Formulae in noncommutative Hodge theory

Mirror symmetry predicts the existence of certain ‘mirror’ pairs of Calabi–Yau Kähler manifolds, $X$ and $Y$, so that the Gromov–Witten invariants of $X$ can be extracted from certain Hodge-theoretic invariants of $Y$.¹¹1We only consider genus-zero Gromov–Witten invariants in this paper. The first thrilling application of mirror symmetry was the prediction of the number of rational curves, in all degrees, on the quintic threefold $X$, in terms of the Hodge theory of a mirror manifold $Y$ [CdlOGP91]. This prediction for the quintic, together with many more examples of mirror symmetry, was later mathematically verified [Giv96, LLY97].

The most conceptually satisfying way of formulating the mirror relationship between numerical invariants of $X$ and $Y$ is as an isomorphism of variations of Hodge structures ($\mathsf{VHS}$) [Mor93]. A $\mathsf{VHS}$ over a complex manifold $\mathcal{M}$ consists of a holomorphic vector bundle $V\to\mathcal{M}$, equipped with a filtration $F^{\geq*}V$ by holomorphic subbundles, and a flat connection $\nabla$ satisfying the condition $\nabla_{v}F^{\geq i}V\subset F^{\geq i-1}V$ known as Griffiths transverality.²²2This is more precisely referred to as a complex variation of Hodge structures. We introduce the Kähler moduli space $\mathcal{M}_{K\ddot{a}h}(X)$, which parametrizes deformations of the Kähler form on $X$, and the complex moduli space $\mathcal{M}_{cpx}(Y)$, which parametrizes deformations of the complex structure on $Y$. The Gromov–Witten theory of $X$ gets packaged into the $A$-model $\mathsf{VHS}$, which lives over the Kähler moduli space: $V^{A}(X)\to\mathcal{M}_{K\ddot{a}h}(X)$. The Hodge theory on deformations of $Y$ gets packaged into the $B$-model $\mathsf{VHS}$, which lives over the complex moduli space: $V^{B}(Y)\to\mathcal{M}_{cpx}(Y)$. Hodge-theoretic mirror symmetry then predicts an isomorphism $V^{A}(X)\cong V^{B}(Y)$, covering an isomorphism $\mathcal{M}_{K\ddot{a}h}(X)\cong\mathcal{M}_{cpx}(Y)$ called the mirror map. Enumerative mirror symmetry, i.e., the explicit formulae relating the numerical invariants, can be deduced from this isomorphism of $\mathsf{VHS}$ (see, e.g., [CK99]).

Kontsevich proposed a generalization of Hodge-theoretic mirror symmetry called homological mirror symmetry [Kon95]. It predicts a quasi-equivalence of $A_{\infty}$ categories $\EuF(X)\simeq\EuD(Y)$, where $\EuF(X)$ is the split-closed derived Fukaya category of $X$, and $\EuD(Y)$ is a $\mathsf{dg}$ enhancement of the bounded derived category of coherent sheaves on $Y$. More precisely, one should think of these as families of categories, parametrized by the Kähler and complex moduli spaces respectively; and the quasi-equivalence matches the Fukaya category living over a point in Kähler moduli space with the derived category living over the corresponding point in complex moduli space, where the correspondence between moduli spaces is given by the same mirror map as before.

Kontsevich also predicted that homological should imply Hodge-theoretic mirror symmetry. This was subsequently made more precise [Bar02, KKP08, Cos09]. The expectation is that the cyclic homology of a family of saturated $A_{\infty}$ categories carries the structure of a $\mathsf{VHS}$; the cyclic homology of the Fukaya category is isomorphic to $V^{A}(X)$; and the cyclic homology of the bounded derived category of coherent sheaves is isomorphic to $V^{B}(Y)$. Therefore, the equivalence of categories implies the isomorphism of $\mathsf{VHS}$.

This paper forms part of a project of the author, joint with Ganatra and Perutz, to carry out this program. Theorem 1.4 implies roughly that the cyclic homology of a family of saturated $\Z$-graded $A_{\infty}$ categories carries the structure of a $\mathsf{VHS}$, and this structure is functorial under $A_{\infty}$ functors; we also give explicit formulae for all of the relevant structures. Ganatra [Gana] has defined a map from the cyclic homology of the Fukaya category to the $A$-model $\mathsf{VHS}$; this map is shown to respect the $\mathsf{VHS}$ structure in [GPSa], using the formulae in the present paper; and the map is shown to be an isomorphism in [GPSb]. The corresponding comparison theorem for the $B$-model is known to experts, although not everything is written in the literature; modulo this $B$-model comparison theorem, the proof is complete.

We refer to [GPSb] for precise statements of the results. One corollary of them is a new proof of the mirror symmetry predictions for Gromov–Witten invariants of the quintic, as a consequence of the proof of homological mirror symmetry for the quintic [She15]. The $B$-model comparison theorem has been established in this specific case by Tu [Tu].

Although mirror symmetry was originally formulated for mirror pairs of Calabi–Yau Kähler manifolds $(X,Y)$, it admits a generalization in which $X$ is allowed to be Fano. In this case the mirror is no longer a manifold, but rather a ‘Landau–Ginzburg model’ $(Y,W)$, which means a variety $Y$ equipped with a function $W:Y\to\C$. The Gromov–Witten invariants of $X$ are related to the singularity theory of $W$ [Giv95].

Once again, the relation between the numerical invariants can be expressed in terms of an isomorphism $V^{A}(X)\cong V^{B}(Y,W)$; however, the structures getting identified are now variations of semi-infinite Hodge structures ($\mathsf{VSHS}$). This notion was introduced by Barannikov [Bar01], but the study of the $B$-model $\mathsf{VSHS}$ associated to $(Y,W)$ goes back to Saito [Sai83].

In Section 2 we define the notion of a graded $\mathsf{VSHS}$. The following notions are equivalent:

	$\Z/2$-graded $\mathsf{VSHS}$	$\displaystyle\longleftrightarrow\quad\text{$\mathsf{VSHS}${} in Barannikov's sense;}$
	$\Z$-graded $\mathsf{VSHS}$	$\displaystyle\longleftrightarrow\quad\text{$\mathsf{VHS}${} in the sense of the previous section.}$

We refer to [GPSb, Lemma 2.7] for a precise statement and proof of the latter equivalence, which is a version of the ‘Rees correspondence’ between filtered bundles over $\mathcal{M}$ and equivariant bundles over $\mathcal{M}\times\mathbb{A}^{1}$ [Sim97]. Thus, Hodge-theoretic mirror symmetry can always be formulated as an isomorphism of graded $\mathsf{VSHS}$; in the Fano case the grading group is $\Z/2$, and in the Calabi–Yau case it is $\Z$. For the rest of the paper, ‘$\mathsf{VSHS}$’ will always mean ‘graded $\mathsf{VSHS}$’, with the grading group implicit.

Homological mirror symmetry admits a generalization to the Fano case: roughly, it predicts a quasi-equivalence of $\Z/2$-graded $A_{\infty}$ categories $\EuD\EuF(X)\simeq MF(Y,W)$, where the latter is the category of matrix factorizations of $W$ [Orl04]. One expects that it should imply Hodge-theoretic mirror symmetry, by a similar argument to the Calabi–Yau case, but with $\mathsf{VHS}$ replaced by $\mathsf{VSHS}$.

Our Theorem 1.4 implies roughly that the cyclic homology of a family of saturated $Y$-graded $A_{\infty}$ categories, which satisfies the degeneration property, carries the structure of a $Y$-graded $\mathsf{VSHS}$. The case $Y=\Z/2$ is the one relevant to Fano mirror symmetry, and the case $Y=\Z$ is the one relevant to Calabi–Yau mirror symmetry. In the latter case the degeneration property holds automatically, by Kaledin’s proof [Kal17] of Kontsevich–Soibelman’s degeneration conjecture [KS08, Conjecture 9.1.2]. In particular, our result implies that the cyclic homology of a family of saturated $\Z$-graded $A_{\infty}$ categories carries the structure of a $\Z$-graded $\mathsf{VSHS}$, which we recall is equivalent to a $\mathsf{VHS}$ (we stated this version of the result in the previous section). We remark that the grading group does not enter into the proof of Theorem 1.4: the cases of relevance to Fano and Calabi–Yau mirror symmetry are handled uniformly.

{rmk}

It should be possible to prove some version of the statement that homological implies Hodge-theoretic mirror symmetry in the Fano case, following the argument in the Calabi–Yau case (and in particular, using the $\Z/2$-graded case of our Theorem 1.4). See [AT] for some recent progress.

Let $\Bbbk\subset\BbK$ be fields. We will write $\mathcal{M}:=\spec\BbK$, and $T\mathcal{M}:=\deriv_{\Bbbk}\BbK$.

We fix a grading group throughout: more precisely, we fix a ‘grading datum’ in the sense of [She15], which is an abelian group $Y$ together with homomorphisms $\Z\to Y\to\Z/2$ whose composition is non-zero. All of our structures are $Y$-graded; when we talk about an element of degree $k\in\Z$, we really mean its degree is the image of $k$ under the map $\Z\to Y$; and when we write a Koszul-type sign $(-1)^{|a|}$, it means that the image of the $Y$-degree of $a$, under the map $Y\to\Z/2$, is $|a|\in\Z/2$.

{rmk}

The two most relevant grading data are $\Z:=\{\Z\xrightarrow{\id}\Z\to\Z/2\}$ and $\Z/2:=\{\Z\to\Z/2\xrightarrow{\id}\Z/2\}$; working with the former is equivalent to working with ordinary $\Z$-gradings, while working with the latter is equivalent to working with ordinary $\Z/2$-gradings.

We define $\BbK[\![u]\!]$ to be the graded ring of formal power series in a formal variable $u$ of degree $2$, and $\BbK(\!(u)\!)$ the graded ring of formal Laurent series. For any $f\in\BbK[\![u]\!]$ or $\BbK(\!(u)\!)$, we denote

$$f^{\star}(u):=f(-u).$$

{rmk}

To be precise, $\BbK[\![u]\!]$ (respectively, $\BbK(\!(u)\!)$) is the degreewise completion of $\BbK[u]$ (respectively, $\BbK[u,u^{-1}]$) with respect to the $u$-adic filtration. If the morphism $\Z\to Y$ is not injective then this includes infinite sums of powers of $u$, but if the morphism is injective then the completion has no effect because all powers of $u$ have different degrees. Thus $\BbK(\!(u)\!)$ contains ‘semi-infinite’ sums of powers of $u$ in the $\Z/2$-graded case; hence Barannikov’s terminology.

Finally, if $\sigma\in\Z/2$, we denote $\sigma^{\prime}:=\sigma-1$.

We define various flavours of $\mathsf{VSHS}$ in Section 2. To give an idea of what they mean, let us explain roughly what they correspond to under the Rees correspondence, in the $\Z$-graded case:

• •

  An unpolarized pre-$\mathsf{VSHS}$ over $\mathcal{M}$ corresponds to an $\mathcal{O}_{\mathcal{M}}$-module equipped with a filtration $F^{\geq*}$ and flat connection $\nabla$ satisfying Griffiths transversality.

• •

  An unpolarized $\mathsf{VSHS}$ is an unpolarized pre-$\mathsf{VSHS}$ such that the $\mathcal{O}_{\mathcal{M}}$-module is a finite-rank vector bundle.

• •

  A polarization for an unpolarized pre-$\mathsf{VSHS}$ is a covariantly constant pairing $(\cdot,\cdot)$ such that $(F^{\geq j},F^{\geq k})=0$ for $j+k<0$, and $(a,b)=(-1)^{n}(b,a)$ for some $n\in\Z/2$ called the weight; a polarized pre-$\mathsf{VSHS}$ is a pre-$\mathsf{VSHS}$ equipped with a polarization.

• •

  A polarized $\mathsf{VSHS}$ is a polarized pre-$\mathsf{VSHS}$, such that the $\mathcal{O}_{\mathcal{M}}$-module is a finite-rank vector bundle, and the pairing is non-degenerate.

{rmk}

Note that a polarized/unpolarized $\mathsf{VSHS}$ is the same thing as a polarized/unpolarized pre-$\mathsf{VSHS}$ satisfying additional properties, rather than equipped with additional data.

Our main results concern the Hochschild invariants of $A_{\infty}$ categories:

{main}

Let $\EuC$ be a $\BbK$-linear graded $A_{\infty}$ category. Then:

1. 1.

   Its negative cyclic homology $\operatorname{\mathsf{HC}}_{\bullet}^{-}(\EuC)$, endowed with the Getzler–Gauss–Manin connection $\nabla$ [Get93], carries the structure of an unpolarized pre-$\mathsf{VSHS}$ over $\mathcal{M}$.

2. 2.

   If $\EuC$ is proper and admits an $n$-dimensional weak proper Calabi–Yau structure (see Section 5.7 for the definition), then $(\operatorname{\mathsf{HC}}_{\bullet}^{-}(\EuC),\nabla)$ admits a natural polarization $\langle\cdot,\cdot\rangle_{res}$ of weight $n$, given by Shklyarov’s higher residue pairing [Shk16].

3. 3.

   These structures are Morita invariant.

4. 4.

   If $\EuC$ is saturated and its noncommutative Hodge-to-de Rham spectral sequence degenerates, then the polarized pre-$\mathsf{VSHS}$ $(\operatorname{\mathsf{HC}}_{\bullet}^{-}(\EuC),\nabla,\langle\cdot,\cdot\rangle_{res})$ is in fact a polarized $\mathsf{VSHS}$.

{rmk}

A conjecture of Kontsevich and Soibelmann [KS08, Conjecture 9.1.2] says that the noncommutative Hodge-to-de Rham spectral sequence degenerates for any saturated $\EuC$. The conjecture has been proved by Kaledin in the case that $\EuC$ is $\Z$-graded [Kal17] (see also [Mat]); so far as the author is aware it remains open if $\EuC$ is, for example, $\Z/2$-graded.

{rmk}

Katzarkov–Kontsevich–Pantev conjecture that the $\mathsf{VSHS}$ of Theorem 1.4 can be endowed with a natural $\Q$-structure (see [KKP08, Section 2.2.6], and also [Bla16]).

Let us comment on the originality of Theorem 1.4. We believe our contribution ranges from ‘writing down explicit formulae for known structures with uniform sign conventions, as a handy reference’ at the low end, to ‘checking that these structures have certain natural (but slightly tricky-to-prove) compatibilities’ at the high end. To be precise:

• •

  Our proof of the Morita invariance of the Getzler–Gauss–Manin connection is new.

• •

  Shklyarov’s construction of the higher residue pairing for $\mathsf{dg}$ categories [Shk16] immediately gives the construction for $A_{\infty}$ categories, because any $A_{\infty}$ category is quasi-equivalent to a $\mathsf{dg}$ category via the Yoneda embedding. On the other hand, for an $A_{\infty}$ category whose morphism spaces are finite-dimensional on the chain level, work of Costello [Cos07] and Konstevich–Soibelman [KS08] implies that there should be an explicit formula for the pairing. We write down the formula and prove that it is equivalent to Shklyarov’s definition (see Proposition 64). Our proof is motivated by Shklyarov’s [Shk16, Proposition 2.6].

• •

  Using the explicit formula for the higher residue pairing, we establish that it is covariantly constant with respect to the Getzler–Gauss–Manin connection: we believe that this result is also new (a related result was proven by Shklyarov in [Shk16], but that was for a different version of Getzler’s connection, namely the one in the $u$-direction rather than in the direction of the base).

Now we give a guide, to help the reader find the proofs of the different parts of Theorem 1.4. Part (1) is proved in Section 3. Part (2) is proved in two parts: covariant constancy of the higher residue pairing, with respect to the Getzler–Gauss–Manin connection, is proved in Corollary 67; and symmetry of the higher residue pairing (which is the only part that requires the weak proper Calabi–Yau structure) is proved in Lemma 70. The tools to prove part (3) are developed in Section 4; Morita invariance of negative cyclic homology and the Getzler–Gauss–Manin connection are proved in Corollary 33, and Morita invariance of the higher residue pairing is proved in Proposition 62. Part (4) is proved in Theorem 72.

The paper involves a lot of long formulae composing multilinear operations in complicated ways, with non-trivial sign factors. We explain a graphical notation for these composition rules in Appendix C, which allows one to check various identities, with signs, in an efficient way; and we draw the graphical notation for some of the trickiest signs that appear in the paper. We omit the proofs of some identities that become trivial using the graphical notation, however we have tried very hard to write down explicitly the correct signs for every operation we define.

{rmk}

It is not usually assumed that a polarization must have degree $0$: we prefer to shift whatever pre-$\mathsf{VSHS}$ we are considering so that this is the case. In this paper, polarizations will arise from Shklyarov’s higher residue pairing on cyclic homology, which has degree $0$ with respect to the standard grading.

A morphism of $\mathsf{VSHS}$ (polarized or unpolarized) is the same thing as a morphism of the underlying pre-$\mathsf{VSHS}$.

{rmk}

What we call a ‘$\Z/2$-graded polarized $\mathsf{VSHS}$’ is usually simply called a $\mathsf{VSHS}$, in particular, the polarization is part of the structure. However, the notion of an unpolarized $\mathsf{VSHS}$ has applications in mirror symmetry so it seems useful to make the distinction.

{rmk}

An unpolarized $\Z$-graded $\mathsf{VSHS}$ is equivalent to a vector bundle over the $\Bbbk$-scheme $\mathcal{M}$, equipped with a filtration and flat connection satisfying Griffiths transversality [GPSb, Lemma 2.7]; in the application to mirror symmetry we take $\Bbbk=\C$. The most relevant choice of $\BbK$ for the application to mirror symmetry is $\BbK=\C\laurent{q}$. One can think of $\mathcal{M}=\spec\BbK$ as a ‘formal punctured disc’. Geometrically, one thinks of this formal punctured disc as mapping into the Kähler/complex moduli space, with the limit $q\to 0$ corresponding to a ‘large volume’/‘large complex structure’ limit point of the moduli space. We consider the pullback of the relevant structures ($\mathsf{VSHS}$, categories) to $\mathcal{M}$. We shall define a ‘family of $A_{\infty}$ categories parametrized by $\mathcal{M}$’ to be a $\BbK$-linear $A_{\infty}$ category. Thus Theorem 1.4 constructs a $\mathsf{VSHS}$ over $\mathcal{M}$ from a family of $A_{\infty}$ categories parametrized by $\mathcal{M}$.

We have assumed that our $\mathsf{VSHS}$ are graded, in the sense that $\EuE$ is a direct sum of its graded pieces. A different notion of ‘graded $\mathsf{VSHS}$’ is used in the literature [Bar01, CIT09], which we will instead refer to as an ‘Euler-graded $\mathsf{VSHS}$’. We depart from the standard terminology in this way to avoid confusion regarding the usual terminology for $A_{\infty}$ categories. Namely, if $\EuC$ is a graded $A_{\infty}$ category (in the usual sense), which is saturated and whose noncommutative Hodge-to-de Rham spectral sequence degenerates, then Theorem 1.4 constructs a graded $\mathsf{VSHS}$ in our sense; it does not construct an Euler-graded $\mathsf{VSHS}$.

{rmk}

For any Euler-graded pre-$\mathsf{VSHS}$, we can extend the connection $\nabla$ to a flat connection that is also defined in the $u$-direction, by setting $\nabla_{u\frac{\partial}{\partial u}}:=\frac{1}{2}\mathsf{Gr}-\nabla_{E}$.

{rmk}

One can also extract an Euler graded $\mathsf{VSHS}$ from an $A_{\infty}$ category $\EuC$, if one assumes that the category itself comes with an ‘Euler grading’. Namely, one assumes that there is an Euler vector field $E$ on the coefficient field $\BbK$, and that $\EuC$ is a $\Z/2$-graded $\BbK$-linear $A_{\infty}$ category. Then an Euler grading on $\EuC$ is a map $\mathsf{Gr}$ on the morphism spaces of the category, compatible with the Euler vector field as in Definition 6, and such that the $A_{\infty}$ structure maps $\mu^{s}$ satisfy

$$\mathsf{Gr}\circ\mu^{s}=\mu^{s}\circ\mathsf{Gr}+(2-s)\cdot\mu^{s}.$$

Our proof of Theorem 1.4 applies to the $\Z/2$-graded $A_{\infty}$ category $\EuC$, to produce $\Z/2$-graded Hochschild invariants: and one easily checks that, if $\EuC$ comes with an Euler grading, then all of the Hochschild invariants admit Euler gradings, compatible with all structures. This is relevant when one studies mirror symmetry for Fano varieties, but not in the Calabi–Yau case. We will not comment further on it in this paper.

We follow the sign conventions of [Get93] and [Sei08a]. A pre-$A_{\infty}$ category $\EuC$ consists of a set of objects, and a graded $\BbK$-vector space $hom^{\bullet}_{\EuC}(X,Y)$ for each pair of objects $X$, $Y$.

We define the convenient notation

$$\EuC(X_{0},\ldots,X_{s}):=hom^{\bullet}(X_{0},X_{1})[1]\otimes\ldots\otimes hom^{\bullet}(X_{s-1},X_{s})[1].$$

For a generator $a_{1}\otimes\ldots\otimes a_{s}$ of $\EuC(X_{0},\ldots,X_{s})$, we define

$$\epsilon_{j}:=|a_{1}|^{\prime}+\ldots+|a_{j}|^{\prime}\in\Z/2.$$

We define the Hochschild cochains of length $s$:

$$CC^{\bullet}(\EuC)^{s}:=\prod_{X_{0},\ldots,X_{s}}\Hom^{\bullet}(\EuC(X_{0},\ldots,X_{s}),\EuC(X_{0},X_{s}))[-1].$$

We then define the Hochschild cochain complex

$$CC^{\bullet}(\EuC):=\prod_{s\geq 0}CC^{\bullet}(\EuC)^{s}$$

(more precisely, the completion of the direct sum in the category of graded vector spaces, with respect to the filtration by length $s$). It admits the Gerstenhaber product:

$$\varphi\circ\psi(a_{1},\ldots,a_{s}):=\sum_{j,k}(-1)^{|\psi|^{\prime}\cdot\epsilon_{j}}\varphi^{*}(a_{1},\ldots,\psi^{*}(a_{j+1},\ldots),a_{k+1},\ldots).$$

An $A_{\infty}$ structure on $\EuC$ is an element $\mu^{*}\in CC^{2}(\EuC)$ satisfying $\mu^{*}\circ\mu^{*}=0$ and $\mu^{0}=0$.

The cohomology category is a graded $\BbK$-linear category $H^{\bullet}(\EuC)$ with the same objects,

	$\displaystyle\mathrm{Hom}^{\bullet}(X,Y)$	$\displaystyle:=H^{\bullet}(hom^{\bullet}(X,Y),\mu^{1}),$
	$\displaystyle[a_{1}]\cdot[a_{2}]$	$\displaystyle=(-1)^{|a_{2}|}\mu^{2}(a_{2},a_{1}).$

We will assume that our $A_{\infty}$ categories are cohomologically unital, i.e., $\mathrm{Hom}^{\bullet}(X,Y)$ admits units.

We recall that an $A_{\infty}$ functor $F:\EuC\to\EuD$ consists of a map on the level of objects, together with maps

$$F^{s}:\EuC(X_{0},\ldots,X_{s})\to\EuD(FX_{0},FX_{s})$$

satisfying

(4)

$$\sum\mu^{*}(F^{*}(a_{1},\ldots),F^{*}(a_{j_{1}+1},\ldots),\ldots,F^{*}(a_{j_{k}+1},\ldots,a_{s}))=\\ \sum(-1)^{\epsilon_{j}}F^{*}(a_{1},\ldots,\mu^{*}(a_{j+1},\ldots),\ldots,a_{s}).$$

{rmk}

If $e\in hom^{0}_{\EuC}(X,X)$ is a cohomological unit in $\EuC$, then $-e\in hom^{0}_{\EuC^{op}}(X,X)=hom^{0}_{\EuC}(X,X)$ is a cohomological unit in $\EuC^{op}$.

{rmk}

This is a different definition of the opposite $A_{\infty}$ category from that given in [Sei08b, Section 1a] (i.e., the two definitions give non-equivalent $A_{\infty}$ categories in general). It was verified in [She, Appendix B] that there is an isomorphism $\EuF(X,\omega)^{op}\cong\EuF(X,-\omega)$ for the exact Fukaya category $\EuF$. We take this as evidence that this definition of the opposite category is most relevant for Fukaya categories. We thank Seidel for drawing our attention to this difference.

{rmk}

Definitions 9, 8 and 10 are compatible, but in a slightly non-trivial way: given a $\mathsf{dg}$ category $\EuC$, there is a strict isomorphism of $A_{\infty}$ categories $A_{\infty}(\EuC^{op})\cong\left(A_{\infty}(\EuC)\right)^{op}$ which sends $x\mapsto-x$ for all morphisms $x$.

We define the Gerstenhaber bracket on $CC^{\bullet}(\EuC)$:

$$[\varphi,\psi]:=\varphi\circ\psi-(-1)^{|\varphi|^{\prime}\cdot|\psi|^{\prime}}\psi\circ\varphi.$$

It is a graded Lie bracket. We define the Hochschild differential $M^{1}:CC^{\bullet}(\EuC)\to CC^{\bullet}(\EuC)[1]$ by $M^{1}:=[\mu^{*},-]$. Because $[\mu^{*},\mu^{*}]=0$, $M^{1}$ is a differential, i.e., $(M^{1})^{2}=0$. Its cohomology is called the Hochschild cohomology, $\HH^{\bullet}(\EuC)$.

For $p\geq 2$, we define $M^{p}\in CC^{2}(CC^{\bullet}(\EuC),CC^{\bullet}(\EuC))^{p}$ by

(5)		$\displaystyle M^{p}(\varphi_{1},\ldots,\varphi_{p})(a_{1},\ldots,a_{s})$	$\displaystyle:=\sum(-1)^{\dagger}\mu^{*}(a_{1},\ldots,\varphi_{1}^{*}(a_{j_{1}+1},\ldots),\ldots,\varphi_{p}^{*}(a_{j_{p}+1},\ldots),\ldots,a_{s}),$
	$\displaystyle\text{ where }\quad\dagger$	$\displaystyle=\sum_{i=1}^{p}|\varphi_{i}|^{\prime}\cdot\epsilon_{j_{i}}.$

By [Get93, Proposition 1.7], the operations $M^{*}\in CC^{2}(CC^{\bullet}(\EuC))$ define an $A_{\infty}$ structure on $CC^{\bullet}(\EuC)$. In particular, the Yoneda product on $\HH^{\bullet}(\EuC)$, defined on the cochain level by

(6)

$$\varphi\cup\psi:=(-1)^{|\psi|}M^{2}(\psi,\varphi),$$

makes $\HH^{\bullet}(\EuC)$ into a graded associative algebra. Together with the Gerstenhaber bracket, this makes Hochschild cohomology into a Gerstenhaber algebra.

{rmk}

We have an isomorphism of $A_{\infty}$ algebras $CC^{\bullet}(\EuC)\to CC^{\bullet}(\EuC^{op})^{op}$, sending $\eta\mapsto\eta_{op}$.

We now define the Kodaira–Spencer map, which is closely related to the Kaledin class [Kal07, Lun10]. We make a choice of $\BbK$-basis for each morphism space $hom^{\bullet}_{\EuC}(X,Y)$. We write each $A_{\infty}$ structure map $\mu^{*}$ in this basis, as a matrix with entries in $\BbK$. We obtain a Hochschild cochain $v(\mu^{*})\in CC^{2}(\EuC)$ by acting with the derivation $v$ on the entries of the matrix for $\mu^{*}$. This Hochschild cochain is closed (as one sees by applying $v$ to the $A_{\infty}$ equations for $\EuC$), so we may define

	$\displaystyle\mathsf{KS}:\deriv_{\Bbbk}\BbK$	$\displaystyle\to\HH^{2}(\EuC),$
	$\displaystyle\mathsf{KS}(v)$	$\displaystyle:=[v(\mu^{*})].$

We will prove in Corollary 30 that the Kodaira–Spencer map is independent of the choice of $\BbK$-bases for the morphisms spaces.

We define the Hochschild chain complex

$$CC_{\bullet}(\EuC):=\bigoplus_{X_{0},\ldots,X_{s}}\EuC(X_{0},\ldots,X_{s},X_{0})[-1].$$

We denote generators by $a_{0}[a_{1}|\ldots|a_{s}]:=a_{0}\otimes\ldots\otimes a_{s}$. For such a generator, we define the sign

$$\varepsilon_{j}:=|a_{0}|^{\prime}+|a_{1}|^{\prime}+\ldots+|a_{j}|^{\prime}$$

(the only difference between $\epsilon_{j}$ and $\varepsilon_{j}$ is that the former starts at $1$, the latter starts at $0$).

We define an operation $t$ on $CC_{\bullet}(\EuC)$ by

$$t(a_{0}[a_{1}|\ldots|a_{s}]):=(-1)^{|a_{s}|^{\prime}\cdot\varepsilon_{s-1}}a_{s}[a_{0}|\ldots|a_{s-1}].$$

We define the Hochschild differential $b:CC_{\bullet}(\EuC)\to CC_{\bullet}(\EuC)[1]$ by

(7)

$$b(a_{0}[a_{1}|\ldots|a_{s}]):=\sum_{j}\mu^{*}(\overbrace{a_{0},\ldots,a_{j}})[a_{j+1}|\ldots|a_{s}]+\sum_{j,k}(-1)^{\varepsilon_{j}}a_{0}[\ldots|\mu^{*}(a_{j+1},\ldots,a_{k})|\ldots|a_{s}].$$

It is a differential, and its cohomology is called the Hochschild homology, $\HH_{\bullet}(\EuC)$.

{rmk}

The convention for Hochschild homology of a $\mathsf{dg}$ category $\EuC$ (see, e.g., [Shk12, Equation (2.1)]) coincides with ours, i.e., there is a natural identification of cochain complexes

$$(C_{\bullet}(\EuC),b)=(CC_{\bullet}(A_{\infty}(\EuC)),b).$$

{rmk}

Definition 12 is compatible with the corresponding map for $\mathsf{dg}$ categories, in the sense that for any $\mathsf{dg}$ category $\EuC$, the following diagram commutes up to an overall sign $-1$:

Here, the horizontal maps are the tautological identifications (see Remark 3.4), the left vertical arrow is the map defined, e.g., in [Shk12, Proposition 4.5], and the right vertical arrow is the map defined in Definition 12, composed with the isomorphism $CC_{\bullet}(A_{\infty}(\EuC)^{op})\cong CC_{\bullet}(A_{\infty}(\EuC^{op}))$ induced by the isomorphism of Remark 3.2.

For $p\geq 1$, we define the operations⁴⁴4 Getzler denotes $b^{p|1}$ by $b\{-,\ldots,-\}$.

$$b^{p|1}:CC^{\bullet}(\EuC)^{\otimes p}\otimes CC_{\bullet}(\EuC)\to CC_{\bullet}(\EuC)[1-p]$$

(8)

$$b^{p|1}(\varphi_{1},\ldots,\varphi_{p}|a_{0},\ldots,a_{s}):=\\ \sum(-1)^{\dagger}\mu^{*}(a_{0},\ldots,\varphi_{1}^{*}(a_{j_{1}+1},\ldots),\ldots,\varphi_{p}^{*}(a_{j_{p}+1},\ldots),\overbrace{\ldots,a_{k}})[\ldots|a_{s}],\quad\text{ where}$$

$$\dagger=\sum_{i=1}^{p}|\varphi_{i}|^{\prime}\cdot\varepsilon_{j_{i}}.$$