ∪ e i ) = δ ij D ( e i ) . Let us write M ( D ( e i )) = (cid:88) j M ji · D ( e j ) . Applying b { e i } to both sides of this equality we obtain M ( b { e i } ( D ( e i ))) = (cid:88) j M ji b { e i } ( D ( e j )) , since b { α } and M commute. Using (9) this now gives (cid:88) j M ji · D ( e j ) = (cid:88) j M ji δ ij D ( e j ) = M ii D ( e i ) , which implies that M ij = 0 if i (cid:54) = j .It remains to prove that M ii is also zero. Lemma 3.3 together with symmetry ofthe Mukai pairing give the identity 2 M ii (cid:104) D ( e i ) , D ( e i ) (cid:105) Muk = 0, since the D ( e j ) arean orthogonal basis. Then non-degeneracy of the pairing gives M ii = 0. (cid:3) We will use the previous results to obtain a canonical splitting of the Hodgefiltration in the semi-simple case. We start with the following definition.
Definition 3.7.
Let A be as in ( † ) . Let ω = D ( ) ∈ HH • ( A ) . A K -linear map s : HH • ( A ) → HC −• ( A ) is called a splitting of the Hodge filtration of A if it satisfies S1. (Splitting condition) s splits the canonical map π : HC −• ( A ) → HH • ( A ) , de-fined as π ( (cid:80) n ≥ α n u n ) = α . S2. (Lagrangian condition) (cid:104) s ( α ) , s ( β ) (cid:105) hres = (cid:104) α, β (cid:105) Muk , ∀ α, β ∈ HH • ( A ) .A splitting s is called a good splitting if it satisfies S3. (Homogeneity) L s := (cid:76) l ∈ N u − l · Im ( s ) is stable under the u -connection ∇ u ddu . This is equivalent to requiring ∇ u ddu s ( α ) ∈ u − Im ( s ) + Im ( s ) , ∀ α .A splitting s is called ω -compatible if S4. ( ω -Compatibility) ∇ u ddu s ( ω ) ∈ r · s ( ω ) + u − · Im ( s ) for some r ∈ K . Corollary 3.8.
Let A be as in ( † ) . Assume that Hochschild cohomology ring HH • ( A ) is semi-simple. Then there exists a good, ω -compatible splitting of theHodge filtration s : HH • ( A ) → HC −• ( A ) uniquely characterized by the equation (10) ∇ u ddu s ( α ) = u − λ · s ( α ) , ∀ α ∈ HH • ( A ) . Proof.
By Corollary 2.5, the Hochschild cohomology is concentrated in even degree,which implies that the Hochschild homology HH • ( A ) is concentrated in degree d (mod 2). This implies that the Hodge-to-de-Rham spectral sequence degenerates atthe E -page for degree reason, which implies that HC −• ( A ) is a free K [[ u ]]-module,of finite rank.Recall from Equation (10) that ∇ ddu = λ · id u + ˜ ∇ ddu , where˜ ∇ ddu = ddu + M u . Due to the vanishing result of the previous theorem, we have that ˜ ∇ is actually aregular connection ˜ ∇ ddu : HC −• ( A ) → HC −• ( A ) . hus Equation (10) is equivalent to requiring ˜ ∇ -flatness of s . This uniquelydetermines a linear map s : HH • ( A ) → HC −• ( A ), which sends an initial vector α ∈ HH • ( A ) = HC −• ( A ) /u · HC −• ( A ) to its unique ˜ ∇ -flat extension s ( α ). Byconstruction, this map s satisfies (S1.)In order to check the Lagrangian condition (S2.) we use Proposition 2.6 tocompute: u ddu (cid:104) s ( x ) , s ( y ) (cid:105) hres = (cid:104)∇ u ddu s ( x ) , s ( y ) (cid:105) hres + (cid:104) s ( x ) , ∇ u ddu s ( y ) (cid:105) hres (11) = (cid:104) u − λs ( x ) , s ( y ) (cid:105) hres + (cid:104) s ( x ) , u − λs ( y ) (cid:105) hres = u − λ ( (cid:104) s ( x ) , s ( y ) (cid:105) hres − (cid:104) s ( x ) , s ( y ) (cid:105) hres ) = 0 , which implies (cid:104) s ( x ) , s ( y ) (cid:105) hres is a constant. Therefore by definition of the higherresidue pairing we have (cid:104) s ( x ) , s ( y ) (cid:105) hres = (cid:104) x, y (cid:105) Muk . Equation (10) immediatelyimplies Homogeneity of the splitting and ω -compatibility with r = 0. (cid:3) We shall refer to the splitting in the above Corollary 3.8 the semi-simple splittingof the Hodge filtration of A , and denote it by s A : HH • ( A ) → HC −• ( A )as it is canonically associated with the A ∞ -algebra A .3.2. Grading operators and good splittings.
In the remaining part of the sec-tion, we shall classify the set of ω -compatible good splittings of the Hodge filtrationof a category C with semi-simple Hochschild cohomology. Recall our setup:( †† ) C is a direct sum of the form C = A (cid:77) · · · (cid:77) A k , with each of A j ( j = 1 , ..., k ) a Z / Z -graded, smooth finite-dimensional cyclic A ∞ -algebra of parity d ∈ Z / Z . Each A j is strict unital, and curvature m ( A j ) = λ j j .Furthermore, we assume that the Hochschild cohomology ring HH • ( C ) is semi-simple, which is equivalent to requiring that each A j has semi-simple Hochschildcohomology. Note that HH • ( C ) ∼ = HH • ( A ) (cid:76) · · · (cid:76) HH • ( A k ). Denote by ξ the diagonaloperator acting on HH • ( C ) by λ j · id on the component HH • ( A j ). In other words, ξ = m ∩ ( − ). Corollary 3.8 applied to such category C states that there is a uniquesplitting s C : HH • ( C ) → HC −• ( C ) satisfying(12) ∇ u ddu s C ( α ) = u − s C ( ξ ( α )) , ∀ α ∈ HH • ( C ) . Definition 3.9.
A grading operator on HH • ( C ) is a K -linear map µ : HH • ( C ) → HH • ( C ) such that (a) It is anti-symmetric: (cid:104) µ ( x ) , y (cid:105) Muk + (cid:104) x, µ ( y ) (cid:105) Muk = 0 . (b) It is zero on the diagonal blocks λ j · id ’s in the matrix ξ . (c) Let ω = D ( ) , where = + . . . + k . Then ω is an eigenvector of µ , i.e. µ ( ω ) = r · ω for some r ∈ K , which we call the weight of µ . e are now ready to prove the main theorem of this section, which gives Theorem1.2 in the Introduction. Theorem 3.10.
Let C be an A ∞ -category as in ( †† ) . Then there exists a naturallydefined bijection between the set of grading operators on HH • ( C ) (Definition 3.9)and the set of ω -compatible good splittings of the Hodge filtration of C (Defini-tion 3.7).Proof. Let s : HH • ( C ) → HC −• ( C ) be a ω -compatible good splitting. By the Homo-geneity condition, we have ∇ u ddu s ( x ) ∈ u − Im ( s ) + Im ( s ) . Use this property to define an operator µ s : HH • ( C ) → HH • ( C ) by requiring that(13) ∇ u ddu s ( x ) − s ( µ s ( x )) ∈ u − Im ( s ) . In other words, the grading operator µ s is simply the regular part of the operator ∇ u ddu s ( x ) pulled back under the isomorphism HH • ( C ) ∼ = Im ( s ). We will show µ s is a grading operator. Property ( a ) of µ s follows from the Lagrangian conditiontogether Proposition 2.6:0 = u ddu (cid:104) x, y (cid:105) Muk = u ddu (cid:104) s ( x ) , s ( y ) (cid:105) hres == (cid:104)∇ u ddu s ( x ) , s ( y ) (cid:105) hres + (cid:104) s ( x ) , ∇ u ddu s ( y ) (cid:105) hres (14) = (cid:104) s ( µ s ( x )) + u − Im ( s ) , s ( y ) (cid:105) hres + (cid:104) s ( x ) , s ( µ s ( y )) + u − Im ( s ) (cid:105) hres = (cid:104) µ s ( x ) , y (cid:105) Muk + (cid:104) x, µ s ( y ) (cid:105) Muk + u − k, for some k ∈ K .For Property ( b ), we write the splitting s in terms of the canonical semi-simplesplitting s C . More precisely we can write s as s ( x ) = s C ( x ) + s C ( R x ) · u + s C ( R x ) · u + · · · , ∀ x ∈ HH • ( C ) , for some matrix operators R j : HH • ( C ) → HH • ( C ) , j ≥
1. We also set R = id fornotational convenience. Applying ∇ u ddu to both sides, and using (12), yields ∇ u ddu s ( x ) = u − (cid:88) k ≥ s C ( ξR k ( x )) u k + (cid:88) k ≥ ks C ( R k ( x )) u k = u − (cid:88) k ≥ s C ( ξR k ( x )) u k + u − (cid:88) k ≥ s C ( R k ξ ( x )) u k − u − (cid:88) k ≥ s C ( R k ξ ( x )) u k + (cid:88) k ≥ ks C ( R k ( x )) u k = u − (cid:88) k ≥ s C ( R k ξ ( x )) u k + u − (cid:88) k ≥ s C ([ ξ, R k ]( x )) u k + (cid:88) k ≥ ks C ( R k ( x )) u k = u − s ( ξx ) + (cid:88) k ≥ s C (cid:0) [ ξ, R k +1 ] x + kR k x (cid:1) u k On the other hand, Equation (13) gives ∇ u ddu s ( x ) = u − Im ( s ) + (cid:88) k ≥ s C ( R k µ s ( x )) u k . omparing these two equations we conclude(15) [ ξ, R k +1 ] = R k ( µ s − k ) , ∀ k ≥ . For k = 0, this equation is [ ξ, R ] = µ s , which proves the Property ( b ) of the gradingoperator µ s . Property ( c ) follow immediately from the ω -compatibility of s .Conversely, if we are given a grading operator µ , then Equation (15) has a uniquesolution inductively obtained as follows.Assume that we had R j for j ≤ k , to obtain R k +1 we use [ ξ, R k +1 ] = R k ( µ − k )to determine the entries in R k +1 outside the big diagonal, that is the entries ( i, j )with λ i (cid:54) = λ j . Explicitly we have( R k +1 ) ij = 1 λ i − λ j (cid:32)(cid:88) l ( R k ) il µ lj − k ( R k ) ij (cid:33) . For entries in the big diagonal we consider the next equation [ ξ, R k +2 ] = R k +1 ( µ − k − R k +1 ) ij = 1 k + 1 (cid:88) l ( R k +1 ) il µ lj . Note that there is no ambiguity on the right hand side of this equation since when λ l = λ i = λ j then µ lj = 0, by Property (b), and when λ l (cid:54) = λ i then ( R k +1 ) il wasdefined previously.Denote this splitting by s µ ( x ) = s C ( x ) + s C ( R x ) · u + s C ( R x ) · u + · · · , ∀ x ∈ HH • ( C ) . We now verify that it automatically satisfies the Lagrangian condition. Indeed, theLagrangian condition is equivalent to R ∗ ( − u ) R ( u ) = id , R ( u ) = (cid:88) n ≥ R n · u n , R ∗ ( u ) = (cid:88) n ≥ R ∗ n · u n , with R ∗ n the adjoint of R n with respect to the Mukai pairing, i.e. it is defined byrequiring that (cid:104) R n x, y (cid:105) Muk = (cid:104) x, R ∗ n y (cid:105) Muk . If we fix a orthonormal basis of HH • ( C ),then the adjoint is simply given by the transpose operation. In such a basis theabove identity R ∗ ( − u ) R ( u ) = id is then equivalent to P n := n (cid:88) j =0 ( − j R tj R n − j = 0 , ∀ n ≥ . or this, we compute[ ξ, P n +1 ] = n +1 (cid:88) j =0 ( − j (cid:0) R tj [ ξ, R n +1 − j ] + [ ξ, R tj ] R n +1 − j (cid:1) = n (cid:88) j =0 ( − j R tj (cid:0) R n − j µ − ( n − j ) R n − j (cid:1) − n +1 (cid:88) j =1 ( − j (cid:0) R j − µ − ( j − R j − (cid:1) t R n +1 − j = n (cid:88) j =0 ( − j R tj R n − j µ + n +1 (cid:88) j =1 ( − j µR tj − R n +1 − j − n (cid:88) j =0 ( − j ( n − j ) R tj R n − j + n +1 (cid:88) j =1 ( − j ( j − R tj − R n +1 − j = P n µ − µP n − nP n = [ P n , µ ] − nP n . Here, on the third equality, we used the fact that in our basis µ t = − µ , by Property(a) of the grading. We now prove P n = 0, by induction on n . Assuming P n = 0 (orin the case n = 0, P = id ), the above computation gives[ ξ, P n +1 ] = 0 , or equivalently ( λ i − λ j ) ( P n +1 ) ij = 0. Hence ( P n +1 ) ij = 0 when λ i (cid:54) = λ j . When λ i = λ j , the ( i, j ) entry of the above computation for n + 2 gives(16) 0 = (cid:88) l ( P n +1 ) il µ lj − (cid:88) l µ il ( P n +1 ) lj − ( n + 1) ( P n +1 ) ij . Now for each l either, λ l (cid:54) = λ j , λ i and therefore we already proved that ( P n +1 ) il =( P n +1 ) lj = 0 or, λ l = λ i = λ j and therefore µ lj = µ il = 0. Hence the first twosums in (16) vanish and we conclude ( P n +1 ) ij = 0 also when λ i = λ j . Therefore s µ satisfies the Lagrangian condition.By design, the splitting s µ satisfies the equation ∇ u ddu s µ ( x ) = s µ ( µ ( x )) + u − s µ ( ξ ( x )) , which immediately implies the Homogeneity and the ω -compatibility conditions, byProperty (c) of µ .To finish the proof, we note that by the uniqueness of solutions of Equation (15),we conclude that the two assignments described above are inverse bijections. (cid:3) Categorical primitive forms
In this section, we prove Theorem 1.1 in the Introduction and describe the Frobe-nius manifolds obtained from the categorical primitive forms.4.1.
VSHS’s from non-commutative geometry.
Here we work in a more gen-eral set-up than in the previous section. The A ∞ -category C will be as in ( †† ),except we do not require the Hochschild cohomology to be semi-simple, only theweaker condition that HH • ( C ) is concentrated in even degree. nder these assumptions the formal deformation theory of C (as strict unital A ∞ -category with finitely many objects ) is therefore unobstructed. assume C Let m ⊂ R be its unique maximal ideal. Following [5], define the completedHochschild and negative cyclic complexes of C by CC • ( C ) := lim ←− CC • ( C / m k ) ,CC −• ( C )[[ u ]] := lim ←− CC • ( C / m k )[[ u ]] . Since we shall only use the completed Hochschild/negative cyclic chain complexes,we choose to not introduce new notations. Similarly, their homology groups will bedenoted by HH • ( C ) and HC −• ( C ).We recall some basic terminologies from [5, Section 3]. By construction, theuniversal family C is given by a R -linear A ∞ -structure on the R -linear category C ⊗ K R . Denote its A ∞ -structure by m k ( t ) for k ≥
0. Define the Kodaira–Spencermap KS : Der ( R ) → HH • ( C ) , KS ( ∂∂t j ) := [ (cid:89) k ≥ ∂ m k ( t ) ∂t j ] . By construction of C , this map is an isomorphism.As in [5], one can construct a polarized VSHS on HC −• ( C ) by considering thefollowing structures • In the t -directions, we consider Getzler’s connection [17], explicitly givenby(17) ∇ ∂∂tj := ∂∂t j − ι ( (cid:81) k ≥ ∂ m k ( t ) dt j ) u , where ι is defined in Equation (1). Getzler proved in loc. cit. this connec-tion is flat. • In the u -direction we take the connection defined in Equation (5). It wasshown in [5, Lemma 3.6] that the u -connection commutes with Getzler’sconnection. • The higher residue pairing (cid:104)− , −(cid:105) hres : HC −• ( C ) ⊗ HC −• ( C ) → R [[ u ]]originally defined in [35]. Here we use the A ∞ version described in [32],and recalled in (4).It is essential here to take the m -adic completed version in order to ensure that HC −• ( C ) is a locally free R [[ u ]]-module (of finite rank). The non-degeneracy of thecategorical higher residue pairing is due to Shklyarov [33]. We denote this VSHSby (cid:0) HC −• ( C ) , ∇ , (cid:104)− , −(cid:105) hres (cid:1) .We define the Euler vector field of the VSHS (cid:0) HC −• ( C ) , ∇ , (cid:104)− , −(cid:105) hres (cid:1) by Eu := KS − (cid:89) k ≥ − k m k ( t ) ∈ Der ( R ) . Since we require C has only finitely many objects, deformations of such A ∞ -category is, bydefinition, given by deformations of the total A ∞ -algebra over the semi-simple ring spanned bythe identity morphisms of objects in C . characteristic property of Eu is that the combined differential operator ∇ u ∂∂u + ∇ GetEu , which a priori has a first order pole at u = 0, is in fact regular at u = 0. Definition 4.1.
An element ζ ∈ HC −• ( C ) is called a primitive form of the polarizedVSHS HC −• ( C ) if it satisfies the following conditions: P1. (Primitivity) The map defined by ρ ζ : Der ( R ) → HC −• ( C ) /uHC −• ( C ) , ρ ζ ( v ) := [ u · ∇ Get v ζ ] is an isomorphism. P2. (Orthogonality) For any tangent vectors v , v ∈ Der ( R ) , we have (cid:104) u ∇ Get v ζ, u ∇ Get v ζ (cid:105) ∈ R. P3. (Holonomicity) For any tangent vectors v , v , v ∈ Der ( R ) , we have (cid:104) u ∇ Get v u ∇ Get v ζ, u ∇ Get v ζ (cid:105) ∈ R ⊕ u · R. P4. (Homogeneity) There exists a constant r ∈ K such that ( ∇ u ∂∂u + ∇ GetEu ) ζ = rζ. The following result exhibits a natural bijection between the set of primitiveforms of the VSHS (cid:0) HC −• ( C ) , ∇ , (cid:104)− , −(cid:105) hres (cid:1) with the set of ω -compatible goodsplittings of the Hodge filtration of its central fiber C . This kind of bijection isoriginally due to Saito [28], and was used to prove the existence of primitive formsin the quasi-homogeneous case. See also the more recent work of Li–Li–Saito [23]. Theorem 4.2.
Let HC −• ( C ) be the polarized VSHS defined as above. Let ω = D ( ) ∈ HH • ( C ) . Then there exists a natural bijection between the following twosets P := (cid:8) ζ ∈ HC −• ( C ) | ζ is a primitive form such that ζ | t =0 ,u =0 = ω. (cid:9) S := (cid:8) s : HH • ( C ) → HC −• ( C ) | s is an ω -compatible good splitting . (cid:9) Proof.
Step 1.
We first define a map Φ :
P→S . Take ζ ∈ P , we will refer to this asa primitive form extending ω . Recall the linear coordinate system t , · · · , t m of R isdual to a basis ϕ , · · · , ϕ m of HH • ( C ). Let us denote by b j := b { ϕ j } ( ω ) ∈ HH • ( C ).It follows from versality of C and Theorem 2.4 that these form a basis of HH • ( C ),since ω = D ( ). We will refer to this fact as the primitivity of ω . The splitting s = Φ( ζ ) is defined by s ( b j ) = (cid:0) u ∇ Get ∂∂tj ζ (cid:1) | t =0 , In other words, s is the unique splitting whose image is Im (cid:0) Φ( ζ ) (cid:1) := span (cid:26)(cid:0) u ∇ ∂∂tj ζ (cid:1) | t =0 , ≤ j ≤ m (cid:27) . By the property P . , the pairing (cid:104) u ∇ ∂∂ti ζ, u ∇ ∂∂tj ζ (cid:105) is inside R , which implies thatits restriction to the central fiber lies inside K . This shows that the splitting Φ( ζ ) atisfies S . . To prove the property S . , observe that ∇ u ∂∂u (cid:0) u ∇ ∂∂tj ζ (cid:1) = u ∇ ∂∂tj ζ + u · ∇ u ∂∂u ∇ ∂∂tj ζ = u ∇ ∂∂tj ζ + u · ∇ ∂∂tj ∇ u ∂∂u ζ = u ∇ ∂∂tj ζ + u · ∇ ∂∂tj (cid:0) rζ − ∇ Eu ζ (cid:1) = (cid:0) (1 + r ) u ∇ ∂∂tj ζ (cid:1) − u ∇ ∂∂tj ∇ Eu ζ. Here the second equality follows from the flatness of the connection. Using P . and P . , the above implies that (cid:104)∇ u ∂∂u (cid:0) u ∇ ∂∂tj ζ (cid:1) , u ∇ ∂∂ti ζ (cid:105) hres ∈ u − R ⊕ R By the non-degeneracy of the higher residue pairing, this gives ∇ u ∂∂u (cid:0) u ∇ ∂∂tj ζ (cid:1) | t =0 ∈ u − Im ( s ) ⊕ Im ( s )which implies S . . Finally, restricting P . to t = 0, gives ∇ u ∂∂u ( ω ) = rω − u − (cid:0) u ∇ GetEu ζ (cid:1) | t =0 , which proves S Step 2.
Next we define a backward map Ψ :
S→P . Let s be an ω -compatiblegood splitting of the Hodge filtration of the central fiber C . It induces a direct sumdecomposition HP • ( C ) = HC −• ( C ) (cid:77) (cid:0) (cid:77) k ≥ u − k · Im ( s ) (cid:1) . Parallel transport Im ( s ) using the Getzler’s connection. We obtain, for each N ≥ HP • ( C ) ( N ) = HC −• ( C ) ( N ) (cid:77) (cid:0) (cid:77) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat (cid:1) . Here for an R -module M , we use M ( N ) to denote M/ m N +1 M . Denote by π ( N ) : HP • ( C ) ( N ) → HC −• ( C ) ( N ) the projection map using the above direct sum decomposition.To this end, starting with ω ∈ HH • ( C ), and apply the splitting s to it yields s ( ω ) ∈ HC • ( C ). Denote by s ( ω ) flat the flat extension of s ( ω ). Since the Getzler’sconnection has a first order pole at u = 0, the flat section in general is inside s ( ω ) flat ∈ HP • ( C ). Denote by s ( ω ) flat , ( N ) ∈ HP • ( C ) its image modulo m N +1 . Wedefine the primitive form associated with the splitting s by ζ = Ψ( s ) = lim ←− π ( N ) (cid:0) s ( ω ) flat , ( N ) (cid:1) . Let us verify that ζ is indeed a primitive form. For condition P . , the primitivityof ζ follows from that of ω by Nakayama Lemma. To prove other properties of ζ ,let us fix a positive integer N . We also choose a basis { s , · · · , s µ } of Im ( s ). Bydefinition we may write ζ ( N ) = s ( ω ) flat , ( N ) − (cid:88) k ≥ u − k (cid:0) µ (cid:88) j =1 f k,j · s flat , ( N ) j (cid:1) . hus for a vector field X ∈ Der ( R ), we have u ∇ Get X ζ ( N ) = (cid:88) k ≥ u − k (cid:0) µ (cid:88) j =1 X ( f k,j ) · s flat , ( N ) j (cid:1) . This shows that for any two vector fields v , v , we have (cid:104) u ∇ Get v ζ ( N ) , u ∇ Get v ζ ( N ) (cid:105) hres ∈ R [ u − ] . On the other hand, since ζ ( N ) ∈ HC −• ( C ) ( N ) and ∇ Get has a simple pole along u = 0, we also have (cid:104) u ∇ Get v ζ ( N ) , u ∇ Get v ζ ( N ) (cid:105) hres ∈ R [[ u ]] . The two together imply ζ satisfies P . . To prove P . , we differentiate ζ ( N ) twiceto get u ∇ Get v u ∇ Get v ζ ( N ) = (cid:88) k ≥ u − k (cid:0) µ (cid:88) j =1 v v ( f k,j ) · s flat , ( N ) j (cid:1) . Taking higher residue pairing, this implies that (cid:104) u ∇ Get v u ∇ Get v ζ ( N ) , u ∇ Get v ζ ( N ) (cid:105) hres ∈ R [ u − ] ⊕ u · R. On the other hand, we have a priori that (cid:104) u ∇ Get v u ∇ Get v ζ ( N ) , u ∇ Get v ζ ( N ) (cid:105) hres ∈ R [[ u ]].Taking intersection yields exactly P . .Finally, to check condition P . , observe that since s ( ω ) flat , ( N ) ∈ ζ ( N ) + (cid:0) (cid:77) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat (cid:1) , we have( ∇ u ∂∂u + ∇ Eu ) s ( ω ) flat , ( N ) ∈ ( ∇ u ∂∂u + ∇ Eu ) ζ ( N ) + (cid:0) (cid:77) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat (cid:1) . Here we used that ∇ u ∂∂u preserves the subspace (cid:0) (cid:76) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat (cid:1) dueto S . . On the other hand, by S . we have ∇ u ∂∂u s ( ω ) ∈ r · s ( ω ) + u − Im ( s ), whichby taking flat extensions on both sides yields ∇ u ∂∂u s ( ω ) flat , ( N ) ∈ r · s ( ω ) flat , ( N ) + (cid:0) (cid:77) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat (cid:1) . Comparing the above two decompositions, and we deduce that( ∇ u ∂∂u + ∇ Eu ) ζ ( N ) = r · ζ ( N ) , as desired. Step 3.
We prove that ΦΨ = id S . Let ν ∈ Der ( R ) be a tangent vector.Differentiating the difference s ( ω ) flat , ( N ) − ζ ( N ) ∈ (cid:76) k ≥ u − k R ( N ) ⊗ K Im ( s ) flat , ( N ) gives − u ∇ Get ν ζ ( N ) ∈ (cid:77) k ≥ u − k +1 R ( N ) ⊗ K Im ( s ) flat , ( N ) . Restricting to the central fiber gives (cid:0) − u ∇ Get ν ζ ( N ) (cid:1) | t =0 ∈ (cid:77) k ≥ u − k +1 Im ( s ) . But the left hand side also lies in HC −• ( C ), which implies that (cid:0) − u ∇ Get ν ζ ( N ) (cid:1) | t =0 ∈ Im ( s ) . his shows that ΦΨ = id S . Step 4.
We prove that ΨΦ = id P . Recall the splitting s = Φ( ζ ) is defined by s ( b j ) = (cid:0) u ∇ Get ∂∂tj ζ (cid:1) | t =0 . Flat extensions of s ( b j ) induces a splitting of the Hodge filtration of HC −• ( C ). Foreach N >
0, we obtain two splittings of the Hodge filtration: L := Ψ( s ) = (cid:77) k ≥ u − k span (cid:110) s ( b j ) flat , ( N ) (cid:111) ,L := (cid:77) k ≥ u − k span (cid:26) u ∇ Get ∂∂tj ζ ( N ) (cid:27) . Note that L is a splitting due to Condition P . . We claim that the two splittingsare equal: L = L . Indeed, by definition the two splittings are clearly the samewhen restricted to the central fiber t = 0. Thus, it suffices to prove that both L and L are preserved by the Getzler-Gauss-Manin connection. Since L is generatedby flat sections, it is obviously preserved by ∇ Get . For any 1 ≤ i, j ≤ µ , we get adecomposition ∇ Get ∂∂ti (cid:0) ∇ Get ∂∂tj ζ ( N ) (cid:1) = u − β − + u − β − + β + uβ + · · · , with β k ∈ L . Using the non-degeneracy of the polarization and Condition P . , wededuce that ∇ Get ∂∂ti (cid:0) ∇ Get ∂∂tj ζ ( N ) (cid:1) = u − β − + u − β − . This shows that the splitting L is preserved by the Getzler connection. This provesour claim that L = L .Now, by definition of ζ (cid:48) = ΨΦ( ζ ) = Ψ( s ), we have s ( ω ) flat , ( N ) − ζ (cid:48) ( N ) ∈ L , which implies that, for any 1 ≤ i ≤ n , we have ∇ Get ∂∂ti ζ (cid:48) ( N ) ∈ L . And it is obviousthat we have ∇ Get ∂∂ti ζ ( N ) ∈ L . Thus we deduce that ∇ Get ∂∂ti (cid:0) ζ (cid:48) ( N ) − ζ ( N ) (cid:1) ∈ L = L , (cid:0) ζ (cid:48) ( N ) − ζ ( N ) (cid:1) | t =0 = 0 . This implies that ζ (cid:48) ( N ) − ζ ( N ) ∈ L = L . But clearly ζ (cid:48) ( N ) − ζ ( N ) ∈ HC −• ( C ) ( N ) since they both are elements of HC −• ( C ) ( N ) , hence we conclude that ζ (cid:48) ( N ) − ζ ( N ) = 0.This finishes the proof of the theorem. (cid:3) Remark 4.3.
The theorem holds for any primitive polarized VSHS’s with the sameexact proof. In particular, it works for any saturated cyclic A ∞ -category such thatthe Hodge-to-de-Rham degeneration property holds. Note that this extra property isto ensure the unobstructedness of deformation theory, which automatically holds inthe semi-simple case by Corollary 2.5. Primitive forms and Frobenius manifolds.
Assume we are given a primi-tive form ζ ∈ HC −• ( C ) of the VSHS (cid:0) HC −• ( C ) , ∇ , (cid:104)− , −(cid:105) hres (cid:1) defined above. We maydefine a formal Frobenius manifold structure on the formal moduli space Spec ( R )parameterizing formal deformations of the A ∞ -category C . We briefly recall thisconstruction here, following the work of Saito–Takahashi [30]. One first defines thefollowing structures: Metric: The R -bilinear form g : Der ( R ) ⊗ Der ( R ) → R as g ( v , v ) = (cid:104) u ∇ Get v ζ, u ∇ Get v ζ (cid:105) hres ∈ R. • Product: On
Der ( R ) one defines v ◦ v to be the unique tangent vectorsuch that u ∇ Get v u ∇ Get v ζ = u ∇ Get v ◦ v ζ + u · HC −• ( C ) . • Unit vector field: Take e = ( ρ ζ ) − ([ ζ ]). • Euler vector field: Eu ∈ Der ( R ) as before.It is proved in [30, Section 7] that the data M ζ := ( Spec ( R ) , g, ◦ , e , Eu ) definesa formal Frobenius manifold, that is, g is a flat metric, ◦ is associative, e is g -flatand a unit for the product and finally there is a potential, meaning a function F on Spec ( R ) satisfying g (cid:18) ∂∂τ i ◦ ∂∂τ j , ∂∂τ k (cid:19) = ∂ F ∂τ i ∂τ j ∂τ k , where ( τ , . . . τ m ) are a system of flat coordinates. Moreover this Frobenius manifoldis conformal: L Eu ( ◦ ) = ◦ , L Eu ( g ) = (2 − d ) g, L Eu ( e ) = − e where L is the Lie derivative and d is called the dimension of the Frobenius manifold.It follows from Equation (20) below that d = − r .By our results in the previous sections, a primitive form ζ µ is determined bydata: the category C and the grading µ . We now describe some features of theFrobenius manifold M ζ µ in terms of this data. (1) The product structure ◦ on Der ( R ) makes (minus) the Kodaira-Spencer map − KS : ( Der ( R ) , ◦ ) → (cid:0) HH • ( C ) , ∪ (cid:1) a ring isomorphism. To see this first note that the order (in u ) zero of the operator u ∇ Get v ( − ) equals − KS ( v ) ∩ ( − ). Therefore given v , v ∈ Der ( R ), their product v ◦ v satisfies the identity( − KS ( v ◦ v )) ∩ ζ | u =0 = KS ( v ) ∩ (cid:0) KS ( v ) ∩ ζ (cid:1) | u =0 = (cid:0) − KS ( v ) ∪ − KS ( v ) (cid:1) ∩ ζ | u =0 By the primitivity of ζ , we conclude that − KS ( v ◦ v ) = − KS ( v ) ∪ − KS ( v ).Similarly we see that − KS ( e ) = . Hence − KS is a ring isomorphism.Furthermore, the map − KS | t =0 also intertwines the metric g with the pairing D ∗ (cid:104)− , −(cid:105) Muk , the pull-back of the Mukai-pairing under the duality map defined byEquation 3. To see this, we observe that g ( v , v ) | t =0 = (cid:104) u ∇ Get v ζ, u ∇ Get v ζ (cid:105) hres = (cid:104)− KS ( v ) | t =0 ∩ ζ | u =0 ,t =0 , − KS ( v ) | t =0 ∩ ζ | u =0 ,t =0 (cid:105) Muk = (cid:104) KS ( v ) | t =0 ∩ D ( ) , KS ( v ) | t =0 ∩ D ( ) (cid:105) Muk = (cid:104) D ( KS ( v ) | t =0 ) , D ( KS ( v ) | t =0 ) (cid:105) = D ∗ (cid:104) KS ( v ) | t =0 , KS ( v ) | t =0 (cid:105) Muk
Thus − KS | t =0 is an isomorphism of Frobenius algebras. Following [30, Proposition 7.3, 7.4], one can define the grading operator N : Der ( R ) → Der ( R ) by(18) (cid:0) ∇ u ∂∂u + ∇ GetEu (cid:1) ( u · ∇ Get v ζ ) = u · ∇ Get N ( v ) ζ + O ( u ) . It is proved in [30, Section 7] that the operator N is flat with respect to the metric.Thus, we can write in flat coordinates τ , · · · , τ m on Spec ( R ), N ( ∂∂τ i ) = (cid:88) j N ji ∂∂τ j . Let us trace through the two bijections exhibited in Theorem 3.10 and Theorem 4.2.First, the image of the splitting s at central fiber corresponding to the primitiveform ζ is spanned by the following vectors in HC −• ( C ):(19) s j := ( u · ∇ Get ∂∂τj ζ ) | τ =0 , j = 1 , · · · , m. Now we have Eu | τ =0 = KS − ([ (cid:81) k − k m k ]) = KS − ([ m ]) = (cid:80) i KS − ( λ i i ), byLemma 3.1. Then restricting Equation (18) to the central fiber yields (cid:0) ∇ u ∂∂u − u − · ξ (cid:1) s j = N ( s j ) . Thus, by the bijection of Theorem 3.10, the matrix N is precisely the grad-ing operator µ corresponding to the splitting s . Hence N ij = µ ij , on the basis s | u =0 , . . . , s m | u =0 . (3) We have the following formula of the Euler vector field written in flat coordi-nates:(20) Eu = ξ − (cid:88) j µ ji τ j ∂∂τ i + (1 + r ) · (cid:88) i τ i ∂∂τ i . This formula can be proved by applying u · ∇ Get ∂∂τj to the identity (cid:0) ∇ u ∂∂u + ∇ GetEu (cid:1) ζ = r · ζ. Then use the following two commutator identities:[ u ∇ Get ∂∂τj , ∇ u ∂∂u ] = − u ∇ Get ∂∂τj [ u ∇ Get ∂∂τj , ∇ GetEu ] = u ∇ Get [ ∂∂τj , Eu ] . We obtain that [ ∂∂τ j , Eu ] = − N ( ∂∂τ j ) + (1 + r ) · ∂∂τ j . Using the above equation together with the initial condition Eu | τ =0 = ξ and theidentity N ij = µ ij (proved above) gives the desired formula.We summarize the observations above in the following proposition. Proposition 4.4.
Let C be an A ∞ -category as in ( †† ) and µ be a grading of weight r in HH • ( C ) . Denote by ζ µ the primitive from determined by the bijections inTheorems 3.10 and 4.2 and let M ζ µ be the associated Frobenius manifold. We havethe following (1) The Frobenius algebra ( M ζ µ ) | t =0 is isomorphic to the Frobenius algebra (cid:0) HH • ( C ) , ∪ , D ∗ (cid:104)− , −(cid:105) Muk (cid:1) . In flat coordinates, the Euler vector field has the expression Eu = ξ − (cid:88) j µ ji τ j ∂∂τ i + (1 + r ) (cid:88) i τ i ∂∂τ i . Remark 4.5.
If we scale the cyclic pairing of C by a constant scalar, i.e. (cid:104)− , −(cid:105) (cid:55)→ c · (cid:104)− , −(cid:105) , then the element ω = D ( ) also scales by c , and similarly the primitiveform ζ µ . The pairing D ∗ (cid:104)− , −(cid:105) Muk of the Frobenius algebra scales by c . In the nextsubsection, we use this extra freedom to fix some constant ambiguity when matchingthe categorical Mukai pairing with the Poincar´e pairing. The class ω is usuallyreferred to as a Calabi-Yau structure of C . The construction of a (versal) VSHSassociated with C is independent the choice of a particular Calabi-Yau structure, andhence is intrinsic to the category C . The primitive form ζ (and hence the Frobeniusmanifold M ζ ), by Definition 3.9, depends on the Calabi-Yau structure. The importance of this proposition lies on the fact, proved by Dubrovin [9] andTeleman [36], that the Frobenius algebra at the central fiber and the Euler vectorfield determine uniquely the Frobenius manifold.We end this section by describing the Frobenius manifold M ζ µ when we considerthe canonical splitting s C which corresponds to the grading operator µ = 0 of weight r = 0. By the above formula, in flat coordinates, the Euler vector field is given by Eu = ξ + (cid:88) i τ i ∂∂τ i . We can further perform a shift of the flat coordinates τ i (cid:55)→ τ (cid:48) i to absorb the extraconstant vector field ξ . Then the Euler vector field takes the form Eu = (cid:80) i τ (cid:48) i ∂∂τ (cid:48) i . Itwas shown in [30] that the potential function F of this Frobenius manifold satisfies Eu ( F ) = (3 + 2 r ) F = 3 F , which implies that the potential is a cubic polynomial in flat coordinates τ (cid:48) i . There-fore the Frobenius manifold M ζ is constant and (point-wise) isomorphic to theFrobenius algebra (cid:0) HH • ( C ) , ∪ , D ∗ (cid:104)− , −(cid:105) Muk (cid:1) .5.
Applications to Fukaya categories
Fukaya category of projective spaces.
Here we will illustrate the resultsof the previous sections in the case of
Fuk ( CP n ) the Fukaya category of CP n . Westart with a description of this category (denoted by C ) which is essentially due toCho [8]. Here we follow the presentation in Fukaya–Oh–Ohta–Ono [14]. Strictlyspeaking, the category below is just a subcategory of Fuk ( CP n ). But upcomingwork by Abouzaid–Fukaya–Oh–Ohta–Ono [2] will extend Abouzaid’s generationcriterion [1] to the compact case and show that C generates the full Fukaya categorymeaning tw π C is quasi-equivalent to tw π Fuk ( CP n ), where tw π denotes the split-closed triangulated envelope of the category. If one restricts to monotone symplecticmanifolds (and Lagrangians), this generation result was already established in [27]for most such manifolds, including CP n .The category C has n +1 orthogonal objects. The endomorphism A ∞ -algebra A k of each of these objects is quasi-isomorphic to a Clifford algebra. More concretely, k has a unit k , odd generators e ( k )1 , . . . e ( k ) n and the following operations: m = λ k k := ( n + 1) T n +1 (cid:15) k k , m ( e ( k ) i , e ( k ) j ) + m ( e ( k ) j , e ( k ) i ) = h ( k ) ij k , (21)where (cid:15) = e πin +1 and h ( k ) ij = (1 + δ ij ) T n +1 (cid:15) k . The algebra A k is then gener-ated (using m ) by the e ( k ) i with only the relations above. All the other A ∞ operations, m and m k , k ≥ A k has rank 2 n ,in fact, there is a convenient basis for this vector space given by: γ ( k ) i ,...,i m := m ( m ( . . . m ( m ( e ( k ) i , e ( k ) i ) , e ( k ) i ) . . . ) , e ( k ) i m ), for any sequence 1 ≤ i < . . . i m ≤ n .The cyclic structure is determined by the relations (cid:104) k , γ ( k ) i ,...,i m (cid:105) = 0 for all i , . . . , i m , except (cid:104) k , γ ( k )1 ,...,n (cid:105) = ( √− n ( n +1)2 ( − n ( n +1)2 . Remark 5.1.
We make this slightly odd choice of cyclic pairing (which differs from [14] ) in order to make the pairing D ∗ (cid:104)− , −(cid:105) Muk match with the Poincar´e pairingunder the closed-open map - see Corollary 5.10.
One clarification is in order: the A ∞ -algebra described in [14, Section 3.6] hasadditional A ∞ operations m k , k ≥
3. However, since H := (cid:110) h ( k ) ij (cid:111) ni,j =1 is a non-degenerate matrix, this algebra is intrinsically formal, as explained in [31, Section6.1], therefore it is isomorphic to the one described above. Remark 5.2.
This A ∞ category is quasi-equivalent to the category of matrix fac-torizations of the Laurent polynomial W = y + . . . y n + Ty ··· y n . Specifically, each A k is the endomorphism algebra of the structure sheaf of one of the ( n + 1 ) sin-gular points p k of W . Moreover , we can label these such that λ k = W ( p k ) and h ( k ) i,j = y i ∂∂y i y j ∂∂y j W | p k , hence h ( k ) i,j is the Hessian matrix of W at the correspondingcritical point. Lemma 5.3.
The Hochschild homology (and therefore the cohomology) of A k is onedimensional. The length zero chain determined by the element γ ( k )1 ,...,n is a generatorof HH • ( A k ) . Moreover (cid:104) γ ( k )1 ,...,n , γ ( k )1 ,...,n (cid:105) Muk = ( − n ( n +1)2 det( H ) = ( − n ( n +1)2 ( n + 1) T nn +1 (cid:15) − k and D ( k ) = ( √− n ( n +1)2 det( H ) γ ( k )1 ,...,n .Proof. The fact about the dimension of HH • ( A k ) is well known, see for example[14, Lemma 3.8.5]. We will compute the Mukai pairing (following [14]), which inparticular implies γ ( k )1 ,...,n is non-zero in HH • ( A k ) and therefore a generator.For convenience, we will use the product a · b := ( − | a | m ( a, b ). It followsfrom the A ∞ relations that this is associative. From Equation (2), we have that (cid:104) γ ( k )1 ,...,n , γ ( k )1 ,...,n (cid:105) Muk = tr ( G ), where G ( c ) = ( − | c | ( n +1) e · · · e n · c · e · · · e n . Let M be an orthogonal matrix that diagonalizes H , that is M t HM equals thediagonal matrix diag ( d , . . . , d n ). We can define a new Clifford algebra, denotedby CL , with generators X i and relations as in (21), with the matrix H replaced by iag ( d , . . . d n ). Then define e (cid:48) i := (cid:80) j M ji e j and construct an algebra isomorphismΦ : CL → A k by setting Φ( X i ) = e (cid:48) i . Hence we have tr ( G ) = tr (Φ − G Φ) = tr (cid:16) c → ( − | c | ( n +1) X · · · X n · c · X · · · X n (cid:17) . Using the defining relations in CL , X i · X i = − d i and X i · X j = − X j · X i for i (cid:54) = j ,it is easy to see that (Φ − G Φ) = ( − n ( n +1)2 d ...d n n Id . So we conclude that tr ( G ) =( − n ( n +1)2 det( diag ( d , . . . d n )) = ( − n ( n +1)2 det( H ). The fact that det( H ) = ( n +1) T nn +1 (cid:15) − k follows from an elementary computation.For the last statement, note that we must have D ( k ) = αγ ( k )1 ,...,n for some α . Bydefinition of D we must have α ( − n ( n +1)2 det( H ) = (cid:104) D ( k ) , γ ( k )1 ,...,n (cid:105) Muk = (cid:104) k , γ ( k )1 ,...,n (cid:105) = ( √− n ( n +1)2 ( − n ( n +1)2 , which immediately implies the statement. (cid:3) Then we can take as a basis for HH • ( C ): θ k := D ( k ), for k = 0 , . . . n . Inthis basis (cid:104) θ k , θ l (cid:105) Muk = δ kl n +1 T − nn +1 (cid:15) k =: δ kl g k . Denote by Z be the diagonalmatrix diag (1 , (cid:15), . . . , (cid:15) n ). Then a grading can be described in this basis as a matrix µ satisfying µ t Z + Zµ = 0(22) µ (1 , . . . ,
1) = r (1 , . . . , , for some r ∈ K . The first condition expresses the anti-symmetry of µ and thesecond is (c) in Definition 3.9. Since the m coefficients are all distinct, Definition3.9(b) follows from (a). Example 5.4.
When n = 1 , we have θ k = ( − k T / e ( k )1 . The semi-simple split-ting must satisfy ∇ u ddu s ( θ k ) = ( − k T / u − s ( θ k ) . It can be easily computed: s C ( θ ) = ∞ (cid:88) n =0 ( − n T − n +12 (2 n − n +1 e (0)1 | ( e (0)1 ) n u n s C ( θ ) = − ∞ (cid:88) n =0 T − n +12 (2 n − n +1 e (1)1 | ( e (1)1 ) n u n Since the homology is rank two, the space of anti-symmetric operators is one dimen-sional. In this case, Definition 3.9(c) doesn’t impose any extra conditions, thereforethe space of gradings is one dimensional, namely is given by matrices of the form µ = (cid:18) rr (cid:19) We can then solve Equation (15) for such µ to find the R -matrix: R n ( r ) = T − n − n r ( n − n − (cid:89) k =1 ( r − k ) (cid:18) rn − n ( − n rn (cid:19) When we take r = − / (in order to get a Frobenius manifold of dimension )we get n = T − n − n n − n − (cid:89) k =1 (4 k − (cid:18) − ( − n n ( − n − n (cid:19) This computation of the R -matrix agrees (up to a sign) with the one in [22,Appendix A] . Example 5.5.
When n = 2 , solving Equation (22) we obtain µ = − (cid:15)r − (cid:15) x (cid:15) x − (cid:15) rr + (cid:15)x − (cid:15)xr − x x where (cid:15) = e πi/ and r, x ∈ K . The grading relevant for Gromov–Witten theory, aswe will see below, is the one given by r = − and x = 1 (cid:15) − . In the next subsection we will see a systematic way to choose the grading relevantfor Gromov–Witten theory. But first we explain an ad hoc method for the case of
Fuk ( CP n ). Let E = (cid:80) nk =0 ( n + 1) T n +1 (cid:15) k k ∈ HH • ( C ) be the Euler vector fieldat the origin, we can easily see that it is a generator of the ring HH • ( C ). This isanalogous to the fact that the first Chern class c ∈ QH • ( CP n ) - which is the Eulervector field at the origin of the Gromov–Witten Frobenius manifold - generates thequantum cohomology ring. Construction 5.6.
As before, let ω = D ( ) = θ + . . . + θ n and E = (cid:80) nk =0 ( n +1) T n +1 (cid:15) k k . Then a simple Vandermonde determinant computation shows that E ∪ m ∩ ω for m = 0 , . . . n forms a basis of HH • ( C ) . We define µ by setting µ ( E ∪ m ∩ ω ) = ( m − n E ∪ m ∩ ω. A straightforward computation shows that this satisfies the first condition in(22) and it obviously satisfies the second with r = − n/
2. The justification for thisconstruction is the following. When taking powers c (cid:63)k in the quantum cohomologyring, for k ≤ n , the quantum product agrees with the classical cup product, there-fore c (cid:63)k is homogeneous of degree 2 k . Finally, in QH • ( CP n ) the grading is definedas µ ( x ) = ( deg ( x ) − n ) x/ Remark 5.7.
The above construction is possible because the quantum cohomologyhas one generator and the minimal Chern number of CP n is very high comparedto the dimension. Therefore we don’t expect this method to be applicable to manyexamples besides CP n . One case however, where we do expect this method to workis the case of a sphere with two orbifold points, whose Gromov–Witten invariantswhere studied in [24] . We can construct the versal deformation C by considering the R -linear category C ⊗ K R , where R = K [[ t , . . . t n , ]], with operations m = (cid:16) ( n + 1) T n +1 (cid:15) k − t k (cid:17) k , (23) m ( e ( k ) i , e ( k ) j ) + m ( e ( k ) j , e ( k ) i ) = h ( k ) ij k . These are called canonical coordinates , because we have ∂∂t i ◦ ∂∂t j = δ ij ∂∂t i . his follows from the fact that − KS is a ring map and − KS (cid:16) ∂∂t i (cid:17) = i .5.2. Closed-open map.
We want to apply our results to the Fukaya category ofa symplectic manifold M , in order to extract the Gromov–Witten invariants of M from the category Fuk ( M ). Our results show that in the semi-simple case, it isenough to pick the correct grading on HH • ( Fuk ( M )). For this purpose we need anextra piece of geometric data, the closed-open map(24) C O : QH • ( M ) → HH • ( Fuk ( M )) . This map, which has been constructed in many cases, is a geometrically definedmap that is expected to be a ring isomorphism for a very wide class of symplecticmanifolds.
Assumption 5.8.
Let M n be a symplectic manifold and let Fuk ( M ) be its Fukayacategory (including only compact and orientable Lagrangians). We assume that Fuk ( M ) is a saturated, unital and Calabi–Yau A ∞ -category. Moreover we assumethat the closed-open map C O : QH • ( M ) → HH • ( Fuk ( M ))(1) is a ring isomorphism; (2) intertwines the Poincar´e pairing (cid:104)− , −(cid:105) PD with D ∗ (cid:104)− , −(cid:105) Muk ; (3) sends the first Chern class c ( M ) to (cid:104)(cid:81) k ≥ − k m k (cid:105) . How reasonable are these assumptions? There is a lot of evidence that theseassumptions will be satisfied for a very wide class of symplectic manifolds. TheFukaya category is proper and as explained in [15] it is smooth whenever it satisfiesAbouzaid’s generation criterion [1]. The endomorphism A ∞ algebra of each objectin the Fukaya category is know to admit a cyclic and (hence Calabi–Yau) structure[11]. In the monotone case it is proved in [31] that Fuk ( M ) admits weak-Calabi–Yau structures. The map C O is constructed in [12] for general M but taking valueson Hochschild cohomology of the endomorphism algebra of a single object in theFukaya category. But the construction should generalize to the full Fukaya categorywithout difficulty. The fact that it is a ring map is proved in [14] for the toric case,but should hold in general. Both the construction and the ring property have beencarried out in the monotone case [31]. The second condition is essentially provedin [14] for toric manifolds (more on this below). The third condition follows from[14] in the toric case and is partially established in [31] in the monotone case. Theorem 5.9.
Let M n be a symplectic manifold that satisfies Assumption 5.8and such that HH • ( Fuk ( M )) is semi-simple. Define µ C O to be the operator on HH • ( Fuk ( M )) which is the pull-back of µ ( x ) = deg ( x ) − n · x under the isomorphism D ◦ C O . Then µ C O is a grading (according to Definition 3.9) and the Frobeniusmanifold M µ C O is isomorphic to the big quantum cohomology of M .Proof. By definition of Poincar´e pairing µ is anti-symmetric and by assumption D ◦C O matches the Poincar´e and Mukai pairings, therefore µ C O = ( D ◦C O ) ◦ µ ◦ ( D ◦C O ) − is anti-symmetric with respect to the Mukai pairing. The third condition onDefinition 3.9 for µ C O follows from D ◦ C O ( ) = ω and µ ( ) = − n . For Definition3.9(b), first note that by assumption C O ( c ( M )) = [ m ] which together with thefact that C O is a ring map and D is a module map gives c ( M ) (cid:63) ( − ) = ( D ◦ C O ) − ◦ ξ ◦ ( D ◦ C O ) . eleman proves in [36, Section 8] that µ satisfies the analogous of condition Defi-nition 3.9(b) with respect to c ( M ) (cid:63) ( − ), therefore part (b) of Definition 3.9 holdsfor µ C O .For the second part of the statement first note that ( C O ) − ◦ ( − KS | t =0 ) definesa Frobenius algebra isomorphism between T t =0 M µ C O and QH • ( M ). Secondly, weargue that ( C O ) − ◦ ( − KS | t =0 ) identifies the Euler vector field in M µ C O with theEuler vector field of QH • ( M ) given in [36, (8.11)]. This follows from the fact thatthe formula (20) for the Euler vector field of M µ C O , in terms of ξ and µ C O , matchesthe one in [36]. This happens since ( C O ) − ◦ ( − KS | t =0 ) intertwines µ with µ C O on T t =0 M µ C O when identified with HH • ( Fuk ( M )) by (19).Therefore by the reconstruction result of Dubrovin [9] and Teleman [36] theFrobenius manifold M µ C O is isomorphic to the big quantum cohomology of M . (cid:3) We will now use the results of Fukaya–Oh–Ohta–Ono to apply the previoustheorem to the case of toric manifolds. Let M be a compact toric manifold and let PO M be its potential function (with bulk b = 0), defined in [14]. This is a Laurentseries, which in the Fano case agrees with the Hori–Vafa potential. As in the caseof CP n we will assume the upcoming generation result of Abouzaid–Fukaya–Oh–Ohta–Ono [2], stating that the subcategory of Fuk ( M ) considered in [14] generates Fuk ( M ). Corollary 5.10.
Let M be a compact toric manifold and let PO M be its potentialfunction. Assume that PO M is a Morse function. Then the category Fuk ( M ) andthe map C O determine a Frobenius manifold M µ C O , which is isomorphic to thebig quantum cohomology of M . In particular, all the genus zero Gromov–Witteninvariants of M are determined by the Fukaya category Fuk ( M ) and the closed-openmap C O .Proof. For each critical point p k , k = 1 , . . . m , of PO M , there is an object in theFukaya category (a torus fiber equipped with a specific bounding cochain) whoseendomorphism algebra is a cyclic A ∞ algebra, which we denote by A k . As a vectorspace, A k is simply the cohomology of an n -dimensional torus which we equip withthe cyclic pairing (cid:104) α, β (cid:105) := ( √− n ( n +1)2 ( − | α || β | (cid:48) (cid:104) α, β (cid:105) T n , where (cid:104)− , −(cid:105) T n is the Poincar´e duality pairing in H • ( T n ). Moreover, if p k is a non-degenerate critical point, A k is quasi-isomorphic to a Clifford algebra as defined in(21) with λ k the critical value PO M ( p k ) and h ( k ) ij the entries of the Hessian at p k as explained in Remark 5.2. These objects are orthogonal and therefore, assumingthe generation result in [2], we can take Fuk ( M ) = ⊕ k A k .Let Jac( PO M ) be the Jacobian ring of PO M , as defined in [14, Section 2.1].Theorem 1.1.1 in [14] defines a ring isomorphism ks : QH • ( M ) → Jac( PO M )(for arbitrary M ). As explained in [14, Section 4.7], composing ks with an iso-morphism from Jac( PO M ) to HH • ( Fuk ( M )) we obtain the closed-open map C O .Then the map C O sends the idempotent e k in QH • ( M ) corresponding to p k (un-der the identification with Jac( PO M )) to k ∈ HH • ( A k ). Since QH • ( M ) isa semi-simple Frobenius algebra and C O is a ring isomorphism it is enough tocheck (cid:104) e k , e k (cid:105) PD = D ∗ (cid:104) k , k (cid:105) Muk to guarantee that Assumption 5.8(2) holds.Let ν k ∈ H • ( T n ) be the Poincar´e dual to the unit k , that is (cid:104) k , ν k (cid:105) T n =1.Then ν k is a generator of the Hochschild homology of A k and we have D ( k ) = √− n ( n +1)2 (cid:104) ν k , ν k (cid:105) − Muk ν k . Hence D ∗ (cid:104) k , k (cid:105) Muk = ( − n ( n +1)2 (cid:104) ν k , ν k (cid:105) − Muk . An ele-mentary computation shows (cid:104) ν k , ν k (cid:105) Muk = ( − n ( n +1)2 Z ( A k ), where Z is the invari-ant defined in [14, (1.3.22)]. Now our claim follows from the fact, proved in [14,Proposition 3.5.2], that (cid:104) e k , e k (cid:105) PD = Z ( A k ) − .Finally, Proposition 2.12.1 in [14], gives ks ( c ( M )) = PO M , hence under theabove identification with HH • ( Fuk ( M )) we get C O ( c ( M )) = (cid:80) k PO M ( p k ) k =[ m ], as required in Assumption 5.8(3). Hence HH • ( Fuk ( M )) and the map C O satisfy all the conditions of Theorem 5.9 which then proves the result. (cid:3) Remark 5.11.
In the case that M is nef, it is proved in [14] that the invariant Z ( A k ) equals the Hessian of PO M at the critical point p k . In the case of CP n wesaw this in Lemma 5.3. Remark 5.12.
As mentioned in the Introduction, when M is Fano and we choosea generic symplectic form, the potential PO M is Morse. In the general case, ifwe consider Fuk b ( M ) the bulk-deformed version of the Fukaya category [14] andpick a “generic” bulk parameter b we expect the corresponding potential PO M, b to be Morse. Therefore Theorem 5.9 would be applicable to this version of theFukaya category. We do not explore this here since the analogue of condition (3)in Assumption 5.8 has not been established in this setting. Applications to mirror symmetry.
Let (
X, W ) be a Landau-Ginzburgmodel, with W ∈ Γ( X, O X ) a non-constant function. Consider the derived categoryof singularities of W , denoted by (cid:89) λ MF ( W − λ )where the coproduct is taken over distinct critical values λ of W . We assume that W has finitely many isolated critical points p , . . . , p k ∈ X . The Jacobian ring of W thus also decomposes as Jac ( W ) ∼ = (cid:89) ≤ i ≤ k Jac ( W p i ) . Here W p i ∈ O X,p i is the localization of W at the point p i . Since the categories MF ( W − λ ) are naturally defined over Jac ( W ), they also naturally decompose usingthe idempotents of each ring Jac ( W p i ): (cid:89) λ MF ( W − λ ) ∼ = (cid:89) ≤ i ≤ k MF ( W p i ) . Taking the corresponding VSHS’s gives an isomorphism (cid:89) λ V MF ( W − λ ) ∼ = (cid:89) ≤ i ≤ k V MF ( W pi ) . The product of VSHS’s is defined as follows. We take the left hand side prod-uct of the above equation as an example. Assume that for each λ , the VSHS V MF ( W − λ ) is defined over a formal deformation space M λ . Then the product VSHS (cid:81) λ V MF ( W − λ ) is defined over (cid:81) λ M λ , the underlying locally free O (cid:81) λ M λ [[ u ]]-module is given by (cid:89) λ π ∗ λ (cid:0) V MF ( W pi ) (cid:1) , ith π λ the canonical projection map onto M λ . The connection operators and thepairing on the product are also defined through the pull-back of π λ ’s.In the following, we prove a direct corollary of the discussion of the previoussubsection that in the semi-simple case, homological mirror symmetry implies enu-merative mirror symmetry. Corollary 5.13.
Let M be a symplectic manifold. Assume that we are given an A ∞ quasi-equivalence F : D b ( Fuk ( M )) → (cid:81) λ MF ( W − λ ) with ( X, W ) a Landau-Ginzburg mirror of M . Then we have (1) F induces an isomorphism of VSHS’s V Fuk ( M ) ∼ = V W where V W stands forthe VSHS constructed by Saito [29][28] . (2) Assume furthermore that
Fuk ( M ) has semi-simple Hochschild cohomology.Then there exists a Saito’s primitive form ζ ∈ V W such that its associatedFrobenius manifold M ζ is isomorphic to the big quantum cohomology of M .Proof. For each p i , there exists an isomorphism of VSHS’s proved in [37]: V MF ( W pi ) ∼ = V W pi . Furthermore, Saito’s VSHS V W also naturally decomposes according to its local-izations at each critical point p i ’s, i.e. we have V W ∼ = (cid:89) ≤ i ≤ k V W pi . Putting all together, we obtain a chain of isomorphism of VSHS’s: V Fuk ( M ) ∼ = (cid:89) λ V MF ( W − λ ) ∼ = (cid:89) ≤ i ≤ k V MF ( W pi ) ∼ = (cid:89) ≤ i ≤ k V W pi ∼ = V W . This proves part (1).To prove part (2), we take the grading operator µ C O on HH • (cid:0) Fuk ( M ) (cid:1) to obtaina primitive form ζ C O of V Fuk ( M ) . In the previous subsection, we have proved thatthe Frobenius manifold M ζ C O is isomorphic to the big quantum cohomology of M .To prove part (2), we may take Saito’s primitive form ζ ∈ V W obtained by pushingforward of ζ C O under the above isomorphism of VSHS’s. (cid:3) Example 5.14.
We compute the primitive form for the mirror of CP . If wenormalize the symplectic form ω so that (cid:82) CP ω = 1 , then its mirror is given by X = Spec K [ x, x − ] and W = x + Tx . One can check that taking the volume form ω = dxx to pull-back the residue pairing on Jac ( W ) · dx to Jac ( W ) gives an isomorphismof Frobenius algebras QH • ( CP ) ∼ = Jac ( W ) which sends (cid:55)→ and the symplecticform ω (cid:55)→ x . Thus the grading operator on Hochschild homology acts by µ (cid:18) dxx (cid:19) = − dxx , µ ( dx ) = 12 dx. Denote by s := dxx and s := dx two cohomology classes in the twisted de Rhamcomplex (cid:0) Ω ∗ X [[ u ]] , dW + ud DR (cid:1) . Saito’s u -connection acts by ∇ ddu = ddu − u − Wu ne verifies that we have ∇ u ddu s = − s − u − (2 s ) ∇ u ddu s = 12 s − u − (2 T s ) By Theorem 3.10, the regular part of the above is given by the grading operator µ ,thus the basis { s , s } is the splitting of the Hodge filtration uniquely determined by µ . Using Theorem 4.2, we may compute the primitive form ζ determined by thisbasis using an algorithm of Li-Li-Saito [23] . The result gives that ζ is simply theconstant form dxx , independent of the deformation parameters and the u parameter. Appendix A. Hochschild invariants of Calabi–Yau A ∞ -algebras In this appendix we prove Theorem 2.4. We start with the following proposition.
Proposition A.1.
Given a Hochschild cochain class ϕ ∈ HH • ( A ) and chains α, β ∈ HH • ( A ) we have (cid:104) ϕ ∩ α, β (cid:105) Muk = ( − | ϕ || α | (cid:104) α, ϕ ∩ β (cid:105) Muk
In other words, capping with a fixed class is a self-adjoint map.Proof.
The proof is identical to that of Lemma 5.39 in [32]. Given a closed cochain ϕ we define the maps H , H , H : CC • ( A ) ⊗ → CC • ( A ), as follows, for α = α | α | . . . | α r and β = β | β | . . . | β s , we set H ( α, β )= (cid:88) tr (cid:2) c → ( − † m ∗ ( α j , ..ϕ ∗ ( α k , .. ) , .., α , .., m ∗ ( α i , .., c, β n , .., β , .. ) , .. ) (cid:3) where † = | c || β | + | ϕ | (cid:48) + | ϕ | ( | α j | (cid:48) + .. + | α k − | (cid:48) )+ | α | (cid:48) + .. + | α i − | (cid:48) + | α k | (cid:48) + .. + | α r | (cid:48) +@. H ( α, β )= (cid:88) tr (cid:2) c → ( − † m ∗ ( α j , .., α , .., m ∗ ( α i , .., c, β n , .., ϕ ∗ ( β p , .. ) , .., β , .. ) , .. ) (cid:3) where † = | c | ( | β | + | ϕ | (cid:48) ) + | ϕ || α | + | ϕ | (cid:48) ( | β n | (cid:48) + .. + | β p − | (cid:48) ) + | α i | (cid:48) + .. + | α j − | (cid:48) + @ . H ( α, β )= (cid:88) tr (cid:2) c → ( − † ϕ ∗ ( α j , .., m ∗ ( α k , .., α , .., m ∗ ( α i , .., c, .., β , .. ) , .. ) , β n , .. ) (cid:3) where † = 1 + | c || β | + | α | (cid:48) + .. + | α i − | (cid:48) + | α k | (cid:48) + .. + | α r | (cid:48) + @.Finally we define H := H + H + H . Then the result follows from the followingstatement: for any chains α and β , (cid:104) ϕ ∩ α, β (cid:105) Muk − ( − | ϕ || α | (cid:104) α, ϕ ∩ β (cid:105) Muk + H ( b ( α ) ⊗ β + ( − | α | α ⊗ b ( β )) = 0This is a direct, albeit long, computation (that we omit) using only the A ∞ rela-tions, the closedness of ϕ and the fact that tr ( A ◦ B ) = tr ( B ◦ A ). (cid:3) Proposition A.2.
Given a Hochschild cochain classes ϕ, ψ ∈ HH • ( A ) we have D ( ϕ ∪ ψ ) = ( − | ϕ | d ϕ ∩ D ( ψ ) . In other words, D is a map of HH • ( A ) -modules (of degree d ). roof. Since the Mukai pairing is non-degenerate it is enough to check(25) (cid:104) D ( ϕ ∪ ψ ) , α (cid:105) Muk = (cid:104) ( − | ϕ | d ϕ ∩ D ( ψ ) , α (cid:105) Muk , for all α ∈ HH • ( A ). By definition of D , we have (cid:104) D ( ϕ ) , α (cid:105) Muk = ( − | α | (cid:48) | α ,n | (cid:48) (cid:104) ϕ ( α , . . . , α n ) , α (cid:105) , where | α ,n | (cid:48) := | α | (cid:48) + . . . + | α n | (cid:48) . Therefore, using commutativity of ∪ , the left-hand side of (25) equals( − | ϕ || ψ | + | α | (cid:48) | α ,n | (cid:48) (cid:104) ( ψ ∪ ϕ )( α , . . . , α n ) , α (cid:105) == ( − ♦ (cid:88) (cid:104) m p ( α , . . . , ψ a ( α i +1 , . . . ) , . . . , ϕ b ( α j +1 , . . . ) , . . . α n ) , α (cid:105) . (26)where ♦ = | ϕ || ψ | + | α | (cid:48) | α ,n | (cid:48) + | ψ | (cid:48) | α ,i | (cid:48) + | ϕ | (cid:48) | α ,j | (cid:48) . On the other hand, usingProposition A.1, the right hand side of (25) equals( − | ϕ | d + | ϕ | ( | ψ | + d +1) (cid:104) D ( ψ ) , ϕ ∩ α (cid:105) Muk == (cid:88) ( − δ (cid:104) D ( ψ ) , m q ( . . . ϕ b ( α j +1 . . . ) , . . . α , . . . ) α i +1 . . . α i + a (cid:105) (27) = (cid:88) ( − δ (cid:104) ψ a ( α i +1 . . . ) , m q ( . . . ϕ b ( α j +1 . . . ) , . . . α , . . . ) (cid:105) . where δ = | ϕ || ψ | + | ϕ | (cid:48) | α i + a +1 ,j | (cid:48) + | α ,i | (cid:48) | α i +1 ,n | (cid:48) and δ = | ϕ || ψ | + | ϕ | (cid:48) | α i + a +1 ,j | (cid:48) + | α ,i | (cid:48) | α i +1 ,n | (cid:48) + | α i +1 ,i + a | (cid:48) ( | ϕ | + | α ,i | (cid:48) + | α i + a +1 ,n | (cid:48) ) . Using cyclic symmetry of the pairing (cid:104)− , −(cid:105) , a straightforward computation showsthat the expressions in (26) and (27) are equal, which proves the desired result. (cid:3) We can now prove Theorem 2.4. Indeed, Part (a) of the theorem is exactlyProposition A.2 proved above. In particular, for any ϕ ∈ HH • ( A ), we have D ( ϕ ) =( − | ϕ | d ϕ ∩ D ( ) = ( − | ϕ | d ϕ ∩ ω . For part (b), it is well known that the cupproduct is associative and graded-commutative. The only condition left to check isthe compatibility between product and pairing. We use Proposition A.1 to compute D ∗ (cid:104) ϕ ∪ ψ, ρ (cid:105) Muk =( − ( | ϕ | + | ψ | ) d (cid:104) D ( ϕ ∪ ψ ) , D ( ρ ) (cid:105) Muk =( − ( | ϕ | + | ψ | ) d + | ϕ || ψ | +( | ϕ | + | ψ | + | ρ | ) d (cid:104) ( ψ ∪ ϕ ) ∩ ω, ρ ∩ ω (cid:105) Muk =( − | ϕ || ψ | + | ρ | d (cid:104) ψ ∩ ( ϕ ∩ ω ) , ρ ∩ ω (cid:105) Muk =( − | ϕ || ψ | + | ρ | d + | ψ | ( | ϕ | + d ) (cid:104) ϕ ∩ ω, ψ ∩ ( ρ ∩ ω ) (cid:105) Muk =( − ( | ψ | + | ρ | ) d (cid:104) ϕ ∩ ω, ( ψ ∪ ρ ) ∩ ω ) (cid:105) Muk =( − | ϕ | d (cid:104) D ( ϕ ) , D ( ψ ∪ ρ ) (cid:105) Muk = D ∗ (cid:104) ϕ, ψ ∪ ρ (cid:105) Muk
Appendix B. Proof of Proposition 2.6
Let H be the map introduced in the proof of Proposition A.1 and extend itsesquilinearly to CC • ( A )[[ u ]] ⊗ . We claim that for any negative cyclic chains α and β ddu (cid:104) α, β (cid:105) hres = (cid:104)∇ ddu α, β (cid:105) hres − (cid:104) α, ∇ ddu β (cid:105) hres + 12 u H (( b + uB )( α ) , β ) + ( − | α | u H ( α, ( b + uB )( β )) , (28)which immediately implies the result. ndeed, writing α = (cid:88) n ≥ α n u n , β = (cid:88) n ≥ β n u n and using the definitions of thepairing and the connection to expand the above expression we see that the left-hand side equals (cid:80) n ≥ (cid:80) n +1 k =0 ( − n − k +1 ( n + 1) (cid:104) α k , β n − k +1 (cid:105) Muk u n . The right handside equals (cid:88) n ≥ n +1 (cid:88) k =0 ( − n − k +1 ( n + 1) (cid:104) α k , β n − k +1 (cid:105) Muk u n + (cid:88) n ≥− n +2 (cid:88) k =0 ( − n − k (cid:104) b { m (cid:48) } α k , β n − k +2 (cid:105) Muk − (cid:104) α k , b { m (cid:48) } β n − k +2 (cid:105) Muk ) u n + (cid:88) n ≥− n +1 (cid:88) k =0 ( − n − k +1 (cid:16) (cid:104) (Γ + B { m (cid:48) } ) α k , β n − k +1 (cid:105) Muk + (cid:104) α k , (Γ + B { m (cid:48) } ) β n − k +1 (cid:105) Muk (cid:17) u n + (cid:88) n ≥− n +2 (cid:88) k =0 ( − n − k (cid:16) H ( b ( α k ) , β n − k +2 ) + ( − | α | H ( α k , b ( β n − k +2 )) (cid:17) u n + (cid:88) n ≥− n +1 (cid:88) k =0 ( − n − k +1 (cid:16) H ( B ( α k ) , β n − k +1 ) − ( − | α | H ( α k , B ( β n − k +1 )) (cid:17) u n Therefore the claim follows from the following two identities (cid:104) b { m (cid:48) } x, y (cid:105) Muk − (cid:104) x, b { m (cid:48) } y (cid:105) Muk + H ( b ( x ) , y ) + ( − | x | H ( x, b ( y )) = 0 , (cid:104) (Γ + B { m (cid:48) } ) x, y (cid:105) Muk + (cid:104) x, (Γ + B { m (cid:48) } ) y (cid:105) Muk ++ H ( B ( x ) , y ) − ( − | x | H ( x, B ( y )) = 0 , for arbitrary Hochschild chains x = x | x . . . x r and y = y | y . . . y s . The firstidentity is exactly the content of the proof of Proposition A.1, since | m (cid:48) | = 0. Forthe second one, we first show by direct computation that (cid:104) B { m (cid:48) } x,y (cid:105) Muk + (cid:104) x, B { m (cid:48) } y (cid:105) Muk + H ( B ( x ) , y ) − ( − | x | H ( x, B ( y )) == − (cid:88) tr (cid:2) c → ( − † m (cid:48)∗ ( x j , .., x , .., m ∗ ( x i , .., c, y n , .., y , .. ) , y m , .. ) (cid:3) − (cid:88) tr (cid:2) c → ( − † m ∗ ( x j , .., x , .., m (cid:48)∗ ( x i , .., c, y n , .., y , .. ) , y m , .. ) (cid:3) (29)where † is as in (2). Then, counting the number of inputs as in the proof of Lemma3.3, we see the right-hand side in (29) gives ( r + s ) (cid:104) x, y (cid:105) Muk . Finally we observethat (cid:104) Γ( x ) , y (cid:105) Muk + (cid:104) x, Γ( y ) (cid:105) Muk = − ( r + s ) (cid:104) x, y (cid:105) Muk which therefore cancels with (29), proving the required identity.
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Department of Mathematics, Kansas State University, 138 Cardwell Hall, 1228 N.17th Street, Manhattan, KS 66506, USA
E-mail address : [email protected] Institute of Mathematical Sciences, ShanghaiTech University, 393 Middle HuaxiaRoad, Pudong New District, Shanghai, China, 201210.
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