Dg Manifolds, Formal Exponential Maps and Homotopy Lie Algebras

Abstract

This paper is devoted to the study of the relation between ‘formal exponential maps,’ the Atiyah class, and Kapranov $L_\infty [1]$ algebras associated with dg manifolds in the $C^\infty$ context. We prove that, for a dg manifold, a ‘formal exponential map’ exists if and only if the Atiyah class vanishes. Inspired by Kapranov’s construction of a homotopy Lie algebra associated with the holomorphic tangent bundle of a complex manifold, we prove that the space of vector fields on a dg manifold admits an $L_\infty [1]$ algebra structure, unique up to isomorphism, whose unary bracket is the Lie derivative with respect to the homological vector field, whose binary bracket is a 1-cocycle representative of the Atiyah class, and whose higher multibrackets can be computed by a recursive formula. For the dg manifold $(T^{0,1}_X[1],{\bar{\partial }})$ arising from a complex manifold X, we prove that this $L_\infty [1]$ algebra structure is quasi-isomorphic to the standard $L_\infty [1]$ algebra structure on the Dolbeault complex $\Omega ^{0,\bullet }(T^{1,0}_X)$.

Appendix A. Fedosov Construction for Graded Manifolds

This section is to give a brief description of the Fedosov construction for graded manifolds. We refer readers to [18, 19, 32] for more details.

Throughout this section, ${\mathcal {M}}$ is a finite-dimensional graded manifold and $\nabla$ is a torsion-free affine connection on ${\mathcal {M}}$. By abuse of notation, the induced linear connection on ${\widehat{S}}(T_{{\mathcal {M}}}^{\vee })$ is denoted by the same symbol. The associated covariant derivative is written $d^\nabla :\Omega ^{\bullet }({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })) \rightarrow \Omega ^{\bullet +1}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))$.

Consider the map $\nabla ^{\lightning }:{\mathfrak {X}}({\mathcal {M}})\times \Gamma \big (S(T_{{\mathcal {M}}})\big ) \rightarrow \Gamma \big (S(T_{{\mathcal {M}}})\big )$ defined by

$$\begin{aligned} \nabla ^{\lightning }_{Y}{\varvec{X}}= ({{\,\mathrm{pbw}\,}}^{\nabla })^{-1} \big ( Y\cdot {{\,\mathrm{pbw}\,}}^{\nabla }({\varvec{X}})\big ) \end{aligned}$$

for all $Y\in {\mathfrak {X}}({\mathcal {M}})$ and ${\varvec{X}}\in \Gamma \big (S(T_{{\mathcal {M}}})\big )$.

Lemma A.1

The above map $\nabla ^{\lightning }$ defines a flat connection on $S(T_{{\mathcal {M}}})$.

Abusing notation, we write the same symbol $\nabla ^{\lightning }$ to denote the induced flat connection on ${\widehat{S}}(T_{{\mathcal {M}}}^{\vee })$. The associated covariant derivative $d^{\nabla ^{\lightning }}:\Omega ^{\bullet }({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })) \rightarrow \Omega ^{\bullet +1}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))$ satisfies $(d^{\nabla ^{\lightning }})^{2}=0$.

We use the identification

$$\begin{aligned} \Omega ^{p}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))\cong \Gamma \big (\Lambda ^{p}(T_{{\mathcal {M}}}^{\vee })\otimes {\widehat{S}}(T_{{\mathcal {M}}}^{\vee })\big ) \cong \Gamma \big ({{\,\mathrm{Hom}\,}}(\Lambda ^{p}(T_{{\mathcal {M}}})\otimes S(T_{{\mathcal {M}}}),{\mathbb {K}})\big ) . \end{aligned}$$

The total degree of $\omega \in \Omega ^{p}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))$ is $p+\left| \omega \right|$, where p is the cohomological degree and $\left| \omega \right|$ is the internal degree of $\omega$.

Define two operators

$$\begin{aligned} \delta :\Omega ^p({\widehat{S}}(T^\vee _{\mathcal {M}})) \rightarrow \Omega ^{p+1}({\widehat{S}}(T^\vee _{\mathcal {M}})) \end{aligned}$$

and

$$\begin{aligned} {\mathfrak {h}}:\Omega ^p({\widehat{S}}(T^\vee _{\mathcal {M}})) \rightarrow \Omega ^{p-1}({\widehat{S}}(T^\vee _{\mathcal {M}})) \end{aligned}$$

by

$$\begin{aligned}&\left( \delta \omega \right) (X_1\wedge \cdots \wedge X_{p+1}; Y_1\odot \cdots \odot Y_{q-1})\\&\quad =\sum _{i=1}^{p+1} (-1)^{i+1} \varepsilon \cdot \omega (X_1\wedge \cdots \wedge {\widehat{X}}_i\wedge \cdots \wedge X_{p+1}; X_i\odot Y_1\odot \cdots \odot Y_{q-1}) \end{aligned}$$

and

$$\begin{aligned}&\left( {\mathfrak {h}}\omega \right) (X_1\wedge \cdots \wedge X_{p-1}; Y_1\odot \cdots \odot Y_{q+1})\\&\quad =\frac{1}{p+q}\sum _{j=1}^{q+1} \varepsilon \cdot \omega (Y_j\wedge X_1\wedge \cdots \wedge X_{p-1}; Y_1\odot \cdots \odot {\widehat{Y}}_j\odot \cdots \odot Y_{q+1}) , \end{aligned}$$

for all $\omega \in \Omega ^{p}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))$ and all homogeneous $X_1,\cdots ,X_{p+1},Y_1,\cdots ,Y_{q+1}\in {\mathfrak {X}}({\mathcal {M}})$. The symbol $\varepsilon$ denotes the Koszul signs: either $\varepsilon (X_1,\cdots ,X_{p+1},Y_1,\cdots ,Y_{q-1})$ or $\varepsilon (X_1,\cdots ,X_{p-1},Y_1,\cdots ,Y_{q+1})$, as appropriate.

Both $\delta$ and ${\mathfrak {h}}$ are $C^\infty ({\mathcal {M}})$-linear, and $\delta$ is the Koszul operator. Observe that $\delta$ has total degree $+1$ and ${\mathfrak {h}}$ has total degree $-1$. However neither $\delta$ nor ${\mathfrak {h}}$ change the internal degree: $\left| \delta \omega \right| =\left| \omega \right|$ and $\left| {\mathfrak {h}}\omega \right| =\left| \omega \right|$.

Remark A.2

In [18, 19, 32], the operator ${\mathfrak {h}}$ is written as $\delta ^{-1}$. We avoid this notation because ${\mathfrak {h}}$ is not an inverse map of $\delta$, and it is rather a homotopy operator.

Lemma A.3

The operator $\delta$ satisfies $\delta ^{2}=0$. That is,

$$\begin{aligned} 0\rightarrow \Omega ^{0}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })) \xrightarrow {\delta } \Omega ^{1}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })) \xrightarrow {\delta } \Omega ^{2}({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })) \xrightarrow {\delta } \cdots \end{aligned}$$

is a cochain complex. Moreover, the operators $\delta$ and ${\mathfrak {h}}$ satisfy

$$\begin{aligned} \delta \circ {\mathfrak {h}}+{\mathfrak {h}}\circ \delta = {{\,\mathrm{id}\,}}- \pi _{0} , \end{aligned}$$

where $\pi _{0}:\Omega ^{\bullet }({\widehat{S}}(T_{{\mathcal {M}}}^{\vee }))\rightarrow C^\infty ({\mathcal {M}})$ is the natural projection.

We have the following theorem

Theorem A.4

[32, Theorem 5.6]. Let ${\mathcal {M}}$ be a finite-dimensional graded manifold and let $\nabla$ be a torsion-free affine connection on ${\mathcal {M}}$. Then the covariant derivative $d^{\nabla ^{\lightning }}$ decomposes as

$$\begin{aligned} d^{\nabla ^{\lightning }}=d^{\nabla }-\delta +{\widetilde{A^\nabla }} , \end{aligned}$$

where the operator ${\widetilde{A^\nabla }}: \Omega ^{\bullet }\big ({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })\big ) \rightarrow \Omega ^{\bullet +1}\big ({\widehat{S}}(T_{{\mathcal {M}}}^{\vee })\big )$ is the derivation of (total) degree $+1$ determined by a certain element $A^\nabla$ of $\Omega ^{1}\big ({\mathcal {M}},{\widehat{S}}^{\ge 2}(T_{{\mathcal {M}}}^{\vee }) \otimes T_{{\mathcal {M}}}\big )$ satisfying

$$\begin{aligned} {\mathfrak {h}}\circ A^\nabla =0 . \end{aligned}$$

Remark A.5

The operator $\widetilde{A^{\nabla }}$ increases the cohomological degree by $+1$ and preserves the internal degree. That is, although the total degree of $\widetilde{A^{\nabla }}$ is $+1$, its internal degree is $\left| \widetilde{A^{\nabla }}\right| =0$.

Write

$$\begin{aligned} A^\nabla = \sum _{n\ge 2}A^\nabla _{n}, \quad \quad A^\nabla _{n} \in \Omega ^{1}({\mathcal {M}},S^{n}(T_{{\mathcal {M}}}^{\vee })\otimes T_{{\mathcal {M}}}). \end{aligned}$$

Let $R^\nabla \in \Omega ^{2}\big ({\mathcal {M}};{{\,\mathrm{End}\,}}(T_{{\mathcal {M}}})\big )$ denote the curvature of $\nabla$.

Proposition A.6

We have the following recursive formula for $A^\nabla _{n}$:

$$\begin{aligned} A^\nabla _{2}= & {} {\mathfrak {h}}\circ R^\nabla \, ,\\ A^\nabla _{n+1}= & {} {\mathfrak {h}}\circ \left( d^{\nabla }A^\nabla _{n} +\sum _{p+q=n}\frac{1}{2}[A^\nabla _{p}, A^\nabla _{q}] \right) , \quad \forall n\ge 2. \end{aligned}$$

Proof

According to Theorem A.4 and Lemma A.1, we have $d^{\nabla ^{\lightning }} = d^{\nabla }-\delta +A^\nabla$ and $(d^{\nabla ^{\lightning }})^{2}=0$.

By Lemma A.3, we know $\delta ^{2}=0$ and $\delta \circ {\mathfrak {h}}+{\mathfrak {h}}\circ \delta = {{\,\mathrm{id}\,}}-\pi _{0}$. Also, $(d^{\nabla })^{2}=R^\nabla$. Since ${\nabla }$ is torsion-free, we have

$$\begin{aligned}{}[\delta , d^{\nabla }]=\delta \circ d^{\nabla } +d^{\nabla }\circ \delta = 0. \end{aligned}$$

As a result, $(d^{\nabla ^{\lightning }})^{2}=0$ implies that

$$\begin{aligned} \delta \circ A^\nabla + A^\nabla \circ \delta = R^\nabla + d^{\nabla }A^\nabla + \frac{1}{2}[A^\nabla , A^\nabla ]. \end{aligned}$$

By applying the operator ${\mathfrak {h}}$, we get

$$\begin{aligned} A^\nabla = {\mathfrak {h}}\circ \delta \circ A^\nabla = {\mathfrak {h}}\circ \left( R^\nabla +d^{\nabla }A^\nabla + \frac{1}{2}[A^\nabla , A^\nabla ] \right) , \end{aligned}$$

because ${\mathfrak {h}}\circ A^\nabla =0$ and $\pi _{0}\circ A^\nabla =0$.

Since ${\mathfrak {h}}\big (\Omega ^2({\widehat{S}}^q(T^\vee _{\mathcal {M}})) \subset \Omega ^{1}({\widehat{S}}^{q+1}(T^\vee _{\mathcal {M}}))$, applying the canonical projections

$$\begin{aligned} \Omega ^{1}({\mathcal {M}},{\hat{S}}(T_{{\mathcal {M}}}^{\vee })\otimes T_{{\mathcal {M}}}) \rightarrow \Omega ^{1}({\mathcal {M}}, S^{n}(T_{{\mathcal {M}}}^{\vee })\otimes T_{{\mathcal {M}}}) \end{aligned}$$

(for each $n\ge 2$) to the equality

$$\begin{aligned} A^\nabla = {\mathfrak {h}}\circ \left( R^\nabla +d^{\nabla }A^\nabla + \frac{1}{2}[A^\nabla , A^\nabla ] \right) \in \Omega ^{1}({\mathcal {M}},{\hat{S}}(T_{{\mathcal {M}}}^{\vee })\otimes T_{{\mathcal {M}}}) \end{aligned}$$

yields the relations

$$\begin{aligned} A^\nabla _{2}&={\mathfrak {h}}\circ R^\nabla , \nonumber \\ A^\nabla _{n+1}&={\mathfrak {h}}\circ \left( d^{\nabla }A^\nabla _{n} +\sum _{p+q=n}\frac{1}{2}[A^\nabla _{p}, A^\nabla _{q}] \right) , \quad \forall n\ge 2. \end{aligned}$$

(65)

This completes the proof. $\square$

Corollary A.7

Under the same hypothesis as in Theorem A.4, the elements $A^\nabla _{n}$ with $n\ge 2$ are completely determined by the curvature $R^\nabla$ and its higher covariant derivatives. In fact, the element $A^\nabla _{n}$ is determined by the elements $A^\nabla _k$ with $k\le n-1$ by way of the recursive formula (65).