The Bogomolov-Tian-Todorov Theorem Of Cyclic $A_\infty$-Algebras

TL;DR

This paper proves a non-commutative analogue of the Bogomolov-Tian-Todorov theorem for cyclic $A__$-algebras, showing smoothness of certain deformation functors under specific conditions.

Contribution

It establishes the smoothness of deformation functors for cyclic $A__$-algebras satisfying the Hodge-to-de Rham degeneration, extending classical deformation theory results.

Findings

Deformation functor of Hochschild cochains is smooth.

Deformation functor of cyclic Hochschild cochains is smooth.

Results apply to finite-dimensional smooth unital cyclic $A__$-algebras.

Abstract

Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of $A$ is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of $A$ is also smooth.

Equations

$${\mathfrak{g}}_X:=\underset{p,q}{⨁}{\mathcal{A}}^{0,q}({\Lambda }^p{T}_X)$$

$$[\alpha ,\beta ]=\Delta (\alpha \wedge \beta )-\Delta (\alpha )\wedge \beta -(-1{)}^{|\alpha |}\alpha \wedge \Delta (\beta ).$$

$$~{}C^{{\sf comb}}_{*}(FD){\rightarrow}{\mathbb{E}{{\sf nd}}}\big{(}C^{-*}(A)\big{)}.$$

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$$h(m_k)$$

$$h(m_{1,k})$$

$$h(m_{p,k})$$

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$$\mathcal{F}$$

$$C_{\ast }^{\mathsf{comb}}(FD)$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}\to \mathsf{Lie}\left\{1\right\}\to BV.$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}\to \mathcal{F}\to \mathcal{X}\to C_{\ast }^{\mathsf{comb}}(FD).$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}\to \mathcal{F}\to B_{\infty }\to C_{\ast }^{\mathsf{comb}}(FD),$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}\to \mathsf{Lie}\left\{1\right\}\to B_{\infty }\to C_{\ast }^{\mathsf{comb}}(FD).$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}$$

$$~{}\rho:{\sf BV}_{\infty}{\rightarrow}\mathcal{X}{\rightarrow}C^{{\sf comb}}_{*}(FD){\rightarrow}{\mathbb{E}{{\sf nd}}}\big{(}C^{-*}(A)\big{)}.$$

$${\mathsf{Lie}}_{\infty }\left\{1\right\}$$

$$\left\{l_k^d|k\ge 1,d\ge 0\right\}$$

$$l_1^0=\delta ,l_2^0=[-,-{]}_G,l_k^0=0\forall k\ge 3,$$

$$l_1^1=\Delta ,l_k^1({\alpha }_1\cdots {\alpha }_k)=0,\forall {\alpha }_1,\cdots ,{\alpha }_k\in \mathsf{Ker}\Delta ,\forall k\ge 2,$$

$$l_k^d=0,\forall d\ge 2.$$

$$\cdots$$

$$\Omega B{C}^{-\ast }(A)\cong C^{-\ast }(A),$$

$$\big{(}sC^{-*}(A)[[u]],\;\delta+u\Delta,\;[-,-]_{G}+u\cdot l^{1}_{2},\;u\cdot l^{1}_{k}\;(k\geq 3)\big{)}$$

$$\iota :s{C}^{\lambda }(A)\hookrightarrow s{C}^{-\ast }(A)[[u]]$$

$$l_k^1(\iota {\alpha }_1,\cdots ,\iota {\alpha }_k)=0,\forall {\alpha }_1,\cdots ,{\alpha }_k\in s{C}^{\lambda }(A),$$

$$\pi \circ \iota :s{C}^{\lambda }(A)\hookrightarrow s{C}^{-\ast }(A)$$

Full text

Institute of Mathematical Sciences, ShanghaiTech University, Shanghai, China, 201210. E-mail: tujw.at.shanghaitech.edu.cn

1. The non-commutative Bogomolov-Tian-Todorov Theorem

Let $X$ be a Calabi-Yau manifold, i.e. a compact complex manifold with trivial canonical bundle. It is a classical result of Bogomolov-Tian-Todorov [11][12] that the formal deformation functor associated with the differential graded Lie algebra (DGLA)

[TABLE]

of the Dolbeault resolution of holomorphic poly-vector fields is smooth. In order to prove this, the key observation was the existence of a BV operator $\Delta:\Lambda^{*}T_{X}{\rightarrow}\Lambda^{*-1}T_{X}$, which “trivializes” the Lie bracket by the Tian-Todorov identity

[TABLE]

With the above formula, the smoothness of the deformation functor follows easily from the classical $\partial\overline{\partial}$-Lemma in Hodge theory.

Following Kontsevich-Soibelman [8] and Katzarkov-Kontsevich-Pantev [7], one can formulate the compactness, smoothness, and the Calabi-Yau property purely in terms of the differential graded category of coherent sheaves on $X$. Thus, a natural question is whether the analogues of the Bogomolov-Tian-Todorov’s theorem holds for any smooth and proper Calabi-Yau categories. This question might have been a folklore theorem for experts in the field. The purpose of this note is to fill some of the missing details in the literature.

A large class of dg categories of interests is compactly generated by a single object. For this reason, instead of considering formal deformations of dg categories (whatever that means), we shall consider deformations of $A_{\infty}$-algebras which is also much more tractable. To state the non-commutative version of the Bogomolov-Tian-Todorov theorem precisely, we first fix some notations and conventions. Throughout the paper, we use the homological degree of chain complexes. If $A$ is a chain complex, its suspension is denoted by $sA$ with $(sA)_{n}:=A_{n-1}$. For a unital $A_{\infty}$-algebra $A$, denote by $C^{-*}(A)$ ($C_{*}(A)$) its reduced Hochschild cochain complex (chain complex respectively). The minus sign is due to that we use homological degree. Let $A$ be a cyclic unital $A_{\infty}$-algebra, denote by $C^{\lambda}(A)\subset C^{-*}(A)$ the sub-complex consisting of cyclic cochains with respect to the pairing on $A$.

Theorem 1.1**.**

Let $A$ be a $\mathbb{Z}/2\mathbb{Z}$-graded, finite-dimensional smooth unital cyclic $A_{\infty}$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. Then we have

• (A.)

The deformation functor ${{\sf Def}}_{sC^{-*}(A)}$ associated with the DGLA $sC^{-*}(A)$ of Hochschild cochains is smooth.

• (B.)

The deformation functor ${{\sf Def}}_{sC^{\lambda}(A)}$ associated with the DGLA $sC^{\lambda}(A)$ of cyclic Hochschild cochains is also smooth.

• (C.)

The natural transformation $f:{{\sf Def}}_{sC^{\lambda}(A)}{\rightarrow}{{\sf Def}}_{sC^{-*}(A)}$ associated with the canonical inclusion map $sC^{\lambda}(A)\hookrightarrow sC^{-*}(A)$ is smooth. In particular, every deformation of the $A_{\infty}$ structure of $A$ lifts to a deformation of the cyclic $A_{\infty}$ structure of $A$.

Remark 1.1**.**

The assumption of the Hodge-to-de-Rham degeneration property automatically holds for any $\mathbb{Z}$-graded smooth and proper $A_{\infty}$-algebra by Kaledin [6]. In the general $\mathbb{Z}/2\mathbb{Z}$-graded case, this remains an open conjecture by [8] [7]. Part $(A.)$ of the above Theorem was proved by Isamu Iwanari [5] with a different method.

2. ${\sf BV}_{\infty}$-algebra structure on $C^{-*}(A)$

To prove the above Theorem 1.1, one follows the same idea as in the proof of Bogomolov-Tian-Todorov’s Theorem. However, the key identity Equation 1 fails to hold. It only holds up to homotopy. Also, the cup product on $C^{-*}(A)$ is only commutative up to homotopy. Thus it is natural to work with a homotopy version of the underlying algebraic structures.

In this section, we first exhibit a homotopy BV algebra structure, or ${\sf BV}_{\infty}$-algebra structure on $C^{-*}(A)$. The definition of ${\sf BV}_{\infty}$-algebras used in this paper is from the article [3]. In fact, it was argued in Loc. Cit. that combining a TCFT structure defined by [1] [8] and the formality of the operad ${\sf BV}$, one easily deduces the existence of a ${\sf BV}_{\infty}$-algebra structure on $C^{-*}(A)$ with $A$ as in Theorem 1.1. However, to make such structure useful in order to deduce Theorem 1.1, one needs to say a bit more about this ${\sf BV}_{\infty}$ structure. For example, its underlying ${\sf Lie}_{\infty}$ algebra is in fact given by the differential graded Lie algebra \big{(}C^{-*}(A),\delta,[-,-]_{G}\big{)}. For this purpose, we need to use a construction of Tamarkin in his proof of the Deligne’s conjecture [10]. We introduce the following notations:

• •

${\sf Lie}$ — The Lie operad.

• •

${\sf Lie}_{\infty}$ — The homotopy Lie operad.

• •

$E_{2}$ — The operad whose representation gives Gerstenhaber algebras.

• •

$G_{\infty}$ — The homotopy $E_{2}$ operad.

• •

${\sf BV}$ — The operad whose representation gives BV algebras.

• •

${\sf BV}_{\infty}$ — The homotopy BV operad.

• •

$B_{\infty}$ — The brace operad [4, Section 5.2].

• •

$\mathcal{F}$ — The operad defined by Tamarkin in [10, Section 6].

• •

$C^{{\sf comb}}_{*}(FD)$ — The operad of black-and-white ribbon trees defined by Kontsevich-Soibelman [8, Section 11.6], see also Wahl-Westerland [14, Section 2]. This operad gives a combinatorial model for the framed little disk operad.

• •

${{\mathcal{X}}}$ — A cofibrant replacement of $C^{{\sf comb}}_{*}(FD)$.

For an operad $\mathcal{O}$, denote by $\mathcal{O}\left\{1\right\}$ its shifted version so that an $\mathcal{O}\left\{1\right\}$-algebra structure on a chain complex $A$ is equivalent to an $\mathcal{O}$-algebra structure on $sA$. The endomorphism operad of a chain complex is denoted by ${\mathbb{E}{{\sf nd}}}(-)$.

The starting point to construct a ${\sf BV}_{\infty}$ structure on $C^{-*}(A)$ is that the operad $C^{{\sf comb}}_{*}(FD)$ naturally acts on $C^{-*}(A)$:

[TABLE]

We refer to Kontsevich-Soibelman [8] and Wahl-Westerland [14] for details of this action. Here we illustrate this action with a few examples. Indeed, the following black-and-white ribbon tree

[TABLE]

inside $C^{{\sf comb}}_{*}(FD)(1)$ gives the pull-back of the Connes operator $B:C_{*}(A){\rightarrow}C_{*}(A)$ under the isomorphism C^{-*}(A)\cong\big{(}C_{*}(A)\big{)}^{\vee}. Denote it by $\Delta:C^{-*}(A){\rightarrow}C^{-*}(A)$. There are also binary operators associated with trees in $C^{{\sf comb}}_{*}(FD)(2)$. For example, consider the following two black-and-white ribbon trees:

[TABLE]

The graph $T$ gives the familiar cup product on $C^{-*}(A)$, while the graph $R$ gives the first brace operator $\circ$ on $C^{-*}(A)$ whose commutator (after shift) is the Gerstenhaber Lie bracket $[-,-]_{G}$.

Lemma 2.1**.**

There is a morphism of operad $h:B_{\infty}{\rightarrow}C^{{\sf comb}}_{*}(FD)$ defined by

[TABLE]

Here $T_{k}$ and $W_{k}$ are given by the following black-and-white ribbon trees:

[TABLE]

Proof.

This is a straight-forward check.∎

In [10], Tamarkin constructed morphisms $s:G_{\infty}{\rightarrow}\mathcal{F}$ and $t:\mathcal{F}{\rightarrow}B_{\infty}$.

Lemma 2.2**.**

There exists a commutative diagram:

[TABLE]

Proof.

This follows the lifting property since ${\mathcal{X}}{\rightarrow}C^{{\sf comb}}_{*}(FD)$ is a trivial fibration, while $\mathcal{F}$ by construction is cofibrant. ∎

The framed little disk operad is known to be formal with cohomology the ${\sf BV}$ operad, which implies that ${\sf BV}\cong C^{{\sf comb}}_{*}(FD)$ in the homotopy category of differential graded operads. Since the operad ${\mathcal{X}}$ is cofibrant, and ${\sf BV}$ is fibrant (as any dg operad is fibrant), we obtain a morphism ${\mathcal{X}}{\rightarrow}{\sf BV}$ such that the roof diagram

[TABLE]

represents an isomorphism in the homotopy category of differential graded operads.

Lemma 2.3**.**

The following diagram is commutative:

[TABLE]

Proof.

It is clear that the left composition factors as

[TABLE]

For the right composition, consider the following composition

[TABLE]

By Lemma 2.2 above, it is equal to

[TABLE]

which by [10] can be factored as

[TABLE]

This shows that both compositions vanish on generators $l_{k}$ of ${\sf Lie}_{\infty}\left\{1\right\}$ with $k\geq 3$. For $k=2$, this is a direct check by definition.∎

By definition of ${\mathcal{X}}$, the right vertical map in the above diagram is a trivial fibration since ${\sf BV}$ is minimal. The left vertical map is a cofibration. Thus by the lifting property, we obtain a map ${\sf BV}_{\infty}{\rightarrow}{\mathcal{X}}$:

[TABLE]

We define a ${\sf BV}_{\infty}$-algebra structure on $C^{-*}(A)$ via the composition:

[TABLE]

Corollary 2.1**.**

Let $C^{-*}(A)$ be endowed with the $BV_{\infty}$-algebra structure defined by the Equation 3. Then its underlying ${\sf Lie}_{\infty}$ structure on the suspension of $C^{-*}(A)$ is given by the differential graded Lie algebra \big{(}sC^{-*}(A),\delta,[-,-]_{G}\big{)}.

Proof.

This follows from Lemma 2.2 and Tamarkin’s commutative diagram:

[TABLE]

∎

Consider a subset of generating operators of a ${\sf BV}_{\infty}$ structure given by

[TABLE]

which label a basis of the convolution between the Koszul dual cooperad of ${\sf Lie}$ and that of the operad generated by the circle operator $\Delta$. We denote the sub-operad generated by $l_{k}^{d}$ in ${\sf BV}_{\infty}$ by ${{\sf q}}{\sf Lie}_{\infty}\left\{1\right\}$. The notation is because that a ${{\sf q}}{\sf Lie}_{\infty}\left\{1\right\}$-algebra structure on a chain complex $V$ is equivalent to an ${\sf Lie}_{\infty}$ structure on $sV[[u]]$ (with $u$ a degree $2$ formal variable), which may be thought of as a “quantum” ${\sf Lie}_{\infty}\left\{1\right\}$ structure.

Corollary 2.2**.**

The induced ${{\sf q}}{\sf Lie}_{\infty}\left\{1\right\}$ structure on $C^{-*}(A)$ is of the form

[TABLE]

Proof.

Property $(3)$ is proven in the previous Corollary 2.1. To prove Property $(4)$ and $(5)$, observe that the degree of the operator $l_{k}^{d}$ is equal to $2d+2k-3$. But, the top dimensional chains in the chain complex $C^{{\sf comb}}_{*}(FD)(k)$ is equal to $2k-1$, which implies the vanishing in $(5)$. The operator $l_{1}^{1}=\Delta$, since the black-and-white graph

[TABLE]

is the unique degree one graph representing the fundamental class in the homology group H_{1}\big{(}C^{{\sf comb}}_{*}(FD)(1)\big{)}\cong H_{1}(\mathbb{S}^{1}). For $k\geq 2$, since the degree of the operator $l_{k}^{1}$ is $2k-1$ which is top dimensional in the chain complex $C^{*}{{\sf comb}}_{*}(FD)(k)$. Thus $l_{k}^{1}$ is a linear combination of operators coming from black-and-white ribbon trees with $k$ white vertices, and of degree $2k-1$. One can show (easy combinatorics) that any such tree must contain at least one white vertex of the form:

[TABLE]

If we input any $\alpha\in{{\sf Ker}}\Delta$ at this type of white vertex, the result gives zero by definition of the action of ribbon trees on $C^{-*}(A)$ (by Equation 2).∎

3. Proof of Theorem 1.1

As explained in the previous section, via the composition in Equation 3, we have a ${\sf BV}_{\infty}$ structure on $C^{-*}(A)$. Thus, we may form its bar-cobar resolution:

[TABLE]

which yields a differential graded BV algebra homotopy equivalent to $C^{-*}(A)$. At this point, we use the following theorem due to Katzarkov-Kontsevich-Pantev [7] and Terilla [13].

Theorem 3.1**.**

Let $S$ be a differential graded BV algebra. Assume that the spectral sequence associated with the complex \big{(}S[[u]],d+u\Delta\big{)} endowed with the $u$-filtration is degenerate at the first page. Then both the DGLA’s \big{(}S,d,[-,-]\big{)} and \big{(}S[[u]],d+u\Delta,[-,-]\big{)} are homotopy abelian.

Since the degeneration of the spectral sequence is a homotopy invariant property (see [2]), we may use the above theorem to deduce the homotopy abelian property of the DGLA $sC^{-*}(A)$. Note that here it is essential that the ${\sf BV}_{\infty}$ structure on $C^{-*}(A)$ extends the DGLA structure of $sC^{-*}(A)$ by Corollary 2.1.

Similarly, we may also restrict the ${\sf BV}_{\infty}$ structure on $C^{-*}(A)$ to the sub-operad ${{\sf q}}{\sf Lie}_{\infty}\left\{1\right\}$. Then theorem above implies that the following ${\sf Lie}_{\infty}$-algebra

[TABLE]

is homotopy abelian. Here the structure maps $l^{1}_{k}$ are as in Corollary 2.2. The following lemma then finishes the proof of $(B.)$ in Theorem 1.1, using the homotopy invariance of deformation functors.

Lemma 3.1**.**

The canonical inclusion map

[TABLE]

is a quasi-isomorphism of ${\sf Lie}_{\infty}$-algebras.

Proof.

This inclusion is a quasi-isomorphism is a classical result, see for example [9]. By Corollary 2.2, the higher brackets $l^{1}_{k}$ vanish on elements inside ${{\sf Ker}}\Delta$ while we certainly have that ${{\sf Im}}\,\iota\subset{{\sf Ker}}\Delta$. This shows that

[TABLE]

implies that $\iota$ is a morphism of ${\sf Lie}_{\infty}$-algebras.∎

Denote by $\pi:sC^{-*}(A)[[u]]{\rightarrow}sC^{-*}(A)$ the projection map defined by setting $u=0$. Part $(C.)$ of Theorem 1.1 easily follows from $(A.)$ and $(B.)$. Indeed, since both functors are smooth, it suffices to check that the inclusion map

[TABLE]

induces a surjection on the tangent space of the associated deformation functors. This is clear as $\iota$ is a quasi-isomorphism, and $\pi$ is a surjective on cohomology by the Hodge-to-de-Rham degeneration assumption.