Categorical Enumerative Invariants, II: Givental formula
CCategorical Enumerative Invariants, II: Givental formula A NDREI
C ˘
ALD ˘ ARARU AND J UNWU T U A BSTRACT : To a pair ( A , s ) consisting of a smooth, cyclic A ∞ -algebra A and asplitting s of the Hodge filtration on its Hochschild homology Costello (2005) as-sociates an invariant which conjecturally generalizes the total descendant Gromov-Witten potential of a symplectic manifold.In this paper we give explicit, computable formulas for Costello’s invariants, asFeynman sums over partially directed stable graphs. The formulas use in a crucialway the combinatorial string vertices defined earlier by Costello and the authors.Explicit computations elsewhere confirm in many cases the equality of categoricalinvariants with known Gromov-Witten, Fan-Jarvis-Ruan-Witten, and Bershadsky-Cecotti-Ooguri-Vafa invariants. Contents
1. Introduction
Gromov-Witten theory associates to a compact symplectic 2 d -dimensional man-ifold X its total descendant potential D X [Giv01], [Coa08]. It is a formal expressionthat encodes the Gromov-Witten invariants of X at all genera and possible insertions a r X i v : . [ m a t h . S G ] S e p C˘ald˘araru and Tu of cohomology classes from X and ψ -classes from the moduli spaces of curves (de-scendants). If we denote by H = H ∗ ( X )[2 d ] the shifted cohomology of X , the dual D X of the total descendant potential lives in a certain completed symmetric algebra D X ∈ (cid:100) Sym (cid:126) (cid:0) H [ u − ] (cid:1) [[ (cid:126) , λ ]]which uses formal variables u , λ, (cid:126) of even homological degree to keep track of ψ -classinsertions, genus, and Euler characteristic, respectively. See Section 2 for a review ofthe above concepts and of the notations we use. In his visionary 1994 address to the International Congress of MathematiciansKontsevich [Kon95] predicted that Gromov-Witten theory is of categorical nature.More precisely he proposed that it should be possible to extract D X directly from the(idempotent completed) Fukaya category Fuk ( X ) of X .The category C = Fuk ( X ) is a Calabi-Yau A ∞ -category. Implicit in Kontsevich’swork was the idea that it should be possible to attach an invariant similar to D X toany category C of this type. In particular if C were the derived category D b coh ( ˇ X )of coherent sheaves on a compact Calabi-Yau manifold ˇ X the resulting genus zeroinvariants were expected to agree with the solutions of the Picard-Fuchs equationgoverning the variation of Hodge structures on ˇ X . The higher genus invariants wouldbe new, B-model analogues of Gromov-Witten invariants; their values would, in somecases, be predicted by Bershadsky-Cecotti-Ooguri-Vafa (BCOV) theory [BCOV94].Costello [Cos09] observed that the proper input for such a construction should includenot just the category C but also a choice of splitting of the non-commutative Hodgefiltration on the Hochschild homology of C . (The reason this was not initially apparentis the fact that the Fukaya category has a canonical choice of splitting; other categories,however, may not.) Costello provided a non-constructive definition of such an invariantin [Cos09]. Our goal is to extend the works of Kontsevich and Costello to make the definitionof the categorical enumerative invariants explicitly computable . Let ( A , s ) be a pairconsisting of a smooth, cyclic A ∞ -algebra A and a splitting s of the Hodge filtrationon its cyclic homology. If d is the Calabi-Yau dimension of A define H = HH ∗ ( A )[ d ] , the shifted Hochschild homology of A . The main contribution of our paper is to givea new definition of the categorical enumerative total descendant potential of the pair( A , s ), D A , s ∈ (cid:100) Sym (cid:126) (cid:0) H [ u − ] (cid:1) [[ (cid:126) , λ ]] . ategorical Enumerative Invariants, II: Givental formula The definition is explicit enough to make direct computations of categorical enumera-tive invariants possible (within limits of computational power).
Our invariant agrees with Costello’s (see Section 3) and it is directly inspired byit. The fundamental difference between the two is in the computability aspect. BothCostello’s original construction and our own rely on the existence of certain chainson the moduli spaces of curves, first introduced by Sen-Zwiebach [SZ94] and called string vertices . Costello’s original definition uses geometrically defined string vertices,which cannot be explicitly computed. We use instead a new, combinatorial model forstring vertices introduced by Costello and the authors in [CCT20]. Using them we areable to bypass certain non-constructive aspects of the construction in [Cos09]. Ourresulting formulas for D A , s are given as Feynman sums over partially directed stablegraphs (a concept we introduce), the vertices of which are labeled by combinatorialstring vertices. Both of these can be explicitly computed. A different approach to categorical enumerative invariants in genus zero wasproposed by Barannikov-Kontsevich [Bar01] using the idea of variation of semi-infinite Hodge structures (VSHS). Their work was extending pioneering work ofSaito [Sai83], [Sai83] on unfoldings of singularities, and was generalized to arbitrarycyclic A ∞ -algebras by Ganatra-Perutz-Sheridan [GPS15]. Our approach accesses allgenera, unlike the VSHS approach which can only recover the genus zero part of thetheory. We emphasize that the relationship between our theory and classical Gromov-Witten theory is so far speculative; it depends on certain standard conjectures insymplectic geometry which are beyond the scope of this paper. However, all evidencepoints to the fact that when the algebra A is Morita equivalent to the Fukaya categoryof a symplectic manifold X the resulting invariant is precisely D A , s = D X . To be more precise, we expect the Fukaya category
Fuk ( X ) to come endowed with anatural splitting s of the Hodge filtration, arising from the fact that Fuk ( X ) is definedover the Novikov ring. It is this choice of splitting s that would be used to get theidentification above. Multiple computations of D A , s confirm this [CT17], [CC20-2]. Our approach works equally well for Z - and Z / Z -graded algebras. As a con-sequence we can use an algebra A that is Morita equivalent to a category of matrixfactorizations. Direct computations [CLT19] strongly suggest that in this case theresulting categorical potential agrees with the potential of the B-model Fan-Jarvis-Ruan-Witten (FJRW) theory. C˘ald˘araru and Tu
Our results can be regarded as giving a unified definition of virtually all knownenumerative invariants in the literature, by taking the algebra A to be Morita equivalentto – the Fukaya category, for Gromov-Witten theory;– the wrapped Fukaya category, for FJRW theory;– the derived category of a Calabi-Yau manifold, for BCOV theory.– the category of matrix factorizations, for B-model FJRW theory;Moreover, all these theories can immediately be generalized to the setting wherea finite group of symmetries is present: the input category is simply replaced bya corresponding smash-product construction. In particular this allows us to defineorbifold FJRW invariants, for which no direct definition exists.From this point of view enumerative mirror symmetry follows tautologically from cat-egorical mirror symmetry. This was Kontsevich’s original explanation for the equalityof numerical invariants on the two sides of mirror symmetry observed by Candelas-dela Ossa-Green-Parkes [COGP91]. However, we cannot yet claim that categorical mir-ror symmetry implies enumerative mirror symmetry for all genera – the main difficultylies in identifying the categorical invariants with the geometric ones. Section 2 collects definitions and results from the literaturethat will be needed in the rest of the paper. This material is not new, but we review it herefor the convenience of the reader. We present certain aspects of Gromov-Witten theory,in particular the definition of the total descendant potential and Givental’s Lagrangianformalism. We also include a short summary of the construction of combinatorialstring vertices from [CCT20] as solutions of the master equation in a certain DGLA (cid:98) g . A choice of cyclic A ∞ -algebra A gives rise to a similar DGLA (cid:98) h and to a map ρ A : (cid:98) g → (cid:98) h .In Section 3 we clarify Costello’s original definition of categorical enumerative invari-ants from [Cos07], [Cos09]. For a pair ( A , s ) consisting of a cyclic A ∞ -algebra A anda splitting s of the non-commutative Hodge filtration we define invariants F A , sg , n ∈ Sym n (cid:0) H [ u − ] (cid:1) for any pair ( g , n ) such that 2 g − + n > , n >
0. The case n = B : L → L in order to obtain an explicit partial trivialization of the DGLA structure on (cid:98) h . ategorical Enumerative Invariants, II: Givental formula Trivializations of circle operators and their relationship with Givental’s formula werealready studied extensively from the operadic point of view in [Dru14, DSV13, DSV15,KMS13].In the final section we prove our main result, Corollary 5.7. It gives a Feynman sumformula for the categorical enumerative invariants F A , sg , n as a sum over partially directed stable graphs (see Definition 4.19) F A , sg , n = (cid:88) G ∈ Γ (( g , , n − wt ( G ) (cid:89) e ∈ E G Cont ( e ) (cid:89) v ∈ V G Cont ( v ) (cid:89) l ∈ L G Cont ( l ) . The vertex contributions
Cont ( v ) in this formula are given by the tensors (cid:98) β Ag , k , l = ρ A ( (cid:98) V comb g , k , l ) obtained from the string vertex in [CCT20]. The contributions of edgesand leaves involve the choice of the splitting s , and the weight wt ( G ) of a partiallydirected stable graph is calculated from the combinatorics of G by a formula involvingits automorphism group.We also prove in Section 5 that our invariants are stable under Morita equivalence.We conclude the paper with two appendices. The first one, Appendix A, gives severalexplicit formulas for low genus invariants (for ξ ≥ − Z -graded case. This is needed because, for ease of notation, we assumethroughout the body of the paper that all our vector spaces are Z / Z -graded, and weignore even degree shifts. In Appendix B we outline how our results can be modifiedto accommodate the Z -graded case. As a consequence we prove that our invariantssatisfy an analogue of the dimension axiom in Gromov-Witten theory. We thank Kevin Costello, Nick Sheridan, Si Li, and DimaArinkin for patiently listening to the various problems we ran into at different stagesof the project and for providing insight.Andrei C˘ald˘araru was partially supported by the National Science Foundation throughgrant number DMS-1811925 and by the Wisconsin Alumni Research Foundationthrough a Vilas Associate Professorship.
2. Preliminaries
Unless otherwise stated all our vector spaces will be Z / Z -graded over a field K of characteristic zero. (The Z -graded case is outlined inAppendix B.) Nevertheless, we think of our chain complexes as being homologically graded: differentials decrease degree. C˘ald˘araru and Tu
Throughout the paper we will work with a cyclic A ∞ -algebra A of Calabi-Yau dimen-sion d , i.e., the cyclic pairing (cid:104) − , − (cid:105) : A ⊗ A → K has degree d . The algebra A willbe assumed to satisfy the conditions below:( † ) A is smooth, finite dimensional, unital, and satisfies the Hodge-deRham degeneration property.(The Hodge-de Rham degeneration property is automatic if A is Z -graded [Kal08].)We will always use the shifted sign conventions described in [Cho08], [She15]. Theshifted degree | x | (cid:48) of an element x is defined to be | x | (cid:48) = | x | + . All the operations in cyclic A ∞ -algebras, including the pairing, and in L ∞ -algebras,including DGLAs, differ by a sign( − (cid:80) nk = ( n − k ) | x k | from the usual ones, when applied to a tensor x ⊗· · ·⊗ x n . In particular a Maurer-Cartanelement in a DGLA has shifted degree zero. We shall frequently use complexes with a circle action, togetherwith a compatible pairing on them. A circle action a chain complex ( C , ∂ ) is given byan operator δ : C → C of degree one such that δ =
0, and [ ∂, δ ] = ∂δ + δ∂ = C Tate = (cid:0) C (( u )) , ∂ + u δ (cid:1) C + = (cid:0) uC [[ u ]] , ∂ + u δ (cid:1) C − = C Tate / C + = (cid:0) C [ u − ] , ∂ + u δ (cid:1) , where u is a formal variable of even degree. Note that for C + this is slightly differentfrom the usual definition used in defining negative cyclic homology – we start with u instead of u . A pairing on a chain complex with a circle action is a symmetric bilinearchain map of even degree (cid:104) − , − (cid:105) : C ⊗ C → K such that the circle operator δ is self-adjoint, i.e., we have (cid:104) δ x , y (cid:105) = ( − | x | (cid:104) x , δ y (cid:105) , for all x , y ∈ C . ategorical Enumerative Invariants, II: Givental formula A pairing like the one described above induces a so-called higher residue pairing onthe associated Tate complex, with values in K (( u )), defined for x = (cid:80) x k · u k and y = (cid:80) x l · u l by (cid:104) x , y (cid:105) hres = (cid:88) k , l ( − k (cid:104) x k , y l (cid:105) · u k + l + . (The choice of shift by u is motivated by our desire for homogeneity in the constructionof the Weyl algebra in (2.9).) Its residue at u = K and defined by (cid:104) x · u k , y · u l (cid:105) res = (cid:40) ( − k (cid:104) x , y (cid:105) if k + l =
10 otherwise.Due to the sign ( − k the residue pairing (cid:104) − , − (cid:105) res is anti-symmetric and of degreezero.Note that the residue pairing induces a natural even map C − → Hom c ( C + , K )which will frequently be a quasi-isomorphism. Here the superscript c denotes contin-uous homomorphisms in the u -adic topology. Thus we will think of C + as the dual of C − . We will often think of it as the odd map ι : C − → Hom c ( C + [1] , K )turning an output into an input as in [CCT20]. The main example of a chain complex with circle action and pairing is providedby the shifted Hochschild chain complex of the A ∞ -algebra A , L = C ∗ ( A )[ d ] . Its homology is the shifted Hochschild homology of A , H = H ∗ ( L ) = HH ∗ ( A )[ d ] . For a smooth and proper Calabi-Yau algebra it is isomorphic to the dual of theHochschild cohomology of A , also known as HH ∗ ( A , A ∗ ). The degree of an ele-ment x | . . . | x n ∈ L is (cid:80) nk = | x k | (cid:48) + ( d − B , and the symmetric pairing on L is the chain-level Mukai pairing (cid:104) − , − (cid:105) Muk : L ⊗ L → K . C˘ald˘araru and Tu
Let X be a compact almost K¨ahler manifold of realdimension 2 d . We will denote by H the graded vector space H = H ∗ ( X )[2 d ], theshifted cohomology of X . Thinking of H as a chain complex with trivial differentialand circle action, the Poincar´e pairing on H extends to a skew-symmetric residuepairing (cid:104) − , − (cid:105) res on H Tate = H (( u )). With respect to this pairing H + = uH [[ u ]] and H − = H [ u − ] are Lagrangian subspaces. Fix a class β ∈ H ( X ) which will be used to define classical Gromov-Witteninvariants of X . In the discussion below we will think of β as being fixed and willignore the dependence of the Gromov-Witten invariants on it.Denote by M g , n ( X ) the moduli space of stable maps to X from curves of genus g with n marked points and class β . It comes equipped with a virtual fundamenta class[ M g , n ( X )] vir . For γ , . . . , γ n ∈ H and i , . . . , i n ≥ (cid:104) τ i ( γ ) , . . . , τ i n ( γ n ) (cid:105) Xg = (cid:90) [ M g , n ( X )] vir ψ i ev ∗ γ · · · ψ i n n ev ∗ n γ n . The genus g invariants are packaged as the Taylor coefficients of a power series F gX ,the genus g descendant potential of X , see [Coa08, 2.1]. It is the function F gX : H + → K [[ λ ]]( λ is another formal even variable) whose value at γ = γ u + γ u + · · · is F gX ( γ ) = (cid:88) n ≥ (cid:88) i ,... i n ≥ n ! (cid:104) τ i − ( γ i ) , . . . , τ i n − ( γ i n ) (cid:105) Xg · λ g − + n . The potentials F gX for g ≥ D X ( γ ) = exp (cid:88) g ≥ (cid:126) g − F gX ( γ ) , another formal function on H + with values in a certain completion of K [ (cid:126) , (cid:126) − , λ ];see (2.12) below. The variable (cid:126) , used to keep track of the genus, is also even. ategorical Enumerative Invariants, II: Givental formula Since the dual of H − with respect to the residue pairing is H + , it is natural to askif F gX satisfies the finiteness condition needed to ensure that it is the dual of an element F Xg ∈ H − [[ λ ]], in other words if there exists an element F Xg = (cid:88) n F Xg , n · λ g − + n ∈ H − [[ λ ]]such that F gX ( γ ) = (cid:88) n (cid:104) F Xg , n , γ (cid:105) for all γ ∈ H + . Such F Xg exists, and we will call it the dual genus g descendant potential of X .A formula for F Xg is most easily written by choosing a basis for H , though the resultis independent of this choice. Let t , . . . , t N be a basis of H , and let t , . . . , t N ∈ H bethe dual basis with respect to the Poincar´e pairing. Define F Xg ∈ H − [[ λ ]] as F Xg = (cid:88) n ≥ (cid:88) i ,..., i n k ,..., k n n ! (cid:104) τ i ( t k ) , . . . , τ i n ( t k n ) (cid:105) Xg · (cid:0) u − i t k (cid:1) · · · (cid:0) u − i n t k n (cid:1) · λ g − + n . Here the sum is over all ψ -class powers i , . . . , i n ≥ k , . . . , k n ∈{ , . . . , N } .The dual total descendant potential D X is then defined as before: D X = exp (cid:88) g ≥ (cid:126) g − F Xg . It lives in the localized and completed symmetric algebra (cid:100)
Sym (cid:126) ( H − )[[ (cid:126) , λ ]] of H − defined below (2.12). It is easy to check that all the expressions (cid:126) g − F Xg have homo-logical degree zero. (This is one of the advantages of defining the residue pairing theway we do.) The dual descendant potentials encode all the Gromov-Witten invariants of X .For example, for targets X = pt and X = T (a two-torus) we have for β = F pt =
16 ([ X ]) λ +
124 ( u − [ X ])([ X ]) λ + · · · F pt =
124 ( u − [ X ]) λ + · · · F T = −
124 ([ X ]) λ + · · · F T = u − [ X ]) λ + · · · . C˘ald˘araru and Tu
Here we write cohomology classes as their Poincar´e duals: for example [ X ] ∈ H ( X )is the identity of this ring. It corresponds to the insertion of a point in the classical(non-dual) invariants. We now review the relationship,discovered by Costello [Cos09], between Batalin-Vilkovisky (BV) algebras and Fockspaces for dg-vector spaces with circle action and pairing. This work builds on earlierwork of Givental [Giv01]. For simplicity, in this exposition we will ignore issues ofcompletion, which will be addressed in (2.12).Let ( C , ∂, δ, (cid:104) − , − (cid:105) ) be a complex with circle action and pairing, as in (2.2) and (2.3).Associated with the symplectic vector space ( C Tate , (cid:104)− , −(cid:105) res ) is the Weyl algebra W (cid:0) C Tate (cid:1) with a formal variable (cid:126) of even degree defined as W (cid:0) C Tate (cid:1) = T (cid:0) C Tate (cid:1) [[ (cid:126) ]] / (cid:0) α ⊗ β − ( − | α || β | β ⊗ α = (cid:126) (cid:104) α, β (cid:105) res (cid:1) . Note that the relation is homogeneous in Z / Z .The positive subspace C + is a subcomplex of C Tate . Hence the left ideal generated bythis subspace W (cid:0) C Tate (cid:1) · C + is a dg-ideal in W ( C Tate ). The quotient F = W (cid:0) C Tate (cid:1) / W (cid:0) C Tate (cid:1) · C + is known as the Fock space of W ( C Tate ); it is a left dg-module of the Weyl algebra.
The linear subspace C − of C Tate , on the other hand, is not a subcomplex: thedifferential ∂ + u δ of an element u x is by definition equal to u ∂ ( x ) when computedin C − , but it equals u ∂ ( x ) + u δ ( x ) in C Tate .Nevertheless, disregarding differentials, C − is still isotropic as a graded vector sub-space of the symplectic space ( C Tate , (cid:104) − , − (cid:105) res ). This yields an embedding ofalgebras (without differentials!) Sym ( C − )[[ (cid:126) ]] (cid:44) → W (cid:0) C Tate (cid:1) . Post-composing with the canonical projection to the Fock space yields an isomorphismof graded vector spaces
Sym ( C − )[[ (cid:126) ]] ∼ −→ F . Costello [Cos09] observed that under the above isomorphism the differential onthe Fock space pulls back to a differential of the form b + uB + (cid:126) ∆ on the symmetricalgebra Sym C − [[ (cid:126) ]]. Here the operator ∆ is a BV differential: it is a degree one,square-zero, second order differential operator of the symmetric algebra. As such it is ategorical Enumerative Invariants, II: Givental formula uniquely determined by its action on Sym ≤ C − ; moreover, it vanishes on Sym ≤ C − ,and on Sym C − it is explicitly given by the formula ∆ ( x · y ) = Ω ( x , y ) = (cid:104) Bx , y (cid:105) , for elements x , y ∈ C − , x = x + x − u − + x − u − + · · · , y = y + y − u − + y − u − + · · · . It is well-known (see for example [Get94]) that such a BV differential ∆ induces aDGLA structure on the shifted Fock space h = ( Sym C − )[1][[ (cid:126) ]] . The differential is b + uB + (cid:126) ∆ and the Lie bracket measures the failure of ∆ to be aderivation: { x , y } = ∆ ( x · y ) − ( ∆ x · y ) − ( − | x | ( x · ∆ y ) . We want to relate solutions of the Maurer-Cartan equation in h with homologyclasses of the operator b + uB + (cid:126) ∆ . The relationship involves computing the expo-nential exp ( β/ (cid:126) ) for a Maurer-Cartan element β . However, this exponential does notmake sense until we impose additional finiteness conditions. These are best expressedby introducing a new formal variable λ of even degree.To this end we modify the definition of the Weyl algebra W (cid:0) C Tate (cid:1) to include thisvariable: W (cid:0) C Tate (cid:1) = T (cid:0) C Tate (cid:1) [[ (cid:126) , λ ]] / (cid:0) α ⊗ β − ( − | α || β | β ⊗ α = (cid:126) (cid:104) α, β (cid:105) res (cid:1) . The rest of the definitions in the above discussion are unchanged (but the variable λ isnow along for the ride).We localize the Weyl algebra at (cid:126) and complete in the λ -adic topology to get thelocalized and completed algebra (cid:99) W (cid:126) (cid:0) C Tate (cid:1) = lim ←− n W (cid:0) C Tate (cid:1) [ (cid:126) − ] / ( λ n ) . Infinite power series of the form (cid:88) k ≥ α k λ k (cid:126) − k exist in (cid:99) W (cid:126) (cid:0) C Tate (cid:1) but not in W (cid:0) C Tate (cid:1) . C˘ald˘araru and Tu
By analogy with the construction of (cid:99) W ( C Tate ), we denote by (cid:99) F (cid:126) and (cid:100) Sym (cid:126) ( C − )[[ (cid:126) , λ ]]the localized and completed versions of the Fock space and the symmetric algebra.In particular, if H is a graded vector space which carries a symmetric bilinear pairing,endow it with trivial differential and circle action. The above construction defines thevector space (cid:100) Sym (cid:126) ( H − )[[ (cid:126) , λ ]]where the dual total descendant potential and the categorical enumerative potentialswill live.With this preparation we can state the following result, which is well known. An element β ∈ λ · h of odd degree satisfies the Maurer-Cartan equationif and only if exp ( β/ (cid:126) ) is b + uB + (cid:126) ∆ -closed.Moreover, two Maurer-Cartan elements β and β are gauge equivalent if and onlyif exp ( β / (cid:126) ) and exp ( β / (cid:126) ) are homologous. All these identities hold in the algebra (cid:100) Sym (cid:126) ( C − )[[ (cid:126) , λ ]] . We will now take the complex ( C , ∂, δ, (cid:104) − , − (cid:105) ) of the previous discussion tobe the shifted Hochschild chain complex( L = C ∗ ( A )[ d ] , b , B , (cid:104) − , − (cid:105) Muk )of a fixed cyclic A ∞ algebra A which satisfies condition ( † ). As in the introduction d denotes the Calabi-Yau dimension of A .Kontsevich-Soibelman [KS09] and Costello [Cos07] sketched the construction of thestructure of a two-dimensional topological field theory with target the complex ( L , b ).In other words they argued that there exists a map from the dg-PROP of normalizedsingular chains on the moduli spaces of framed curves M fr g , k , l to the endomorphismdg-PROP of L . More specifically this field theory is given by explicit even degreechain maps ρ Ag , k , l : C ∗ ( M fr g , k , l ) → Hom (cid:0) L ⊗ k , L ⊗ l (cid:1) satisfying natural gluing/composition relations. These maps are constructed by first re-placing the source chain complex C ∗ ( M fr g , k , l ) by a combinatorial version C comb ∗ ( M fr g , k , l )of it, and then explicitly describing the corresponding maps combinatorially. A com-plete description of these maps, following the ideas sketched in [KS09], will be givenin full detail (including compatible sign conventions) in the upcoming paper [CC20-1]. ategorical Enumerative Invariants, II: Givental formula In [CCT20] Costello and the authors constructed two morphisms of DGLAs g + → (cid:98) g , h + → (cid:98) h . Both are denoted by ι . The former is defined purely combinatori-ally, and it is a quasi-isomorphism; the latter is associated to a cyclic A ∞ -algebra A ,and it is a quasi-isomorphism if we assume condition ( † ). We will call (cid:98) g and (cid:98) h theKoszul resolutions of g + and h + , respectively.The first pair of DGLAs is constructed using chains on the moduli spaces of curveswith framed incoming and outgoing marked points. While the construction of g + requires the use of geometric singular chains, (cid:98) g can be defined using the ribbon graph,combinatorial model of moduli spaces of curves. More precisely, with notations asin [CCT20] we have g + = (cid:77) g , l ≥ C ∗ ( M g , , l ) hS [1][[ (cid:126) , λ ]]and (cid:98) g = (cid:77) g , k ≥ , l C comb ∗ ( M fr g , k , l ) hS [2 − k ][[ (cid:126) , λ ]] . The DGLA g + is a quotient of a larger DGLA g which includes the l = L of the A ∞ -algebra A . They are denoted by h + = (cid:77) l ≥ Sym l ( L − )[1][[ (cid:126) , λ ]]and (cid:98) h = (cid:77) k ≥ , l Hom c (cid:0) Sym k ( L + [1]) , Sym l ( L − ) (cid:1) [[ (cid:126) , λ ]] . The notation
Hom c stands for the space of u -adically continuous K -linear homomor-phisms. As before, h + is a quotient of a larger DGLA h which includes the case l = (cid:98) h means.) The two dimensional field theory structure on L gives a morphism of DGLAs ρ A : (cid:98) g → (cid:98) h . C˘ald˘araru and Tu (We include in this map the sign corrections from [CCT20, (1.13)].) We showedin [CCT20, Theorem 3.6] that in the DGLA (cid:98) g there exists a special Maurer-Cartanelement (cid:98) V comb , (cid:98) V comb = (cid:88) g , k ≥ , l (cid:98) V comb g , k , l (cid:126) g λ g − + k + l which is unique up to gauge equivalence. This element is called the combinatorialstring vertex following its original geometric definition of Sen-Zwiebach [SZ94] andCostello [Cos09].The push-forward of (cid:98) V comb under ρ A yields a Maurer-Cartan element (cid:98) β A ∈ (cid:98) h of theform (cid:98) β A = ρ A ∗ ( (cid:98) V comb ) = (cid:88) g , k ≥ , l ρ A ( (cid:98) V comb g , k , l ) (cid:126) g λ g − + k + l . The tensors (cid:98) β Ag , k , l = ρ A ( (cid:98) V g , k , l ) ∈ Hom c (cid:0) Sym k ( L + [1]) , Sym l L − (cid:1) [[ (cid:126) , λ ]]form the starting point of the current paper. They will be used to define and computethe categorical enumerative invariants of the cyclic A ∞ -algebra A and of a splitting s of the Hodge filtration.
3. Definition of the categorical enumerative invariants
In this section we use the Maurer-Cartan element (cid:98) β A ∈ (cid:98) h to define, for a pair A , s consisting of a cyclic A ∞ algebra and a splitting s of its non-commutative Hodgefiltration, the categorical enumerative invariant D A , s . We begin by sketching the construction of D A , s ,ignoring for the sake of clarity two technical points:– the distinction between h + and h ; and– the localization and completion aspect of the construction (2.12).The pre-image β A ∈ h of (cid:98) β A ∈ (cid:98) h under the quasi-isomorphism ι : h → (cid:98) h is a Maurer-Cartan element in h , unique up to gauge. Lemma 2.13 gives a well-defined homologyclass D A abs = [ exp ( β )] ∈ H ∗ ( h , b + uB + (cid:126) ∆ ) ategorical Enumerative Invariants, II: Givental formula because the DGLA h is a particular case of the construction in (2.11). We call D A abs theabstract categorical enumerative potential D A abs of A ; it only depends on the algebra A and not on the splitting s .The construction of h in (2.11) depends only on the data of (cid:0) L , b , B , (cid:104) − , − (cid:105) Muk (cid:1) .Moreover, it is functorial with respect to homotopies (this statement will be madeprecise in Section 4). A splitting s of the Hodge filtration is a quasi-isomorphism ofmixed complexes ( L , b , B ) ∼ = ( L , b ,
0) which respects pairings. In particular it gives ahomotopy-trivialization of the operator B . The operator ∆ and the bracket { − , − } are defined using B , so they inherit homotopy trivializations. It follows that the choiceof s induces an L ∞ quasi-isomorphism of DGLAs( h , b + uB + (cid:126) ∆ , { − , − } ) ∼ = ( h , b + uB , . This quasi-isomorphism will be constructed explicitly in Section 4. Using the splitting s once again, the latter complex is quasi-isomorphic to ( h , b , D A , s ∈ H ∗ ( h , b ) = Sym ( H − )(( (cid:126) , λ ))is defined as the image of the homology class D A abs under the composite quasi-isomorphism ( h , b + uB + (cid:126) ∆ , { − , − } ) ∼ = ( h , b , . Our construction is essentially the same as the original one of Costello [Cos09],differing from it in the following two aspects:– we use the Koszul resolution (cid:98) g instead of g , which allows us to compute thecombinatorial string vertex explicitly;– we use explicit formulas in Section 4 to trivialize the DGLA( h , b + uB + (cid:126) ∆ , { − , − } );the original construction used a non-explicit argument relying on deformationtheory (rigidity of Fock modules). We will now carry out the aboveconstruction in detail. Condition ( † ) is assumed to hold and therefore the map ι : h + → (cid:98) h is a quasi-isomorphism of DGLAs ([CCT20, Lemma 4.4]). The Maurer-Cartan modulispace is invariant under such quasi-isomorphisms. Thus there exists a Maurer-Cartanelement β A ∈ h + , β A = (cid:88) g , n ≥ β Ag , n (cid:126) g λ g − + n , C˘ald˘araru and Tu such that ι ( β A ) is gauge equivalent to (cid:98) β A . It is unique up to homotopy.Recall that h decomposes as a vector space (but not as a DGLA!) as h = K [1][[ (cid:126) , λ ]] ⊕ h + . More precisely, the space of shifted scalars K [1][[ (cid:126) , λ ]] ⊂ h forms a central subalgebrain h , and h + is the DGLA quotient of h by it, i.e., we have a short exact sequence ofDGLAs 0 → K [1][[ (cid:126) , λ ]] → h → h + → . The vector space direct sum decomposition h = K [1][[ (cid:126) , λ ]] ⊕ h + allows us to regard the Maurer-Cartan element β A ∈ h + as an element in h , by takingits K [1][[ (cid:126) , λ ]]-component to be zero . Even though h + is not a subalgebra of h ,we will prove in Lemma 4.14 that β A ∈ h still satisfies the Maurer-Cartan equation.Lemma 2.13 then yields a ( b + uB + (cid:126) ∆ )-closed element exp ( β A / (cid:126) ) ∈ (cid:100) Sym (cid:126) L − [[ (cid:126) , λ ]]in the localized and completed symmetric algebra. The fact that the string vertex isunique up to homotopy [CCT20, Theorem 3.6] shows that the cohomology class ofthis element depends only on the cyclic A ∞ -algebra A . We denote it by D A abs = (cid:2) exp ( β A / (cid:126) ) (cid:3) ∈ H ∗ (cid:0) (cid:100) Sym (cid:126) L − [[ (cid:126) , λ ]] , b + uB + (cid:126) ∆ (cid:1) ∼ = H ∗ ( (cid:99) F (cid:126) )and call it the abstract total descendent potential of A . It has shifted degree zero. To obtain invariants of A that are similar to those from Gromov-Witten theory we need a further ingredient: achoice of splitting s of the Hodge filtration.We define a splitting of the (non-commutative) Hodge filtration of A to be a gradedvector space map s : HH ∗ ( A ) → HC −∗ ( A )satisfying the following two conditions:S1. (Splitting condition.) s splits the canonical projection HC −∗ ( A ) → HH ∗ ( A ).S2. (Lagrangian condition.) (cid:104) s ( x ) , s ( y ) (cid:105) hres = (cid:104) x , y (cid:105) Muk , for any x , y ∈ HH ∗ ( A ). This choice is only made for the sake of proving Proposition 3.11. A more reasonablechoice would be to force the dilaton equation. But this would present additional difficulties inProposition 3.11. ategorical Enumerative Invariants, II: Givental formula Remark.
In many circumstances it is useful to put further restrictions on the allowedsplittings. For example one may impose certain homogeneity conditions, or ask for thesplitting to be compatible with the cyclic structure of the A ∞ -algebra. In this paper weonly need conditions S1 and S2, but we refer the reader to [AT19, Definition 3.7] formore information on these possible restrictions. By the Hodge-to-de Rham degeneration property of A the homology H = H ∗ ( L )is endowed with the trivial circle action. According to our conventions on circle actionswe have the following graded vector spaces with trivial differentials: H Tate = H (( u )) , H + = uH [[ u ]] , H − = H [ u − ] . Shifting a splitting s by d allows us to view it as a map s : H → u − H ∗ ( L + ) . This map can then be repackaged by extending it u -linearly to an isomorphism ofsymplectic vector spaces s : (cid:0) H Tate , (cid:104)− , −(cid:105) res (cid:1) → (cid:0) H ∗ ( L Tate ) , (cid:104)− , −(cid:105) res (cid:1) which maps the Lagrangian subspaces H + , H ∗ ( L + ) to each other. We will need later a chain level analogue of the homology splitting above. Givena splitting s , a chain level lift of s is a chain map R : ( L , b ) → ( u − L + , b + uB )which associates to x ∈ L a formal series R ( x ) = x + x u + x u + · · · such that its induced map in homology is s (after shifting degrees by d ). We will writesuch a splitting R as R = id + R u + R u + · · · where R i : L → L is an even map. Any splitting on homology (3.4) can be lifted to a splitting at chainlevel (3.6). All liftings of a fixed homology splitting are homotopy equivalent to oneanother.
Proof.
This is the classical fact that for complexes of vector spaces, the homology ofthe
Hom complex
Hom ∗ (( L , b ) , ( L [[ u ]] , b + uB )) C˘ald˘araru and Tu computes homomorphisms between the homology groups
Hom ( H , u − H + ). The sec-ond statement follows from the fact that the difference R − R (cid:48) between two chain maps R and R (cid:48) is exact in the complex Hom ∗ (( L , b ) , ( L [[ u ]] , b + uB )) if and only if R and R (cid:48) are homotopic. ( A , s ) . The symplectomorphism s : H Tate → H ∗ ( L Tate ) induces an isomorphism of Weyl algebras Φ s : (cid:99) W (cid:126) (cid:0) H Tate (cid:1) → (cid:99) W (cid:126) (cid:0) H ∗ ( L Tate ) (cid:1) . Define Ψ s to be the composition of the other three maps in the diagram below (cid:99) W (cid:126) (cid:0) H Tate (cid:1) Φ s (cid:45) (cid:99) W (cid:126) (cid:0) H ∗ ( L Tate ) (cid:1) = H ∗ (cid:0) (cid:99) W (cid:126) ( L Tate ) (cid:1)(cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] i (cid:54) Ψ s (cid:45) H ∗ (cid:0) (cid:99) F (cid:126) (cid:1) . p (cid:63) It is a graded vector space isomorphism. Here the left vertical map is the inclusion ofthe symmetric algebra generated by the Lagrangian subspace H − into the Weyl algebra,while the right vertical map is the canonical quotient map from the Weyl algebra tothe Fock space. The abstract total descendant potential D A abs lives in the lower rightcorner. The total descendant potential D A , s ∈ (cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] of a pair ( A , s ) is the pre-image of the abstract total descendant potential D A abs under the map Ψ s , D A , s = ( Ψ s ) − ( D A abs ) . The n -point function of genus g , F A , sg , n ∈ Sym n H − , is defined by the identity (cid:88) g , n F A , sg , n · (cid:126) g λ g − + n = (cid:126) · ln D A , s . The right hand term (cid:126) · ln D A , s will be denoted by F A , s . Denote by G A the Givental groupof the pair ( H + , (cid:104) − , − (cid:105) Muk ). Abstractly, it is the subgroup of automorphisms of thesymplectic vector space ( H Tate , (cid:104) − , − (cid:105) res ) preserving the Lagrangian subspace H + and acting as the identity on H . Explicitly it consists of elements g of the form g = id + g · u + g · u + · · · with each g j ∈ End ( H ) required to satisfy (cid:104) g · x , g · y (cid:105) res = (cid:104) x , y (cid:105) res for any x , y ∈ H Tate . ategorical Enumerative Invariants, II: Givental formula If the set of splittings of the non-commutative Hodge filtration is nonempty, then it isa left torsor over the Givental group, by letting an element g ∈ G A act on a splitting s : H + → u − H ∗ ( L + ) by pre-composing with g − : g · s : H + g − −→ H + s −→ u − H ∗ ( L + ) . The Givental action on the Fock space (cid:100)
Sym (cid:126) H − [[ (cid:126) , λ ]] is by definition the automor-phism (cid:98) g of the Fock space induced from the symplectic transformation g : (cid:99) W (cid:126) (cid:0) H Tate (cid:1) Φ g −−−−→ (cid:99) W (cid:126) (cid:0) H Tate (cid:1) i (cid:120) (cid:121) π (cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] (cid:98) g −−−−→ (cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] The construction of the total descendant potential is compatiblewith the action of the Givental group G A . Explicitly, for a splitting s and an element g ∈ G A we have D A , g · s = (cid:98) g ( D A , s ) . Proof.
Consider the following diagram: (cid:99) W (cid:126) (cid:0) H Tate (cid:1) Φ g − (cid:45) (cid:99) W (cid:126) (cid:0) H Tate (cid:1) Φ s (cid:45) H ∗ (cid:0) (cid:99) W (cid:126) ( L Tate ) (cid:1)(cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] i (cid:54) (cid:100) g − (cid:45) (cid:100) Sym (cid:126) H − [[ (cid:126) , λ ]] π (cid:63) i (cid:54) Ψ s (cid:45) H ∗ (cid:0) (cid:99) F (cid:126) (cid:1) . p (cid:63) We claim that p Φ g · s i = p Φ s i π Φ g − i . The left hand side equals p Φ s Φ g − i . The composition i π in the middle is not theidentity, but rather the projection onto the negative subspace (image of i ). Its kernelis by definition the left ideal generated by the positive subspace H + . Thus, we have i π = id ( mod H + ).Now observe that Φ s preserves the ideal H + , just because by definition a splitting s only contains non-negative powers in the variable u . Followed by the projection map p gives zero, that is, p Φ s i π (cid:0) H + (cid:1) = . We conclude that p Φ s = p Φ s i π , and hence p Φ g · s i = p Φ s Φ g − i = p Φ s i π Φ g − i . C˘ald˘araru and Tu
This implies that D A , g · s = ( p Φ g · s i ) − ( D A abs ) = ( π Φ g − i ) − ( p Φ s i ) − ( D A abs ) = ( π Φ g − i ) − ( D A , s ) = (cid:98) g ( D A , s )
4. Trivializing the DGLAs associated to a cyclic algebra
One of main difficulties in understanding the categorical invariants F A , sg , n in Defini-tion 3.9 is caused by the fact that the differential b + uB + (cid:126) ∆ on the Fock space is nothomogeneous in the symmetric power degree in Sym H − . Indeed, b + uB preservesthis degree (it acts on each element of a symmetric product individually), while ∆ reduces it by 2. In this section we shall explicitly trivialize the operator ∆ using an L ∞ quasi-isomorphism. Since the Lie bracket is essentially defined using ∆ we shallin fact simultaneous trivialize both the operator ∆ and the Lie bracket. We need some notation to state this result more precisely. Recall from [CCT20]that there are three DGLA’s h , h + and (cid:98) h associated to the cyclic A ∞ -algebra A ,together with a quasi-isomorphism ι : h + → (cid:98) h .We introduce partial trivializations of h , h + and (cid:98) h , denoted by h triv , h triv , + and (cid:98) h triv ,respectively. Their underlying vector spaces are the same as those of the original Liealgebras, but their differentials are given by b + uB for the first two and b + uB + ι forthe third one (instead of b + uB + (cid:126) ∆ for the first two and b + uB + (cid:126) ∆ + ι for thethird one). Moreover, they are all endowed with zero bracket.Let A be a cyclic A ∞ -algebra satisfying condition ( † ) and let R be a chain levelsplitting of its non-commutative Hodge filtration. This data will be fixed for the rest ofthis section. (A) There exists an L ∞ quasi-isomorphism K : h → h triv , depending on R andconstructed explicitly in the proof below. The map K restricts to an L ∞ quasi-isomorphism of the positive parts of both sides. We shall denote this restrictionby K as well.(B) There exists an L ∞ quasi-isomorphism (cid:99) K : (cid:98) h → (cid:98) h triv , also depending on R . ategorical Enumerative Invariants, II: Givental formula (C) These quasi-isomorphism are compatible with the Koszul resolution maps ι , inthe sense that the following diagram commutes: h + ι −−−−→ (cid:98) h K (cid:121) (cid:121) (cid:99) K h triv , + ι −−−−→ (cid:98) h triv . We begin with part (A) of the above theorem.Recall (2.11) that the restriction of the operator ∆ : h → h to Sym L − is given by Ω ( x , y ) = (cid:104) Bx , y (cid:105) Muk where x = x + x − u − + · · · and y = y + y − u − + · · · are elements of L − . In order to trivialize the operator ∆ we will first constructa homotopy operator for the chain map Ω : Sym L − → K using the chain-levelsplitting R of the Hodge filtration.Extend R to a u -linear isomorphism of chain complexes R : (cid:0) L [[ u ]] , b (cid:1) → (cid:0) L [[ u ]] , b + uB (cid:1) which we still denote by R . The inverse operator of R is another operator of the form T = id + T u + T u + · · · . By definition we have the following identity (cid:88) i + j = k T i S j = (cid:40) id if k =
00 if k ≥ . Solving the above recursively yields formulas for the T j ’s in terms of the R i ’s. Forexample we have T = − R , T = − R + R R , etc.Since R and T are chain maps, we have[ b , R n ] = − BR n − , [ b , T n ] = T n − B for all n ≥ . We now define an even linear map H : L − ⊗ L − → K . It is the sum, over all i ≥ j ≥
0, of maps H i , j : u − i L ⊗ u − j L → K defined by H i , j ( u − i x , u − j y ) = (cid:104) ( − j j (cid:88) l = R l T i + j + − l x , y (cid:105) . C˘ald˘araru and Tu
For any two elements α, β ∈ L − we have H (cid:0) ( b + uB ) α, β (cid:1) + ( − | α | H (cid:0) α, ( b + uB ) β (cid:1) = − Ω ( α, β ) . We write this identity as [ b + uB , H ] = Ω . In other words the operator H is a bounding homotopy of Ω . Proof.
We begin with the first case, when i = j =
0. Then H , ( u x , u y ) = (cid:104) T x , y (cid:105) . We need to verify that H , ( bx , y ) + ( − | x | H , ( x , by ) = (cid:104) Bx , y (cid:105) . This is a straight-forward computation: H , ( bx , y ) + ( − | x | H , ( x , by ) == (cid:104) T bx , y (cid:105) + ( − | x | (cid:104) T x , by (cid:105) = (cid:104) T bx , y (cid:105) − (cid:104) bT x , y (cid:105) = −(cid:104) [ b , T ] x , y (cid:105) = −(cid:104) Bx , y (cid:105) . In the general case we would like to prove that H i , j ( u − i bx , u − j y ) + ( − | x | H i , j ( u − i x , u − j by ) ++ H i − , j ( u − i + Bx , u − j y ) + ( − | x | H i , j − ( u − i x , u − j By ) = . Again, this is a straightforward computation using the commutator relations of the R ’sand the T ’s. We have H i , j ( u − i bx , u − j y ) = (cid:104) ( − j j (cid:88) l = R l T i + j + − l bx , y (cid:105) = (cid:104) ( − j j (cid:88) l = R l bT i + j + − l x , y (cid:105) − (cid:104) ( − j j (cid:88) l = R l T i + j − l Bx , y (cid:105) . We also have( − | x | H i , j ( u − i x , u − j by ) = −(cid:104) ( − j b j (cid:88) l = R l T i + j + − l x , y (cid:105) = −(cid:104) ( − j j (cid:88) l = R bT i + j + − l x , y (cid:105) + (cid:104) j (cid:88) l = ( − j BR l T i + j + − l x , y (cid:105) . ategorical Enumerative Invariants, II: Givental formula Adding together these two equations yields H i , j ( u − i bx , u − j y ) + ( − | x | H i , j ( u − i x , u − j by ) == −(cid:104) ( − j j (cid:88) l = R l T i + j − l Bx , y (cid:105) + (cid:104) ( − j l (cid:88) l = BR l − T i + j + − l x , y (cid:105) = − H i − , j ( u − i + Bx , u − j y ) − ( − | x | H i , j − ( u − i x , u − j By ) . This proves the proposition.
The homotopy operator H induces a first order trivialization of ∆ . Indeed, since B is self-adjoint with respect to the Mukai pairing, we can symmetrize the homotopyoperator H to obtain a homotopy operator H sym : Sym L − → K , H sym ( xy ) = (cid:16) H ( x , y ) + ( − | x || y | H ( y , x ) (cid:17) . The degrees here are the degrees in the complex L − .The operator H sym bounds ∆ : Sym L − → K . Extending it as a second orderdifferential operator to the full symmetric algebra Sym L − yields an operator ∆ H .To see that [ b + uB , ∆ H ] = ∆ we note that both sides are second order differentialoperators. Thus it suffices to prove that they are equal on Sym ≤ L − , in which casethe statement follows from Proposition 4.5. The construction of the morphisms K and (cid:99) K involvessumming over certain types of graphs. We need to introduce some terminology aboutthese graphs. We refer to [GK98, Section 2] for details.A labeled graph shall mean a graph G endowed with a genus labeling g : V G → Z ≥ on its set of vertices V G . The genus of a labeled graph is defined to be g ( G ) = (cid:88) v ∈ V G g ( v ) + dim H ( G ) . We will use the following notations for a labeled graph G :– V G denotes the set of vertices of G ;– L G denotes the set of leaves of G ;– the valency of a vertex v ∈ V G is denoted by n ( v );– the graph G is called stable if 2 g ( v ) − + n ( v ) > v ∈ V G . C˘ald˘araru and Tu – if G has m vertices, a marking of G is a bijection f : { , · · · , m } → V G . An isomorphism between two marked and labeled graphs is an isomorphism ofthe underlying labeled graphs that also preserves the markings.We will denote various classes of graphs as follows:– Γ ( g , n ) denotes the set of isomorphism classes of graphs of genus g and with n leaves;– using double brackets as in Γ (( g , n )) requires further that the graphs be stable;– a subscript m as in Γ ( g , n ) m or Γ (( g , n )) m indicates that the graphs in discussionhave m vertices;– adding a tilde as in (cid:94) Γ ( g , n ) m or (cid:94) Γ (( g , n )) m signifies that we are looking at marked graphs and isomorphisms. L ∞ -morphism K . We will now construct the L ∞ -morphism K : h → h triv claimed in part (A) of Theorem 4.2 by “exponentiating" thefirst order trivialization H sym .The morphism K will be a collection of even linear maps K m : Sym m ( h [1]) → h triv [1]defined for each m ≥
1. Each of these maps will be defined as a Feynman-type sumover all the graphs in (cid:94) Γ ( g , n ) m for arbitrary g , n . (Recall that m denotes the number ofvertices in a given graph, so graphs with m vertices contribute to the m -th map in the L ∞ -morphism K .)Since h = Sym ( L − )[1][[ (cid:126) , λ ]] already includes a shift by one, the parity of the elementsin h [1] is the same as in Sym L − . Hence in order to define K m we will associate to amarked graph ( G , f ) with m vertices a K -linear map K G , f : (cid:0) Sym L − [[ (cid:126) ]] (cid:1) ⊗ m → Sym L − [[ (cid:126) ]] . The map K m will be the λ -linear extension of the map K m = (cid:88) g , n (cid:88) ( G , f ) ∈ (cid:94) Γ ( g , n ) m | Aut ( G , f ) | · K G , f . ategorical Enumerative Invariants, II: Givental formula Let ( G , f ) ∈ (cid:94) Γ ( g , n ) m be a marked graph. We will denote the i -th marked vertexof G by v i = f ( i ), its genus marking by g i = g ( v i ), and its valency by n i = n ( v i ). Forelements γ , . . . , γ m ∈ Sym L − the expression K G , f ( γ · (cid:126) t , . . . , γ m · (cid:126) m )will be non-zero only if t i = g i and γ i ∈ Sym n i L − for all i . If this is the case, theresult will be an element in Sym n ( L − ) · (cid:126) g .For an element γ = x x · · · x i ∈ Sym i L − we let (cid:101) γ ∈ (cid:0) L − (cid:1) ⊗ i be its symmetrization, (cid:101) γ = (cid:88) σ ∈ Σ i (cid:15) · x σ (1) ⊗ · · · ⊗ x σ ( i ) ∈ L ⊗ i − . Here (cid:15) is the Koszul sign for permuting the elements x , . . . , x i (with respect to thedegree in L − ). For elements γ ∈ ( L − ) ⊗ n , . . . , γ m ∈ ( L − ) ⊗ n m we define K G , f ( γ · (cid:126) g , . . . , γ n · (cid:126) g m ) = γ · (cid:126) g where γ is computed by the following Feynman-type procedure:(1) Decorate the half-edges adjacent to each vertex v i by (cid:101) γ i . Tensoring together theresults over all the vertices we obtain a tensor of the form m (cid:79) i = (cid:101) γ i ∈ L ⊗ ( (cid:80) n i ) − . The order in which these elements are tensored is the one given by the marking.(2) For each internal edge e of G contract the corresponding components of theabove tensor using the even symmetric bilinear form H sym : L ⊗ − → K . When applying the contraction we always permute the tensors to bring thetwo terms corresponding to the two half-edges to the front, and then apply thecontraction map. The ordering of the set E G does not matter since the operator H sym is even; also the ordering of the two half-edges of each edge does notmatter since the operator H sym is (graded) symmetric.(3) Read off the remaining tensor components (corresponding to the leaves of thegraph) in any order, and regard the result as the element γ in Sym n L − via thecanonical projection map L ⊗ n − → Sym n L − .With these assignments one can check that the maps K m are even. C˘ald˘araru and Tu K . The following figures illustrate a few graphs thatcontribute to the map K . Their common feature is that they have only one vertex; inthis case the marking f is unique.The contribution of the first type of graphs (the star graphs) is the identity map h [1] = Sym L − [[ (cid:126) , λ ]] → h triv = Sym L − [[ (cid:126) , λ ]] . To see this, note that symmetrization followed by projection produces a factor of n !,which is canceled by the size of the automorphism group of the star graph. We alsonote that it is necessary to use the set Γ ( g , n ) of all labeled graphs, as opposed to usingjust stable graphs. For instance, the ( g , n ) = (0 ,
0) star graph (just a vertex with agenus label of 0) contributes to the identity map
Sym L − = K → Sym L − = K .In a similar way one checks that the middle type of graph acts precisely by the operator (cid:126) ∆ H , while the third type of graph contributes ( (cid:126) ∆ H ) . Here the extra factor 1 / K = (cid:88) l ≥ l ! ( (cid:126) ∆ H ) l = e (cid:126) ∆ H . And indeed, the first L ∞ -morphism identity is ( b + uB ) K = K ( b + uB + (cid:126) ∆ ), whichcan be verified directly using the above formula. First we note that the maps K m are symmetricin their inputs. Indeed, if we permute two inputs γ i with γ j and switch the correspondingentries in the marking of the graph, it results in a Koszul sign change in the formationof the tensor (cid:78) k (cid:101) γ k , precisely showing that K m is graded symmetric. Furthermore,the contraction maps associated with the edges are all even maps, so K m is also even.Therefore the maps { K m } m ≥ form a pre- L ∞ homomorphism.In order to show that the collection of maps { K m } m ≥ form an L ∞ -morphism h → h triv ategorical Enumerative Invariants, II: Givental formula we need to check that for every m ≥ b + uB , K m ]( γ · · · γ m ) = (cid:126) (cid:88) i ( − (cid:15) i K m ( γ · · · ∆ γ i · · · γ m ) + (cid:88) i < j ( − (cid:15) ij K m − ( { γ i , γ j } γ · · · (cid:98) γ i · · · (cid:98) γ j · · · γ m ) , where the signs are Koszul signs (cid:15) i = | γ | + · · · + | γ i − | ,(cid:15) ij = | γ i | ( | γ | + · · · + | γ i − | ) + | γ j | ( | γ | + · · · + | γ i − | + | γ i + | + · · · + | γ j − | ) . (Recall that we use shifted sign conventions, and the shifted degrees in our algebrasare the ordinary degrees in Sym L − .)For each marked graph ( G , f ) consider the commutator [ b + uB , K G , f ]. Using theidentity [ b + uB , H sym ] = Ω from Proposition 4.5 it is easy to check that[ b + uB , K G , f ] = (cid:88) e K eG , f where the sum is over the internal edges e of G , and the operator K eG , f is defined in thesame way as K G , f , but the contraction corresponding to e uses the operator Ω insteadof H sym .We will see below that when e is a loop we will obtain the first type of term on theright hand side of the L ∞ identities above, while when e is a non-loop we will obtainthe second type of term. In the remainder of the proof we will explain this in moredetail, primarily in order to match up the coefficients given by graph automorphisms.First we deal with the loop case. Let e denote a loop at a vertex v i , and write G \ e for the result of deleting e from G . Assume that in a computation of K G , f we needto insert at v i a symmetric tensor γ i = x · · · x N + ∈ Sym N + L − . (Here N + n i of vertex i .) The total number of terms in (cid:101) γ i is ( N + γ i .Now consider the result of inserting ∆ ( γ i ) at v i in G \ e . The expression ∆ ( γ i ) has (cid:0) N + (cid:1) terms, corresponding to the number of choices of picking the two x j ’s to apply ∆ to. Thus the total number of terms in (cid:103) ∆ γ i ∈ Sym N L − (to be inserted at v i ) is (cid:18) N + (cid:19) · N ! =
12 ( N + . C˘ald˘araru and Tu
The extra factor of 1 / | Aut ( G , f ) | / | Aut ( G \ e , f ) | = e has an extra non-trivial automorphism that switches its two ends.Next we deal with the non-loop case. The combinatorics in this case are more involved.Let e be a non-loop edge in a marked graph ( G , f ), connecting vertices v i and v j . Theresult of contracting e will be denoted by ( G / e , f (cid:48) ), where the marking f (cid:48) is obtainedfrom the original marking of f by marking the new vertex (the result of collapsing v i and v j ) to be vertex “1”, and relabeling the remaining vertices to have the originalorder, skipping v i and v j We illustrate such a contraction in the following figure.Since an automorphism of ( G , f ) must preserve the marking of the vertices, the cardi-nality of Aut ( G , f ) is given by | Aut ( G , f ) | = (cid:89) v l v ! · c ( v ) · c ( v )! (cid:89) v (cid:54) = w n ( v , w )! , where l v is the number of leaves at v , c ( v ) is the number of loops at v , and n ( v , w ) isthe number of edges between two different vertices v and w . From this formula wewill deduce the matching of coefficients between the expressions arising in K G , f and K G / e , f (cid:48) .We will compare the combinatorial coefficients when there is only one extra vertex v k , as illustrated in the picture, leaving the general case to the reader. Let us considerdecorations γ i ∈ Sym N i + L − and γ j ∈ Sym N j + of the vertices v i and v j . Let us alsofix a marked graph ( G (cid:48) , f (cid:48) ) with m − L ∞ -morphism identities is given by I = Aut ( G (cid:48) , f (cid:48) ) K ( G (cid:48) , f (cid:48) ) ( { γ i , γ j } γ k ) . It suffices to match the above term with II = (cid:88) ( G , f ) , e Aut ( G , f ) K e ( G , f ) ( γ i · γ j · γ k ) ategorical Enumerative Invariants, II: Givental formula where the sum is over labeled and marked graphs ( G , f ) such that ( G / e , f ) = ( G (cid:48) , f (cid:48) )with e an edge between v i and v j .For the graph ( G , f ) let us write c i , l i for the number of loops and leaves at v i , andsimilarly for v j . We also write n ij , n jk , n ik for the number of edges between twovertices as indicated by the indices. Using the above formula of the automorphismgroup, we deduce that the ratio Aut ( G (cid:48) , f (cid:48) ) Aut ( G , f ) = n ij − n ij (cid:18) n ik + n jk n ik (cid:19) · (cid:18) l i + l j l i (cid:19) · (cid:18) c i + c j + n ij − c i , c j , n ij − (cid:19) . Using this to compute II we obtain II = Aut ( G (cid:48) , f (cid:48) ) (cid:88) ( G , f ) , e n ij − n ij (cid:18) n ik + n jk n ik (cid:19) · (cid:18) l i + l j l i (cid:19) · (cid:18) c i + c j + n ij − c i , c j , n ij − (cid:19) · K e ( G , f ) ( γ i · γ j · γ k )Since choosing e in ( G , f ) has exactly n ij choices, the above simplifies to II = Aut ( G (cid:48) , f (cid:48) ) (cid:88) ( G , f ) (cid:18) n ik + n jk n ik (cid:19) · (cid:18) l i + l j l i (cid:19) · n ij − · (cid:18) c i + c j + n ij − c i , c j , n ij − (cid:19) · K ij ( G , f ) ( γ i · γ j · γ k )Here K ij ( G , f ) means we choose any edge between v i and v j , and contract it by Ω .Next we use the combinatorial identity (with n ik + l i + c i + n ij − = N i and n jk + l j + c j + n ij − = N j ): (cid:88) n ij n ij − (cid:18) c i + c j + n ij − c i , c j , n ij − (cid:19) = (cid:18) N i + N j − n ik − n jk − l i − l j N i − n ik − l i (cid:19) , which is proved in Lemma 4.13. This implies that II = Aut ( G (cid:48) , f (cid:48) ) (cid:88) ( G , f ) (cid:18) n ik + n jk n ik (cid:19) · (cid:18) l i + l j l i (cid:19) · (cid:18) N i + N j − n ik − n jk − l i − l j N i − n ik − l i (cid:19) · K ij ( G , f ) ( γ i · γ j · γ k )This expression is exactly I , since the combinatorial product (cid:18) n ik + n jk n ik (cid:19) · (cid:18) l i + l j l i (cid:19) · (cid:18) N i + N j − n ik − n jk − l i − l j N i − n ik − l i (cid:19) is precisely the number of ways to split the tensor { γ i , γ j } ∈ Sym N i + N j L − into twoparts with l i / l j , n ik / n jk prescribed by the graph ( G , f ). Thus the proof of Part (A) ofTheorem 4.2 is complete. C˘ald˘araru and Tu
Let M , N be two positive integers. Assume that M ≡ N ( mod andthat M ≥ N . Denote by δ = M − N . Then we have (cid:88) k ≥ , k + l + δ = M + N k · (cid:18) M + N k , l , l + δ (cid:19) = (cid:18) M + NM (cid:19) . Proof.
The left hand side is the coefficient of z − δ in the product (2 + z + z ) M + N . Butobserve that (2 + z + z ) M + N = ( √ z + √ z ) M + N . The coefficient of z − δ in the latter expression is given by (cid:0) M + NM (cid:1) .As an application of Part (A) of Theorem 4.2 we prove the following lemma whichwas used in the previous section to define categorical enumerative invariants. The element β A of (3.3) is a Maurer-Cartan element in h , via thecanonical inclusion h + ⊂ h . Proof.
The construction of the L ∞ -morphism K makes it obvious that it restricts toan L ∞ -morphism K + : h + → ( h + ) triv . We have a commutative diagram h π (cid:45) h + (cid:45) h triv K (cid:63) (cid:45) ( h + ) triv K + (cid:63) (cid:45) b + uB ) x =
0, thelifting property of the Maurer-Cartan element K + ( β A ) is obvious. But the two verticalmaps are both L ∞ quasi-isomorphisms, so it follows that there exists a Maurer-Cartanelement α ∈ h such that π ( α ) is gauge equivalent to β A , i.e. for some degree zero g ∈ h + , we have exp ( g ) .π ( α ) = β A . Since π is surjective, let (cid:101) g ∈ h be a lift of g in h . Then the Maurer-Cartan element exp ( (cid:101) g ) .α lifts the Maurer-cartan element β A . Butmodifying such an element in its scalar part (since K [1][[ (cid:126) , λ ]] is central in h ) does notaffect the property of it being Maurer-Cartan, so we can choose the lift to have trivialscalar part. ategorical Enumerative Invariants, II: Givental formula (cid:98) h . Our next goal is to prove Part(B) of Theorem 4.2. More precisely, we would like to use a chain-level splitting R of the non-commutative Hodge filtration, as defined in (3.6), to partially trivialize theDGLA (cid:98) h = (cid:77) k ≥ , l ≥ Hom c (cid:0) Sym k ( L + [1]) , Sym l ( L − ) (cid:1) [[ (cid:126) , λ ]] . To simplify notation we will write L k , l = Hom c (cid:0) Sym k ( L + [1]) , Sym l ( L − ) (cid:1) . We think of an element γ ∈ L k , l as an operation with k inputs and l outputs. Theinputs are elements in L + [1]; the outputs are in L − .For a map γ ∈ L k , l we define its symmetrization (cid:101) γ ∈ Hom c (cid:0) ( L + [1]) ⊗ k , L ⊗ l − (cid:1) as the composition( L + [1]) ⊗ k → Sym k ( L + [1]) γ −→ Sym l ( L − ) → L ⊗ k − . Denote by S : L − → L + [1] the circle action map defined by S ( α ) = uB ( α )for α ∈ L − of the form α = α + α − u − + · · · . Note that S is even.We think of S as the operation of turning an output in L − to an input in L + [1], with an S twist. Combining this with composition of maps is the building block of the bracketoperations. There exists an odd operator F : L − → L + [1] of degree one whichis a bounding homotopy of S . That is, we have [ b + uB , F ] = S . Furthermore, it is compatible with the homotopy operator H constructed in Proposi-tion 4.5 in the sense that the following diagram is commutative L − ⊗ L − H (cid:45) K L − ⊗ Hom c ( L + [1] , K ) id ⊗ ι (cid:63) F ⊗ id (cid:45) L + [1] ⊗ Hom c ( L + [1] , K ) , ev (cid:54) where ev is the evaluation map and ι is the map discussed in (2.3). C˘ald˘araru and Tu
Proof.
We set the component F i , j of F that maps L · u − i to (cid:126) − L · u j + (for i ≥ j ≥
0) to be F i , j ( x · u − i ) = ( − j j (cid:88) l = R l T i + j + − l x · u j + . The above properties of F can be verified as in Proposition 4.5. The construction of the L ∞ morphism (cid:99) K : (cid:98) h → (cid:98) h triv is similar to the construc-tion of K in that it also involves the use of labeled graphs. However, as is evidentfrom the definition of (cid:98) h , we need to have extra information to distinguish inputs andoutputs on these graphs. The particular type of graphs that are relevant for us will becalled partially directed graphs , as defined below. A partially directed graph of type ( g , k , l ) is given by a quadruple ( G , L in G (cid:96) L out G , E dir , K ) consisting of the following data:– A labeled graph G of type ( g , k + l ) .– A decomposition L G = L in G (cid:97) L out G of the set of leaves L G such that | L in G | = k and | L out G | = l . Leaves in L in G will becalled incoming, while leaves in L out G will be called outgoing.– A subset E dir ⊂ E G of edges of G whose elements are called directed edges,and a direction is chosen on them. Edges in E G − E dir are called un-directed.– A spanning tree K ⊂ E dir of the graph G .Let us denote by G dir the graph obtained from G by removing all un-directed edges.Then we require the following properties to hold:– The directed graph G dir is connected.– Each vertex has at least one incoming half-edge.– For every e ∈ E dir there exists a non-empty, directed path in K joining the samevertices as e , in the same direction.A partially directed graph is called stable if the underlying labeled graph is. ategorical Enumerative Invariants, II: Givental formula To simply notation we sometimes use the nota-tion G to denote a partially directed graph ( G , L in G (cid:96) L out G , E dir , K ). An isomorphism ϕ : G → G of partially directed graphs is an isomorphism of the underlying labeledgraphs G and G such that the directions of leaves and edges are preserved, and suchthat ϕ ( K ) = K . Denote by Γ ( g , k , l ) the set of isomorphism classes of partiallydirected graphs of type ( g , k , l ) and by Γ (( g , k , l )) the subset of those that are stable.A marking of a partially directed graph G is a bijection f : { , · · · , m } → V G onto the set of vertices of G . We shall denote such a marked partially directedgraph by ( G , L in G (cid:96) L out G , E dir , K , f ), or sometimes simply ( G , f ). An isomorphism ofmarked partially directed graphs is an isomorphism of partially directed graphs thatalso preserves the marking maps. Denote by (cid:94) Γ ( g , k , l ) m the set of isomorphism classesof marked partially directed graphs with m vertices. Let G be a partially directed graph. The spanning tree K defines a partialordering on V G : we set v < w if there exists a non-empty directed path in K from v to w .The last condition in the definition of a partially directed graph can be rephrased assaying that an edge in E dir may only join a vertex v with one of its descendants in K .Another way of stating this is as follows. Define a new relation ≺ on V G by declaring x ≺ y if there exists a directed path in G dir from x to y . Then x ≺ y if and only if x < y . In particular ≺ is an order relation, i.e., G dir has no directed loops.Yet a third way to understand this condition is as follows. For e ∈ K define G / e tobe the result of deleting all directed edges in G which are parallel to e (have the samestarting and ending points as e ), and then contracting e . The condition is equivalent tothe fact that after contracting all of K (by successively contracting its edges as above)we are left with a graph with no directed loops. (cid:99) K m . We will define for each m ≥ (cid:99) K m : Sym m ( (cid:98) h [1]) → (cid:98) h triv [1] , in a way similar to the construction of K m in (4.8). The map (cid:99) K m will be the λ -linearextension of the sum over all partial directed graphs ( G , f ) with m vertices of maps (cid:99) K G , f taken with weights wt ( G , f ): (cid:99) K m = (cid:88) g , k , l ( G , f ) ∈ (cid:94) Γ ( g , k , l ) m wt ( G , f ) · (cid:99) K G , f : (cid:98) h [1] ⊗ m → (cid:98) h triv [1] C˘ald˘araru and Tu
The maps (cid:99) K G , f will be defined in (4.23), while the weight of ( G , f ) will be definedin (4.25). Let G ∈ (cid:94) Γ ( g , k , l ) m be a partially directed graph of type ( g , k , l ) endowed with amarking f . Let v ∈ V G . We define two integers k v ≥ l v ≥ v , according to the following rules:– Each directed half-edge of a directed edge e ∈ E dir is considered incom-ing/outgoing as indicated by the direction of e .– Each leaf at v is considered incoming/outgoing as indicated by its direction.– Each un-directed half-edge at v is considered outgoing.For the vertex v i marked by i ∈ { , . . . m } its genus label and numbers of inputs andoutputs will be denoted by g i , k i and l i , respectively.With this preparation we define the operator (cid:99) K G , f : m (cid:79) i = L k i , l i · (cid:126) g i → L k , l · (cid:126) g . It maps γ · (cid:126) g ⊗ · · · ⊗ γ m · (cid:126) g m to γ · (cid:126) g according to the following rules:(1) Each internal vertex v i is decorated by the symmetrization (cid:101) γ i of γ i . The resultsare tensored together in the order prescribed by the marking, yielding the product (cid:101) γ ⊗ · · · ⊗ (cid:101) γ m of symmetrized linear maps.(2) The above product of linear maps is composed according to the directed graph G dir , using the operators S and T of (4.16) and Proposition 4.17 to convertoutputs into inputs: T on the edges of K , S on the edges of E dir \ K . The resultis a single linear map f with ( (cid:80) k i ) − E dir inputs and ( (cid:80) l i ) − E dir outputs.The fact that ≺ is a partial order ensures that composition according to the “flow"of the graph G dir is well-defined. This is well-known in the literature, mainly asa tool to establish the bar construction for PROPs, see [EE05], [MV09], [Val07].(3) Pairs of outputs of f are contracted according to the undirected edges in E G \ E dir ,using the operator H sym of (4.6). ategorical Enumerative Invariants, II: Givental formula (4) The resulting homomorphism is regarded as a map γ ∈ Hom c (cid:0) Sym k ( L + [1]) , Sym l ( L − ) (cid:1) by symmetrizing the inputs and the outputs, respectively.The picture below illustrates how the compositions and contractions are performed.The levels are from the partial ordering of vertices, and the blue directed edges formthe spanning tree K ⊆ G dir . Formally we set (cid:99) K G , f ( γ · (cid:126) g , . . . , γ m · (cid:126) g m ) = π (cid:89) e ∈ E G \ E dir τ e (cid:89) e ∈ G dir τ e m (cid:89) i = (cid:101) γ i · (cid:126) g . Here π : Hom c (cid:0) L + [1] ⊗ k , L ⊗ l − (cid:1) → Hom c (cid:0) Sym k ( L + [1]) , Sym l L − (cid:1) is the natural projection map, and τ e is the contribution of edge e as above.We remark that the above expression is purely symbolic, and cannot be though of as away to compute the compositions successively, edge by edge. This would be possibleif E dir \ K were empty, in which case exactly one output of an operation would beplugged into an input of another at each step. However, edges in E dir \ K indicatefurther outputs that need to be plugged into inputs of operations that have already beencomposed; if we were dealing with finite dimensional spaces this would be a trace-likeoperation, but our spaces of inputs and outputs are infinite dimensional.Nonetheless one can see that the overall composition is an even operation, because each“step” above is even: for an edge e ∈ K τ e is an even operation h [1] ⊗ h [1] → h [1] C˘ald˘araru and Tu because it uses the odd map T in the middle; while the contractions corresponding toedges in E G \ K can be thought of as “maps” τ e : h [1] → h [1] using the even maps S or H sym . Let G = ( G , L in G (cid:96) L out G , E dir , K ) be a partially directedgraph. We shall now define its associated weight, wt ( G ) ∈ Q .To do this we first assign to each vertex v ∈ V G its number of inputs and outputs( k v , l v ). This gives a map v G : V G → N + × N which we call the vertex type of G . We define a new set PD ( G ) consisting of all thepartially directed graph structures on ( G , L in G (cid:96) L out G ) which have vertex type v G . Herea partially directed graph structure on ( G , L in G (cid:96) L out G ) means a partially directed graph G (cid:48) = ( G , L in G (cid:96) L out G , E dir , K ). We emphasize that we are not taking isomorphismsclasses of partially directed graphs in the above definition.The rational weight to G is defined by the formula wt ( G ) = | Aut ( G ) | · | PD ( G ) | . We shall also need the marked version analogue of the above definition. For a markedpartially directed graph ( G , f ) ∈ (cid:94) Γ ( g , k , l ) m , we define the set PD ( G , f ) as the set ofmarked partially directed graph structures on ( G , L in G (cid:96) L out G ) which have vertex type v G and marking f . The associated weight of ( G , f ) is given by wt ( G , f ) = | Aut ( G , f ) | · | PD ( G , f ) | . Note that the sets PD ( G ) and PD ( G , f ) are naturally isomorphic by simply keepingtrack of the marking f . Thus the two weights are related by wt ( G ) = | Aut ( G , f ) || Aut ( G ) | wt ( G , f ) . We illustrate these definitions with a few examples. Consider the followinggraph: ategorical Enumerative Invariants, II: Givental formula Assume there are d directed edges between the two vertices, with the spanning tree K given by the blue edge. The partially directed graph G has k = l = v , v are of types (1 , d ), ( d ,
0) respectively. The automorphismgroup
Aut ( G , f ) has order ( d − d − K .On the other hand, the set PD ( G ) has exactly d elements: all the d edges must pointfrom v to v to preserve the numerical type, but any of them can be chosen as thespanning tree K . Putting the two calculations together, the weight of G is given by wt ( G , f ) = / d !. Note that this is precisely the factor needed in the operator thathomotopy bounds the d -th Lie bracket d ! { γ , γ } d .As another example consider the marked graph ( G , L in (cid:96) L out , f ) with g = k = l = m =
3. .The possible partially directed structures on it are depicted below: .Since these partially directed graphs have no automorphisms, each of them has weight wt ( G , f ) = / By the same reasoning as in the proof of Part(A) the maps (cid:99) K m are symmetric. Moreover, we have argued in (4.24) that the maps (cid:99) K G , f are even. C˘ald˘araru and Tu
Next we need to prove that the maps { (cid:99) K m } m ≥ defined above form an L ∞ homomor-phism (cid:99) K : (cid:98) h → (cid:98) h triv . In other words we need to prove that for each m ≥ b + uB + ι, (cid:99) K m ]( γ · · · γ m ) = (cid:126) (cid:88) ≤ i ≤ m ( − (cid:15) i (cid:99) K m ( γ · · · ∆ γ i · · · γ m ) + (cid:88) i < j ( − (cid:15) ij (cid:99) K m − ( { γ i , γ j } (cid:126) · γ · · · (cid:98) γ i · · · (cid:98) γ j · · · γ m ) , where the signs are again given by the Koszul rule, (cid:15) i = | γ | + · · · + | γ i − | (cid:15) ij = | γ i | ( | γ | + · · · + | γ i − | ) + | γ j | ( | γ | + · · · + | γ i − | + | γ i + | + · · · + | γ j − | ) . We first consider the term [ b + uB , (cid:99) K m ]( γ · · · γ m ). By definition, we have (cid:99) K m = (cid:80) wt ( G , f ) · (cid:99) K G , f . Using the identities[ b + uB , H sym ] = Ω , [ b + uB , F ] = S , [ b + uB , S ] = b + uB to a loop edge; the result correspondsto a term (cid:126) (cid:99) K m ( γ · · · ∆ γ j · · · γ m ) on the right hand side.(2) the second type is obtained by applying b + uB to an edge in K ; the resultcorresponds to a term (cid:99) K m − ( { γ i , γ j } (cid:126) · γ · · · (cid:98) γ i · · · (cid:98) γ j · · · γ m ) on the right.(3) the third type is obtained by applying b + uB to an un-directed, non-loop edge;the result is a similar graph evaluation, but where the contraction operator H sym on that edge is changed to Ω .The cancellation of the terms of types (1) and (2) with the corresponding terms on theright hand side can be proved in the same way as in the proof of part (A). Thus thereare two types of terms left which need to cancel out: the terms of type (3), and theexpression [ ι, (cid:99) K m ]( γ · · · γ m ).Let us understand the terms in [ ι, (cid:99) K m ]( γ · · · γ m ) in more detail. Observe that the termsin ι (cid:99) K m ( γ · · · γ m ) are obtained by switching an output leaf to an input leaf. However,these are only part of the terms in (cid:80) mj = (cid:99) K m ( γ · · · ιγ j · · · γ m ). Indeed, the extra termsin the latter expression come from switching the in/out label of a half-edge that is partof an edge (i.e., not a leaf). Call e the edge whose end label is being switched. Thereare three possible cases: ategorical Enumerative Invariants, II: Givental formula – if the edge e is a loop, since we always consider half-edges of a loop as outgoing,switching either end of the loop results in an invalid graph (now we would havea directed cycle);– if the edge e is already a directed edge, then switching the outgoing half-edgeof e to “incoming” yields a directed edge with two ends that are both incoming,which is also an invalid configuration in a graph;– this is the essential case: if e is an un-directed edge which is not a loop, thenthere are two possible contributions illustrated in the following picture:By the commutativity of the diagram in Proposition 4.17, the resulting term inthis case precisely corresponds to a term of type (3).This finishes the proof of Part (B) of Theorem 4.2. We need to prove that (cid:99) K m ( ιγ · · · ιγ m ) = ι K m ( γ · · · γ m ) . We first claim that the left hand side contains no contributions from graphs with morethan one input. Indeed, assume that G ∈ Γ ( g , k , l ) is a labeled partially directed stablegraph with k ≥
2. The spanning tree K has | V G | − k + l directed leaves, of which k are incoming, which implies that it has at least k + | V G | − ≥ | V G | + | V G | manyinternal vertices. By the pigeonhole principle we conclude that at least one of thevertices must have more than one incoming half-edges. Since all the tensors ιγ i haveonly one input, the contribution from such a graph G must be zero. C˘ald˘araru and Tu
Thus the left hand side equals (cid:88) ( G , f ) | Aut ( G , f ) | · | PD ( G , f ) | (cid:99) K G , f ( ιγ · · · ιγ m )where the summation is over the set of marked, partially directed graphs ( G , f ) thathave only k = v the number of incoming halfedges k v is one. We need to compare this expression with (cid:88) ( G , f ) | Aut ( G , f ) | ι K G , f ( γ · · · γ m )We may rewrite this as (cid:88) ( G , f , l ) | Aut ( G , f , l ) | ι l K G , f ( γ · · · γ m ) , where we sum over isomorphism types of triples ( G , f , l ) where ( G , f ) is a markedgraph and l ∈ L G is a fixed leaf (output). In the summand we apply the operator ι onlyat the leaf l ; this operator is denoted by ι l . The notation Aut ( G , f , l ) stands for the setof automorphisms that also preserve the decomposition L G = { l } ∪ ( L G \ { l } ).By the definition of the operators K G , f and (cid:99) K G , f and the commutativity of the diagramin Proposition 4.17 the two expressions (cid:80) (cid:99) K G , f ( ιγ · · · ιγ m ) and (cid:80) ι l K G , f ( γ · · · γ m )contain the same terms. Thus it suffices to match up their coefficients; in other words,we need to show that the following identity holds for a fixed triple ( G , f , l ) (cid:88) ( G , f ) | Aut ( G , f ) | · | PD ( G , f ) | = | Aut ( G , f , l ) | . The summation on the left hand side is over isomorphism classes of marked partiallydirected graphs ( G , f ) whose underlying marked graph is ( G , f ), and where l is theonly incoming leaf (all others are outgoing) and the vertex type is (1 , l v ) at each vertex v . Note that all these graphs have the same PD ( G , f ); we will denote this set by PD ( G , f , l ).To show the above identify consider the action of Aut ( G , f , l ) on the set PD ( G , f , l ),and observe that its orbits are precisely the isomorphism classes of the marked partiallydirected graphs ( G , f ) that appear in the above summation. By the orbit-sum formulawe obtain that | PD ( G , f , l ) | = (cid:88) ( G , f ) | Aut ( G , f , l ) || Aut ( G , f ) | which is equivalent to the desired identity. ategorical Enumerative Invariants, II: Givental formula
5. Feynman sum formulas for the categorical invariants
In this section we use the L ∞ -morphisms of Theorem 4.2 to derive explicit formulas forthe categorical invariants F A , sg , n . These formulas are given as summations over partiallydirected stable graphs, with vertices labeled by the tensors (cid:98) β Ag , k , l . We also prove thatour invariants depend only on the Morita equivalence class of the pair ( A , s ). Let s : H → H ∗ ( L + ) be a splitting of the Hodge filtration,see (3.4). As in the previous section we fix a chain-level splitting R : ( L , b ) → ( L [[ u ]] , b + uB ) that lifts s , of the form R = id + R u + R u + · · · , R j ∈ End ( L ) . Denote by L Triv = ( L , b ) the same underlying chain complex as L , but endowed withtrivial circle action. The chain map R induces an isomorphism L Triv − → L − , whichfurther induces an isomorphism of DGLA’s (which we still denote by R ) R : h Triv = Sym ( L Triv − )[1][[ (cid:126) , λ ]] → h triv = Sym ( L − )[1][[ (cid:126) , λ ]] . Here the differential on the left hand side is just b while on the right hand side is b + uB ; they both have zero Lie bracket. As in the previous section we denote by T the inverse of R . It analogously induces an inverse isomorphism of DGLA’s T : h triv → h Triv . We also have the corresponding hat-version of the morphism (cid:98) T : (cid:98) h triv → (cid:98) h Triv . Consider the following commutative diagram of DGLA’s: h + K + −−−−→ h triv , + T −−−−→ h Triv , + ι (cid:121) ι (cid:121) ι (cid:121)(cid:98) h (cid:99) K −−−−→ (cid:98) h triv (cid:98) T −−−−→ (cid:98) h Triv . This induces a commutative diagram of isomorphisms of the associated Maurer-Cartanmoduli spaces: MC ( h + ) K + ∗ −−−−→ MC ( h triv , + ) T ∗ −−−−→ MC ( h Triv , + ) ι ∗ (cid:121) ι ∗ (cid:121) ι ∗ (cid:121) MC ( (cid:98) h ) (cid:99) K ∗ −−−−→ MC ( (cid:98) h triv ) (cid:98) T ∗ −−−−→ MC ( (cid:98) h Triv ) . C˘ald˘araru and Tu
Consider the two moduli spaces on the right. Since the Maurer-Cartan equation forboth h Triv , + and (cid:98) h Triv is the linear equation bx =
0, there are natural inclusions, where H = H ∗ ( L Triv ) = HH ∗ ( A ) MC ( h Triv , + ) ⊆ Sym H − [1][[ (cid:126) , λ ]] MC ( (cid:98) h Triv ) ⊆ (cid:77) k ≥ , l Hom c (cid:0) Sym k ( H + [1]) , Sym l H − (cid:1) [[ (cid:126) , λ ]] . The images of both inclusions are simply the odd homology groups of the correspond-ing spaces (because Maurer-Cartan elements are odd). The categorical enumerativeinvariant F A , s of Definition 3.9 lies inside the upper-right corner space MC ( h Triv , + ).The Maurer-Cartan element (cid:98) β A lies inside the lower-left corner MC ( (cid:98) h ). We have ι ∗ F A , s = (cid:98) T ∗ (cid:99) K ∗ (cid:98) β A . Before proving this result we need some preparation. In the previous diagram we haveby definition ι ∗ β A = (cid:98) β A , thus the commutativity of the diagram yields (cid:98) T ∗ (cid:99) K ∗ (cid:98) β A = (cid:98) T ∗ (cid:99) K ∗ ι ∗ β A = ι ∗ T ∗ K + ∗ β A . From this, it is clear that Theorem 5.3 would follow from the identity F A , s = T ∗ K + ∗ β A . The inverse of T is given by R , thus the above identity is equivalent to R ∗ F A , s = K + ∗ β A . A remarkable property of the construction of the L ∞ -morphism K is thatone can construct an inverse morphism K − in the same way as in the constructionof K (4.8) with the only change that edges in a labeled graph are all contracted by − H Sym instead of H Sym . Denote the resulting L ∞ morphism by K − . The followinglemma justifies this notation. We have K − ◦ K = id . Proof.
Observe that in the composition K − ◦ K we are summing over labeledgraphs with edges either decorated by − H Sym (from K − ) or by H Sym (from K ).This is equivalent to summing over labeled graphs with decorations of edges by (cid:0) H Sym + ( − H Sym ) (cid:1) , which of course is zero. Thus the only nonzero contributionto the composition K − ◦ K is from graphs with no edges at all, i.e., the star graphs.These yield the identity map on Maurer-Cartan spaces as discussed in (4.8). ategorical Enumerative Invariants, II: Givental formula The above discussion reduces the problem to proving thefollowing identity: K − ∗ R ∗ F A , s = β A . According to Definition 3.9, the Maurer-Cartan element F A , s is determined by theidentity Ψ s (cid:0) exp ( F A , s / (cid:126) ) (cid:1) = exp ( β A / (cid:126) ) . Using the Feynman rules (see [Giv01] and [Pan18]), the term β A = (cid:126) · ln Ψ s (cid:0) exp ( F A , s / (cid:126) ) (cid:1) is given by a stable graph sum with vertices labeled by F A , sg , n ’s, legs labeled by R ’s, andedge propagator given by Givental’s formula: Giv : L triv − ⊗ L triv − → K Giv ( u − i · x , u − j · y ) = j (cid:88) l = ( − j − l (cid:104) R i + j − l + ( x ) , R l ( y ) (cid:105) . (Note that we are re-indexing so that i , j ≥
0, rather than using the more classicalconvention that i , j ≥ K − ∗ R ∗ F A , s is also given by a sum over stablegraphs, with vertices labeled by F A , sg , n ’s, legs labeled by R ’s, and edge propagator givenby − H Sym ◦ ( R ⊗ R ) : L triv − ⊗ L triv − → K . Thus it remains to identify Givental’s propagator
Giv with − H Sym ◦ ( R ⊗ R ). Theelements F A , sg , n are Maurer-Cartan elements of h Triv so they are all b -closed. The resultthen follows from the following lemma. The two propagators above are equal when restricted to the subspace ( ker b ) − ⊂ L − . In other words we have Giv ( u − i · x , u − j · y ) = − H Sym ◦ (cid:0) R ( u − i · x ) ⊗ R ( u − j · y ) (cid:1) for x , y ∈ ker b . Proof.
By the symplectic property of the splitting s and the fact that R is a chain levellift of s , one can verify that Givental’s propagator Giv is symmetric when restricted to( ker b ) − . Thus it suffices to prove that we have Giv ( u − i · x , u − j · y ) = − H (cid:0) R ( u − i · x ) , R ( u − j · y ) (cid:1) for x , y ∈ ker b . C˘ald˘araru and Tu
Next we explicitly calculate the right hand side. Take two elements u − i · x and u − j · y in L triv − with i , j ≥
0. We have R ( u − i · x ) = i (cid:88) k = u − i + k R k ( x ) , R ( u − j · y ) = j (cid:88) l = u − j + l R l ( y ) . Plugging in the formula for H from Proposition 4.5 yields H (cid:0) R ( u − i · x ) , R ( u − j · y ) (cid:1) = H (cid:0) i (cid:88) k = u − i + k R k ( x ) , j (cid:88) l = u − j + l R l ( y ) (cid:1) = i (cid:88) k = j (cid:88) l = H (cid:0) u − i + k R k ( x ) , u − j + l R l ( y ) (cid:1) = i (cid:88) k = j (cid:88) l = ( − j − l (cid:104) j − l (cid:88) r = R r T i + j − k − l + − r R k ( x ) , R l ( y ) (cid:105) = j (cid:88) r = j − r (cid:88) l = ( − j − l (cid:104) R r i (cid:88) k = T i + j − k − l + − r R k ( x ) , R l ( y ) (cid:105) = j (cid:88) r = j − r (cid:88) l = ( − j − l + (cid:104) R r i + j − l − r + (cid:88) k = i + T i + j − k − l + − r R k ( x ) , R l ( y ) (cid:105) = j (cid:88) l = ( − j − l + i + j − l − (cid:88) k = i + (cid:104) i + j − k − l + (cid:88) r = R r T i + j − k − l + − r R k ( x ) , R l ( y ) (cid:105) . The crucial step is the fifth equality above, which uses the fact that for n ≥ (cid:80) k T k R n − k =
0. We use this in the form i (cid:88) k = T i + j − k − l + − r R k = − i + j − l − r + (cid:88) k = i + T i + j − k − l + − r R k . Using the fact that R ◦ T = id it follows that the summation over r in the last expressionis zero unless r = k = i + j − l +
1, in which case it is R k ( x ). We conclude that H (cid:0) R ( u − i · x ) , R ( u − j · y ) (cid:1) = j (cid:88) l = ( − j − l + (cid:104) R i + j − l + ( x ) , R l ( y ) (cid:105) = − Giv ( u − i · x , u − j · y ) . This completes the proof. ategorical Enumerative Invariants, II: Givental formula Theorem 5.3 yields an explicit formula for the categorical enumerative invariants F A , sg , n as a Feynman sum over partially directed stable graphs, using the formula of (cid:99) K . The following formula holds for any g ≥ , n ≥ such that g − + n > : ι F A , sg , n = (cid:88) G ∈ Γ (( g , , n − wt ( G ) (cid:89) e ∈ E G Cont ( e ) (cid:89) v ∈ V G Cont ( v ) (cid:89) l ∈ L G Cont ( l ) The contributions of vertices, edges and legs are as follows:(i) Vertices are decorated by tensors in (cid:98) β A . More precisely, a vertex v is dec-orated with the tensor (cid:98) β Ag ( v ) , k ( v ) , l ( v ) , where the genus g ( v ) and the number ofincoming/outgoing half-edges k ( v ) and l ( v ) of v are as defined in (4.23).(ii) Incoming leaves are decorated by R , outgoing leaves are decorated by T .(iii) Edges are decorated by the contraction operators τ e defined in (4.24). Proof.
By Theorem 5.3 the left hand side equals (cid:98) T (cid:16) (cid:88) m ≥ m ! (cid:99) K m ( (cid:98) β A , . . . , (cid:98) β A ) (cid:17) , n − = (cid:98) T (cid:16) (cid:88) m ≥ m ! (cid:88) ( G , f ) ∈ (cid:94) Γ ( g , , n − m wt ( G , f ) · (cid:99) K G , f ( (cid:98) β A , . . . , (cid:98) β A ) (cid:17) . The fact that we apply (cid:98) T explains the leg contribution as in (ii), and the constructionof (cid:99) K ( G , f ) explains the contributions from vertices and edges as in (i) and (iii). We onlyneed to match up the coefficients given by graph weights.Consider the forgetful map π : (cid:94) Γ ( g , , n − m → Γ ( g , , n − m that forgets themarking f on the set of vertices. Fix a partially directed graph G ∈ Γ ( g , , n − m .Denote the set of all possible markings of G by Mark ( G ), The group Aut ( G ) acts on Mark ( G ). Observe that we have π − ( G ) ∼ = Mark ( G ) / Aut ( G ) . Furthermore, the stabilizer of this action is exactly given by
Aut ( G , f ) for each markedpartially directed graph ( G , f ). This implies that the size of the orbit containing( G , f ) is | Aut ( G ) | / | Aut ( G , f ) | , which by the formula at the end of (4.25) equals C˘ald˘araru and Tu wt ( G , f ) / wt ( G ). We thus obtain1 m ! (cid:88) ( G , f ) ∈ (cid:94) Γ ( g , , n − m wt ( G , f ) · (cid:99) K ( G , f ) ( (cid:98) β A , . . . , (cid:98) β A ) == m ! (cid:88) G ∈ Γ ( g , , n − m (cid:88) ( G , f ) ∈ π − ( G ) wt ( G , f ) · (cid:99) K ( G , f ) ( (cid:98) β A , . . . , (cid:98) β A ) = m ! (cid:88) G ∈ Γ ( g , , n − m (cid:88) f ∈ Mark ( G ) wt ( G , f ) · | Aut ( G , f ) || Aut ( G ) | (cid:99) K ( G , f ) ( (cid:98) β A , . . . , (cid:98) β A ) = (cid:88) G ∈ Γ ( g , , n − m wt ( G ) · (cid:99) K G ( (cid:98) β A , . . . , (cid:98) β A ) . In the last equality we use the fact that the inputs (cid:98) β A are invariant under permutation.There are exactly m ! markings in the set Mark ( G ), and this cancels the coefficient1 / m !. Here the notation (cid:99) K G means any (cid:99) K ( G , f ) , since the result is independent of themarking f . Finally, we also observe that since the input tensor (cid:98) β A is only non-zerofor stable triples ( g , k , l ), the terms in the summation are non-zero only when thecorresponding graph is stable. Let C be a cyclic A ∞ -category. We assume that C satisfiesthe categorical version of condition ( † ) and that it is compactly generated: there exists asplit generator E of C . These conditions are satisfied, for example, for C = D b coh ( X ),the derived category of coherent sheaves on a smooth projective Calabi-Yau variety.Given the data of such a pair ( C , E ) and a choice of splitting s : H ∗ ( L C ) → H ∗ ( L C + )(see Definition 3.4) we can define enumerative invariants in two steps: first, we replacethe category C by the cyclic A ∞ algebra A E = End C ( E ) , and then we compute categorical enumerative invariants of ( A E , s ). (Note that thesplitting s induces a splitting, also denoted by s , for the Hodge filtration of A E . )Ideally, the above construction should not depend on the choice of E : if this were thecase, we could then define F C , sg , n = F A E , sg , n for some choice of generator E . Since the algebras A E and A F are Morita equivalentfor different generators E , F of C , the problem reduces to the problem of showingthat the invariants we defined are constant under Morita equivalences. ategorical Enumerative Invariants, II: Givental formula Let F be another generator. We then obtain inclusions A E (cid:44) → A E (cid:96) F ← (cid:45) A F . We then reduce the problem of proving Morita invariance to the problem of showingthat F A E , sg , n = F A E (cid:96) F , sg , n .To see this, observe that these invariants depend on two parts of the data:– The tensors (cid:98) β g , k , l ’s obtained from the map ρ , see (2.14).– The chain-level splitting R : L → L + that lifts the splitting s .To prove that F A E , sg , n = F A E (cid:96) F , sg , n it suffices to match up the above two pieces of data.For the first part, note that the restriction of ρ A E (cid:96) F to the subspaces Sym k ( L A E + [1]) (forvarious k ≥
1) equals to ρ A E . As for the second part, let us start with a chain-levelsplitting for E , a chain map R : ( L A E , b ) → ( L A E + , b + uB ) . Since we are over a field we can extend it to obtain a chain-level splitting (cid:101) R : ( L A E (cid:96) F , b ) → ( L A E (cid:96) F + , b + uB ) . This matches the second part: the restriction of the map (cid:101) R to the subspace L A E ⊂ L A E (cid:96) F is given by R . The formula in Corollary 5.7 then shows that indeed we have F A E , sg , n = F A E (cid:96) F , sg , n . A. Explicit formulas
In this appendix we list explicit formulas for categorical invariants of Euler character-istic χ ≥ − G , L in G (cid:96) L out G , E dir , K ) are as follows:– we shall omit the genus decoration of a vertex if it is clear from the combinatoricsof the graph;– we shall omit the drawing of the spanning tree K ⊂ E dir if there is a uniquechoice of it; otherwise, the spanning tree K will be drawn in blue;– when drawing ribbon graphs vertices decorated by u will not be marked;– the orientation of ribbon graphs is the one described in [CC20-1]. C˘ald˘araru and Tu
A.1. The formula for the (0 , , -component. We begin with the case when g = n =
3. In this case there is a unique partially directed stable graph. Thus we have ι ∗ F A , s , = is due to the automorphism that switches the two outputs in the stablegraph. Its vertex is decorated by the tensor (cid:98) β A , , = ρ A ( (cid:98) V , , ) = − ρ A (cid:0) (cid:1) using the action ρ A on the first combinatorial string vertex (cid:98) V , , , see [CCT20]. Notethat the latter “T"-shaped graph is a ribbon graph, not be confused with the first graphwhich is a (partially directed) stable graph. (The negative sign appears due to ourchoice of orientation of ribbon graphs.) A.2. The formula for the (1 , , -component. In this case there are two stablegraphs. We have ι F A , s , is given byFor the first graph the unique vertex is decorated by the image under ρ A of the com-binatorial string vertex (cid:98) V , , . It was computed in [CT17] and it is explicitly given bythe following linear combination of ribbon graphs: (cid:98) V comb , , = − u − + . A.3. The formula for the (0 , , -component. In this case we have ι ∗ F A , s , is equalto ategorical Enumerative Invariants, II: Givental formula Note that the coefficient disappears due to the symmetry of the two outgoing leaveson the right hand side of the stable graph. The combinatorial string vertex (cid:98) V comb , , iscomputed explicitly in [CCT20] and it is given by (cid:98) V comb , , = − − + u − + u − A.4. The formula for the (1 , , -component. In this case we get ι ∗ F A , s , = • g = + • + • • g = ++ •• + •• + • Observe that in the first graph of the second line, there are 2 directed edges betweenthe two vertices. This explains how the tensors (cid:98) β Ag , k , l with k ≥ A.5. The formula for the (0 , , -component. In this case we have ι ∗ F A , s , is givenby C˘ald˘araru and Tu
A.6. The formula for the (1 , , -component. In this case, we have ι ∗ F A , s , is givenbyNote that in the above graphs the genus decoration is also omitted since in this caseit is evident from the graph itself. For example, in the first graph there is a uniquevertex of genus 1. In the third graph the genus 1 decoration is forced on the left vertex,otherwise it would not be stable. A.7. A partial formula for the (2 , , -component. We list a few formulas in ι ∗ F A , s , according to the number of edges in stable graphs. There is a unique star graph in Γ ((2 , , ategorical Enumerative Invariants, II: Givental formula The terms with two edges are
B. The integer-graded case
So far we have assumed that all our vector spaces were Z / Z -graded, and we didnot concern ourselves with even graded shifts (as these do not affect signs). In thisappendix we will sketch how our results need to be modified in the Z -graded case, sothat all the maps involved will be of a well-defined homogeneous degree. This allowsus to obtain an analogue of the dimension axiom of Gromov-Witten theory in this case. B.1.
The choices we make are as follows. The formal variables u and λ have homo-logical degree −
2. If the Calabi-Yau degree (i.e., the homological degree of the cyclicpairing) of the algebra A is d , then (cid:126) has degree − + d . When the variable (cid:126) is usedin the context of chains on moduli spaces of curves, d will be assumed to be zero (sincethe chain level operator giving the Mukai pairing has degree zero), so deg (cid:126) = − (cid:104) − , − (cid:105) Muk : C ∗ ( A ) ⊗ C ∗ ( A ) → K . Thus when viewed as an operator on L = C ∗ ( A )[ d ] the Mukai pairing has degree − d . B.2.
With the above conventions the Weyl algebra is Z -graded, because the generatorsof the ideal defining it are of homogeneous degree 2 d −
2. This is the reason we choosethe definition of L − to start with u and not u − , and we take the residue pairing totake the coefficient of u and not u − . B.3.
Taking into account the shifts in the definition of (cid:98) h in [CCT20] we can rewrite itsdefinition as (cid:98) h = (cid:77) k ≥ , l Hom c (cid:0) Sym k ( L + [1 − d ]) , Sym l L − (cid:1) [2 − d ][[ (cid:126) , λ ]] . C˘ald˘araru and Tu
Now assume that the splitting s , its chain-level lift R , and the inverse T of R allpreserve degrees. This implies that– H Sym : Sym L − → K has degree 2 − d in Proposition 4.5;– S : L − → L + [1 − d ] has degree 2 − d in Proposition 4.17;– F : L − → L + [1 − d ] has degree 3 − d in Proposition 4.17.Using these facts it is easy to verify that the maps (cid:99) K m have degree zero, as desired. B.4.
We can now use Theorem 5.3 to prove that the categorical enumerative invariants F A , sg , n satisfy the dimension axiom of Gromov-Witten theory when the A ∞ -algebra is Z -graded. More precisely we have the following. B.5. Theorem.
Assume that A is Z -graded, of Calabi-Yau dimension d , and assumethat the splitting s preserves degrees. Then deg F A , sg , n = g − − d ) + n as an element of Sym n H − . Proof.
The element (cid:98) V = (cid:88) g , k ≥ , l (cid:98) V g , k , l (cid:126) g λ g − + k + l is a solution of the Maurer-Cartan equation in (cid:98) g , so in particular its degree is −
1. Thusthe degree of its component (cid:98) V g , k , l of genus g , k inputs and l outputs is deg (cid:98) V g , k , l = g − + k + l . Its image under ρ A will have degree deg (cid:98) β Ag , k , l = (6 g − + k + l ) + d (2 − g − k ) . A simple inspection shows that in (cid:98) h (with the further shift by 2 − d + dk − k , andwith (cid:126) of degree 2 d −
2) the element (cid:98) β Ag , k , l (cid:126) g λ g − + k + l has degree − (cid:98) β A is a Maurer-Cartan element in the Z -graded DGLA (cid:98) h . Thesplitting s is degree preserving, so we can find a chain-level lift of it R that alsopreserves degrees. This implies that the inverse T of R also preserves degrees, andhence the L ∞ morphism (cid:99) K has degree zero.Theorem 5.3 implies that F A , s = (cid:88) g , n F A , sg , n (cid:126) g λ g − + n , ategorical Enumerative Invariants, II: Givental formula as a Maurer-Cartan element of h Triv = Sym H − [[ (cid:126) , λ ]][1 − d ], also has degree − F A , sg , n as an element of Sym n H − its degree is given by deg ( F A , sg , n ) = ( − + (2 d − − deg ( (cid:126) ) · g − deg ( λ ) · (2 g − + n ) = (2 d − − (2 d − g + g − + n ) = g − − d ) + n . This indeed matches with the virtual dimension formula in Gromov-Witten theory.
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