A Microlocal Category Associated to a Symplectic Manifold

Abstract

For a symplectic manifold subject to certain topological conditions a category enriched in $A_{\infty }$ local systems of modules over the Novikov ring is constructed. The construction is based on the category of modules over Fedosov’s deformation quantization algebra that have an additional structure, namely an action of the fundamental groupoid up to inner automorphisms. Based in large part on the ideas of Bressler-Soibelman, Feigin, and Karabegov, it motivated by the theory of Lagrangian distributions and is related to other microlocal constructions of a category starting from a symplectic manifold, such as those due to Nadler-Zaslow and Tamarkin. In the case when our manifold is a flat two-torus, the answer is very close to both the Tamarkin microlocal category and the Fukaya category as computed by Polishchuk and Zaslow.

In memory of Boris Vasilievich Fedosov and Moshé Flato

Appendices

12 Appendix. Metaplectic and Metalinear Groups

We recall the classical material that is contained, for example, in [15, 36].

1.1 12.1 Metalinear Groups and Metalinear Structures

Recall [15] that the metalinear group is by definition

$$\begin{aligned} {\text {ML}}(n,{\mathbb R})=\{(g,\, z)| g\in {\text {GL}}(n, {\mathbb R}),\, z^2=\det (g)\} \end{aligned}$$

(12.1.1)

This is a twofold cover of ${\text {GL}}(n, {\mathbb R})$. There is a morphism

$$\begin{aligned} {\det } ^{\frac{1}{2}}: {\text {ML}}(n,{\mathbb R})\rightarrow {\mathbb C}^{\times }; \; (g, z)\mapsto z. \end{aligned}$$

(12.1.2)

Denote by $\mathrm{MO}(n)$ the preimage of $\mathrm{{O}}(n)$ in ${\text {ML}}(n).$ Let also

$$\begin{aligned} \mathrm{{MU}}(n)=\{(u,\, \zeta )| u\in \mathrm{{U}}(n, {\mathbb C}),\, \zeta ^2=\det (u)\} \end{aligned}$$

(12.1.3)

Definition 12.1

Let ${\text {Mp}}(2n, {\mathbb R})$ be the universal twofold cover of ${\text {Sp}}(2n, {\mathbb R}).$ We call this group the metaplectic group.

There is a commutative diagram

where the horizontal embeddings are homotopy equivalences.

A metalinear structure on a real vector bundle E is a lifting of the transition automorphisms $g_{jk}^E$ to an ${\text {ML}}(n,{\mathbb R})$-valued cocycle ${\widetilde{g}}_{jk}^E$. For a real bundle E with a metalinear structure, the complex line bundle $\wedge ^{\frac{1}{2}}E$ is by definition given by the transition automorphisms ${\det }^{\frac{1}{2}}({\widetilde{g}}_{jk}^E)$, cf. (12.1.2).

A metaplectic structure on a symplectic vector bundle E is a lifting of the transition automorphisms $g_{jk}^E$ to an ${\text {Mp}}(n,{\mathbb R})$-valued cocycle ${\widetilde{g}}_{jk}^E$. A metalinear structure on a manifold (resp. a metaplectic structure on a symplectic manifold) is by definition the corresponding structure on its tangent bundle.

Lemma 12.2

A manifold X has a metalinear structure if and only if $T^*X$ has a metaplectic structure. If a symplectic manifold has a metaplectic structure then any Lagrangian submanifold of M has a metalinear structure.

Proof

The obstruction to existence of a metalinear, resp. metaplectic, structure is as follows. Pick any transition isomorphisms $g_{jk}$ for the tangent bundle. Lift them to a cochain ${\widetilde{g}}_{jk}$ with values in ${\text {ML}}$, resp. in ${\text {Mp}}.$ Then compute the two-cocycle $a_{jk\ell }={\widetilde{g}}_{jk}{\widetilde{g}}_{k\ell }{\widetilde{g}}_{j\ell }^{-1}$ with values in ${\mathtt {\mathbb Z}}/2{\mathtt {\mathbb Z}}.$ The cohomology class of this cocycle is the obstruction. If $M=T^*X$, this cohomology class is determined by its restriction to X. But on X the symplectic transition functions $g_{jk}$ for TM can be chosen as the image of ${\text {GL}}(n)$-valued transition functions for TX under the embedding ${\text {GL}}\rightarrow {\text {Sp}}.$ This proves the first statement of the Lemma. Now, for a Lagrangian submanifold L of M, the transition isomorphisms for TM|L are cohomologous to an ${\text {Mp}}$-valued cocycle $p_{jk}:$ $g_{jk}=h_j p_{jk} h_k^{-1}.$ Lift $h_j$ to ${\text {Mp}}(2n)$ somehow. Put

$$\begin{aligned} {\widetilde{p}}_{jk}={\widetilde{h}}_j^{-1} {\widetilde{g}}_{jk} {\widetilde{h}}_k. \end{aligned}$$

(12.1.4)

This is a cocycle cohomologous to ${\widetilde{g}}_{jk}|L.$ It takes values in the preimage of the subgroup of ${\text {Sp}}(2n)$ consisting of matrices preserving the Lagrangian submanifold $L_0=\{\widehat{\xi }=0\}.$ The image of this cocycle under the projection to ${\text {GL}}$ via ${\text {ML}}$ is a cocycle defining the bundle TX.   $\square$

1.2 12.2 The Maslov Class of a Lagrangian Submanifold

1.2.1 12.2.1 The Case $c_1(M)=0$

Consider the cohomology class of the two-cocycle $a_{jk\ell }$ constructed as in the proof of Lemma 12.2 above but when we use the universal cover $\widetilde{{\text {Sp}}}(2n,{\mathtt {\mathbb R}})$ instead of ${\text {Mp}}(n).$ This is now a class in $H^2(M,{\mathtt {\mathbb Z}})$ that represents $c_1(M)$, the first Chern class of TM viewed as a complex bundle after we reduce the structure group ${\text {Sp}}$ to the maximal compact subgroup U(n). Indeed, $\widetilde{{\text {Sp}}}$ is homotopy equivalent to

$${\widetilde{U}}(n)=\{(u,x)|u\in U(n),\, x\in {\mathtt {\mathbb R}}, \det (u)=e^{2\pi i x}\}.$$

The proof of Lemma 12.2 applied to this case establishes the following fact.

Consider the group

$$\begin{aligned} {\widetilde{{\text {GL}}}}(n,{\mathtt {\mathbb R}})=\{(g,x)|x\in {\text {GL}}(n,{\mathtt {\mathbb R}});x\in {\mathtt {\mathbb R}}; \det (g)=e^{2\pi i x}\} \end{aligned}$$

(12.2.1)

(Of course ${\widetilde{{\text {GL}}}}$, unlike ${\widetilde{U}}$ or $\widetilde{{\text {Sp}}}$, has nothing to do with the universal cover).

Lemma 12.3

A trivialization of $c_1(M)$ defines a ${\widetilde{GL}}(n)$-structure on any Lagrangian submanifold L of M, i.e. a lifting of the transition automorphisms of TL to a ${\widetilde{{\text {GL}}}}(n)$-valued cocycle.

Assume that L is oriented. Then there is another ${\widetilde{GL}}(n)$-structure on L, due to the fact that $\mathrm{{SL}}(n)$ is a subgroup of ${\widetilde{GL}}(n).$ The two liftings differ by a class in $\lambda (L)\in H^1(L,{\mathtt {\mathbb Z}}).$ We will call this class the Maslov class of an oriented Lagrangian submanifold of a symplectic manifold M with a trivialization of $c_1(M).$

1.2.2 12.2.2 The Case $2c_1(M)=0$

Now consider the group

$$\begin{aligned} {\widetilde{U}}^{(2)} (n)=\{(g,x)|g\in U(n);x\in {\mathtt {\mathbb R}}; \det (g)^2=e^{2\pi i x}\} \end{aligned}$$

(12.2.2)

Note that

$$\begin{aligned} \{(g,x)|x\in {\text {GL}}(n,{\mathtt {\mathbb R}});x\in {\mathtt {\mathbb R}}; \det (g)^2=e^{2\pi i x}\}\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {GL}}(n,{\mathtt {\mathbb R}})\times {\mathtt {\mathbb Z}}\end{aligned}$$

(12.2.3)

Arguing exactly as before, we get

Lemma 12.4

A trivialization of $2c_1(M)$ defines a ${{{\text {GL}}}}(n)\times {\mathtt {\mathbb Z}}$-structure on any Lagrangian submanifold L of M.

Projecting to ${\mathtt {\mathbb Z}}$, we get a class $\mu (L)\in H^1(L,{\mathtt {\mathbb Z}}).$ We call $\mu (L)$ the Maslov class of a Lagrangian submanifold of a symplectic manifold M with a trivialization of $2c_1(M).$

Note that

$$\begin{aligned} \mu (L)=2\lambda (L) \end{aligned}$$

(12.2.4)

for a trivialization of $c_1$, the induced trivialization of $2c_1$, and an oriented L.

Remark 12.5

Let ${\widetilde{\Lambda }}(n)$ be the universal cover of the Lagrangian Grassmannian $\Lambda (n).$ Define the group $\widetilde{{\text {Sp}}}^{(2)}(2n,{\mathtt {\mathbb R}})$ by the condition that the following square be Cartesian.

Then ${\widetilde{U}}^{(2)}$ is a homotopy equivalent subgroup of $\widetilde{{\text {Sp}}}^{(2)}(2n,{\mathtt {\mathbb R}}) .$

Example 12.6

For $n=1$, $U(1) \,{\mathop {\rightarrow }\limits ^{\sim }}\,S^1;$ also $\Lambda (1)\,{\mathop {\rightarrow }\limits ^{\sim }}\,S^1.$ Under these identifications, the projection $U(1)\rightarrow \Lambda (1)$ becomes the map $\zeta \mapsto \zeta ^2.$

1.3 12.3 The Groups ${\text {Sp}}^N$

Here we use definitions and notation from [36]. For $N\ge 1$, let $\Lambda ^N (n)$ be the universal N-fold cover of $\Lambda (n).$ Define the group ${\text {Sp}}^N(2n,{\mathtt {\mathbb R}})$ by requiring the following diagram to be Cartesian:

In other words, ${\text {Sp}}^N(2n)=\widetilde{{\text {Sp}}}^{(2)}(2n)/({\mathtt {\mathbb Z}}/N)$. Define also

$$U^N (n)=\{(u, \zeta )| u\in U(n), \zeta \in {\mathtt {\mathbb C}}, \det (u)^2=\zeta ^N\}={\widetilde{U}}^{(2)}/({\mathtt {\mathbb Z}}/N)$$

This is a subgroup of ${\text {Sp}}^N(n)$ and the embedding is a homotopy equivalence.

A ${\text {Sp}}^N(2n)$-structure on M is the same as a trivialization of $2c_1(M)$ in $H^2(M,{\mathtt {\mathbb Z}}/N).$

The universal N-fold cover of ${\text {Sp}}(2n)$ is a subgroup of $Sp^{2N}(2n).$ In particular, the metaplectic group ${\text {Mp}}(2n)$ is a subgroup of ${\text {Sp}}^4(2n).$ The latter is generated by ${\text {Mp}}(2n)$ and the central subgroup $\{\pm 1,\pm i\}.$ The intersection of the two is $\{\pm 1\},$ the kernel of ${\text {Mp}}\rightarrow {\text {Sp}}.$

The following makes sense for any N. We fix $N=4$ just to fix the notation for the rest of the paper.

Definition 12.7

(a) Define $P(n,{\mathtt {\mathbb R}})$ as the subgroup of ${\text {Sp}}(2n,{\mathtt {\mathbb R}})$ consisting of pairs (A, z) where $A=\left[ \begin{array}{cc} b&{}a\\ 0&{} (b^{-1})^t\end{array}\right]$ is a symplectic matrix. In other words, P(n) is the subgroup of ${\text {Sp}}(2n)$ consisting of matrices preserving the Lagrangian submanifold $L_0=\{\widehat{\xi }=0\}.$

(b) Define ${\text {MPar}}(n,{\mathtt {\mathbb R}})$ as the subgroup of ${\text {Sp}}^4(2n,{\mathtt {\mathbb R}})$ consisting of pairs (A, z) where $A=\left[ \begin{array}{cc} b&{}a\\ 0&{} (b^{-1})^t\end{array}\right]$ is a symplectic matrix, z is a complex number, and $\det (b)^2=z^4.$ In other words, this is the lifting to ${\text {Sp}}^4(2n)$ of P(n).

Lemma 12.8

(a) ${\text {MPar}}(n,{\mathtt {\mathbb R}})\,{\mathop {\rightarrow }\limits ^{\sim }}\,P(n,{\mathtt {\mathbb R}})\times \{\pm 1,\pm i\}$

(b) If a symplectic manifold M has an ${\text {Sp}}^4$ structure and L is a Lagrangian submanifold then formulas (12.1.4) define an ${\text {MPar}}(n)$-valued cocycle cohomologous to the transition isomorphisms of TM|L.

(c) If M has a real polarization then it has an ${\text {Sp}}^4(2n)$-structure.

Definition 12.9

The projection of the cohomology class from Lemma 12.8, (b) to $H^1(L,{\mathtt {\mathbb Z}}/4{\mathtt {\mathbb Z}})$ is called the Maslov class of L.

When the trivialization of $2c_1(M)$ modulo 4 comes from a trivialization of $2c_1(M)$ then the Maslov class defined above is equal to $\exp (\frac{i\pi }{2}\mu (L))$ that was defined in Sect. 12.2.2.

13 Appendix. The Algebraic Metaplectic Representation

Most of the material of this section is contained in [40]. Recall the algebra ${\mathtt {\mathcal A}}$ from Sect. 4.1 and the ${\mathtt {\mathcal A}}$-module from Definition 9.5. In this section we give an interpretation of this module in terms of the metaplectic representation.

1.1 13. 1 Symmetries of the Deformation Quantization Algebra of a Formal Neighborhood

Any continuous automorphism g of ${\widehat{\mathbb A}}$ induces a symplectic linear transformation $g_0$ of ${\mathbb C}^{2n}.$ Denote by G the group of those g whose linear part $g_0$ preserves the real structure. We have

$$\begin{aligned} G={\text {Sp}}(2n,{\mathbb R})\ltimes \exp (\mathfrak {g}_{\ge 1}) \end{aligned}$$

(13.1.1)

Define the central extension

$$\begin{aligned} {\widetilde{\mathbf G}}=\exp \left( \frac{1}{i\hbar }{\mathbb C}\oplus {\mathbb C}\right) \times \mathrm {Sp}^4(2n,{\mathbb R})\ltimes \exp (\widetilde{ \mathfrak g}_{\ge 1}) \end{aligned}$$

(13.1.2)

where $\widetilde{{\text {Sp}}}(2n,{\mathtt {\mathbb R}})$ is the universal cover of ${\text {Sp}}(2n,{\mathtt {\mathbb R}}).$ One has an exact sequence

$$\begin{aligned} 1\rightarrow \frac{{\mathbb Z}}{4}\times \exp \left( \frac{1}{i\hbar }{\mathbb C}[[\hbar ]]\right) \rightarrow {\widetilde{\mathbf G}}\rightarrow G\rightarrow 1 \end{aligned}$$

(13.1.3)

Define also P to be the subgroup of G consisting of elements g whose linear part preserves the Lagrangian subspace

$$\begin{aligned} L_0=\{\widehat{\xi }_1=\ldots =\widehat{\xi }_n=0\} \end{aligned}$$

(13.1.4)

Let ${\widetilde{\mathbf P}}$ be the preimage of P in ${\widetilde{\mathbf G}}$.

1.2 13.2 The Algebraic Fourier Transform

Let $\widehat{y}=(\widehat{y}_1,\ldots ,\widehat{y}_n)$ be n formal variables. For a symmetric real $n\times n$ matrix a, put

$$\begin{aligned} {\mathcal H}_a^{\widehat{y}}=\exp \left( \frac{a\widehat{y}^2}{2i\hbar }\right) \widehat{\mathbb C}[[\widehat{y}, \hbar ]]((e^{\frac{c}{i\hbar }}|c\in {\mathbb C})) \end{aligned}$$

(13.2.1)

Here

$$\begin{aligned} \widehat{\mathbb C}[[\widehat{y}, \hbar ]]=\left\{ \sum _{k=-N}^\infty v_k|v_k\in {\mathbb C}[[\widehat{y}]]((\hbar ))_k\right\} \end{aligned}$$

(13.2.2)

with respect to the grading (3.1.3); for any vector space V, we define

$$\begin{aligned} V((e^{\frac{c}{i\hbar }}|c\in {\mathbb C}))=\left\{ \sum _{k\in {\mathbb N};\mathrm{{Re}}(c_k)\rightarrow +\infty }e^{\frac{c_k}{i\hbar }}v_k\right\} , \end{aligned}$$

(13.2.3)

$v_k\in V.$ In particular, the operator of multiplication by h is automatically invertible.

For a nondegenerate a, define the Fourier transform (cf. [22])

$$\begin{aligned} F: {\mathcal H}_a^{{\widehat{y}}}\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\mathcal H}_{-a^{-1}}^{\widehat{\eta }} \end{aligned}$$

(13.2.4)

as follows. Heuristically,

$$\begin{aligned} (Ff)(\widehat{\eta })=\frac{e^{-\frac{\pi i n}{4}}}{(2\pi i \hbar )^{n/2}} \int e^\frac{\widehat{y}\widehat{\eta }}{i\hbar }f(\widehat{y})d\widehat{y}; \end{aligned}$$

(13.2.5)

To give the above formula a rigorous meaning, put

$$ F\left( f(\widehat{y})\exp \left( \frac{a\widehat{y}^2}{2i\hbar }\right) \right) (\widehat{\eta }) =f\left( i\hbar \frac{\partial }{\partial \widehat{\eta }}\right) F\left( \exp \left( \frac{a\widehat{\eta }^2}{2i\hbar }\right) \right) = $$

$$ f\left( i\hbar \frac{\partial }{\partial \widehat{\eta }}\right) \frac{e^{-\frac{\pi i n}{4}}}{\det (\sqrt{ia})} \exp \left( \frac{-a^{-1}\widehat{\eta }^2}{2i\hbar }\right) =\frac{e^{-\frac{\pi i p(a)}{2}}}{\det \sqrt{|a|}}f\left( i\hbar \frac{\partial }{\partial \widehat{\eta }}\right) \exp \left( \frac{-a^{-1}\widehat{\eta }^2}{2i\hbar }\right) $$

Here p(a) is the number of positive eigenvalues of a. We used the branch of the square root for which ${\sqrt{x}}>0$ if $x>0$; it is defined on the complex plane with the line $\{x<0, \; x\in {\mathbb R}\}$ removed. The final term in the above chain of equalities can be viewed as the definition of the first term.

Remark 13.1

The definition of the Fourier transform F extends to elements of the form

$$\begin{aligned} \mathbf{f}(\widehat{y})=\exp \left( \frac{a\widehat{y}^2}{2 i\hbar }+i\widehat{y}\widehat{z}\right) f(\widehat{y}) \end{aligned}$$

(13.2.6)

where a is nondegenerate and $\widehat{z}$ is another formal parameter:

$$\begin{aligned} F(\mathbf{f})(\widehat{\eta })=F\left( \exp \left( \frac{a\widehat{y}^2}{2i \hbar }\right) f(\widehat{y})\right) (\widehat{\eta }+\widehat{z}) \end{aligned}$$

(13.2.7)

One has

$$\begin{aligned} F^2 (\mathbf{f})(\widehat{y})=i^{-n} \mathbf{f}(-\widehat{y});\;F\widehat{y}F^{-1}=i\hbar \frac{\partial }{\partial \widehat{\eta }};\;Fi\hbar \frac{\partial }{\partial \widehat{y}}F^{-1}=-\widehat{\eta }\end{aligned}$$

(13.2.8)

1.3 13.3 The Two-Dimensional Case

For the readers convenience, we first present the case $n=1.$

$$\begin{aligned} {\mathtt {\mathcal H}}=\bigoplus _{a\in {\mathtt {\mathbb R}}} {\mathtt {\mathcal H}}^{\widehat{x}}_a \bigoplus \bigoplus _{a\in {\mathtt {\mathbb R}}} F{\mathtt {\mathcal H}}^{\widehat{x}}_a /\sim \end{aligned}$$

(13.3.1)

where

$$\begin{aligned} F f(\widehat{x})\exp \left( {\frac{a\widehat{x}^2}{2i\hbar }}\right) \sim \frac{e^{-\frac{\pi i}{2}p(a)}}{\sqrt{|a|}} f\left( i\hbar \frac{\partial }{\partial \widehat{x}}\right) \exp \left( {-\frac{a^{-1}\widehat{x}^2}{2i\hbar }}\right) \end{aligned}$$

(13.3.2)

for $a\ne 0.$ Here $p(a)=1$ if $a>0$ and $p(a)=0$ otherwise.

1.3.1 13.3.1 The Action of ${\widehat{{\widehat{\mathbb A}}}}$ on ${\mathcal H}$

The algebra ${\widehat{{\widehat{\mathbb A}}}}$ acts on the space ${\mathcal H}$ as follows. If ${\mathbf f}$ is in the first summand in (13.3.1), then $\widehat{x}$ acts on it by multiplication and $\widehat{\xi }$ by $i\hbar \frac{\partial }{\partial \widehat{x}}$, the latter defined by

$$\frac{\partial }{\partial \widehat{x}}\left( \exp \left( \frac{a\widehat{x}^2}{2i\hbar }f(\widehat{x})\right) \right) =\exp \left( \frac{a\widehat{x}^2}{2i\hbar }\right) \left( \frac{\partial }{\partial \widehat{x}}+a\widehat{x}\right) f(\widehat{x}).$$

As for $F{\mathbf f}$, $\widehat{x}$ sends it to $-i\hbar F\frac{\partial }{\partial \widehat{x}}{\mathbf f}$ and $\widehat{\xi }$ sends it to $F\widehat{x}{\mathbf f}.$

1.3.2 13.3.2 Some Operators on ${\mathtt {\mathcal H}}$

The operator $F:{\mathtt {\mathcal H}}\rightarrow {\mathtt {\mathcal H}}.$ Define for $\mathbf{f}(\widehat{x})=\exp (\frac{a\widehat{x}^2}{2i\hbar })f(\widehat{x})$

$$F: \mathbf{f}\mapsto F\mathbf{f}\mapsto i^{-1} \mathbf{f}(-\widehat{x})$$

The operator $\exp (\frac{a\widehat{x}^2}{2i\hbar }):{\mathtt {\mathcal H}}\rightarrow {\mathtt {\mathcal H}}.$ (1)

$$\exp \left( \frac{a\widehat{x}^2}{2i\hbar }\right) : \exp \left( \frac{c\widehat{x}^2}{2i\hbar }\right) f(\widehat{x})\mapsto \exp \left( \frac{(a+c)\widehat{x}^2}{2i\hbar }\right) f(\widehat{x})$$

for $c\in {\mathtt {\mathbb R}};$

(2)

$$F\exp \left( \frac{c\widehat{x}^2}{2i\hbar }\right) f(\widehat{x})\mapsto \frac{e^{\frac{-\pi i}{2}p(c)}}{{\sqrt{|c|}}} f\left( -i\hbar \frac{\partial }{\partial \widehat{x}}+a\widehat{x}\right) \exp \left( \frac{(a-c^{-1})\widehat{x}^2}{2i\hbar }\right) $$

for $c\ne 0;$

(3)

$$F\exp \left( \frac{c\widehat{x}^2}{2i\hbar }\right) f(\widehat{x})\mapsto iFf\left( \widehat{x}-ai\hbar \frac{\partial }{\partial \widehat{x}}\right) \frac{e^{{-\frac{\pi i}{2}}(p(c)+p(\frac{-c}{1-ac}))}}{\sqrt{|1-ac|}} \exp \left( \frac{c}{1-ac}\frac{\widehat{x}^2}{2i\hbar }\right) $$

for $c\ne a^{-1}.$ These maps preserve the equivalence relation and therefor define operators on ${\mathtt {\mathcal H}}$.

1.3.3 13.3.3 The Action of ${\text {Sp}}^4(2,{\mathbb R})$ on ${\mathcal H}$

The group ${\text {Sp}}^4(2,{\mathbb R})$ acts by the algebraic version of the metaplectic representation that we are going to describe next.

1.4 13.4 The Metaplectic Projective Representations of $\mathrm{{SL}}_2({\mathtt {\mathbb R}})$

Define the action of generators of $\mathrm{{SL}}_2({\mathtt {\mathbb R}})$ by exactly the same formula as the usual metaplectic representation

$$\begin{aligned} T:\left[ \begin{array}{cc} 1&{}0\\ a&{}1\end{array}\right] \mapsto \exp \left( \frac{a\widehat{x}^2}{2i\hbar }\right) ; \; \left[ \begin{array}{cc} 0&{}1\\ -1&{}0\end{array}\right] \mapsto F; \end{aligned}$$

(13.4.1)

$$\begin{aligned} \left[ \begin{array}{cc} b&{}0\\ 0&{} b^{-1}\end{array}\right] \mapsto T_b;\;(T_bf)(x)=\frac{1}{\sqrt{\det (b)}}f(b^{-1}x) \end{aligned}$$

(13.4.2)

The corresponding representation of ${\mathfrak {sl}}(2)$ is given by

$$\begin{aligned} X_-=\frac{\widehat{x}^2}{2i\hbar };\, H=-\frac{\widehat{x}\widehat{\xi }}{i\hbar }=-\frac{\widehat{x}*\widehat{\xi }}{i\hbar }-\frac{1}{2}; \, X_+=-\frac{\widehat{\xi }^2}{2i\hbar } \end{aligned}$$

(13.4.3)

1.4.1 13.4.1 The Bruhat Relations

The following are well known to be the defining relations of $\mathrm{{SL}}_2$ (together with the requirement that $a\mapsto \left[ \begin{array}{cc} 1&{}0\\ a&{}1 \end{array}\right]$ is a morphism from the additive group and $b\mapsto \left[ \begin{array}{cc} b&{}0\\ 0&{}b^{-1} \end{array}\right]$ is a morphism from the multiplicative group).

$$\begin{aligned} \left[ \begin{array}{cc} 0&{}1\\ -1&{}0 \end{array}\right] \left[ \begin{array}{cc} 1 &{}0\\ a&{}1\end{array}\right] \left[ \begin{array}{cc} 0&{}1\\ -1 &{}0 \end{array} \right] ^{-1}=\left[ \begin{array}{cc} 1&{}-a\\ 0 &{}1 \end{array}\right] \end{aligned}$$

(13.4.4)

$$\begin{aligned} \left[ \begin{array}{cc} 0&{}1\\ -1&{}0 \end{array}\right] \left[ \begin{array}{cc} b &{}0\\ 0&{}b^{-1}\end{array}\right] \left[ \begin{array}{cc} 0&{}1\\ -1 &{}0 \end{array} \right] ^{-1}=\left[ \begin{array}{cc} b^{-1}&{}0\\ 0&{}b \end{array}\right] \end{aligned}$$

(13.4.5)

$$\begin{aligned} \left[ \begin{array}{cc} b&{}0\\ 0&{}b^{-1} \end{array}\right] \left[ \begin{array}{cc} 1&{}0\\ a&{}1\end{array}\right] \left[ \begin{array}{cc} b&{}0\\ 0 &{}b^{-1} \end{array}\right] ^{-1}=\left[ \begin{array}{cc} 1&{}0\\ b^{-2} a&{}1\end{array}\right] \end{aligned}$$

(13.4.6)

$$\begin{aligned} \left[ \begin{array}{cc} 1&{}0\\ a&{}1\end{array}\right] \left[ \begin{array}{cc} 0&{}1\\ -1&{}0\end{array}\right] \left[ \begin{array}{cc} 1&{}0\\ a^{-1}&{}1\end{array}\right] = \left[ \begin{array}{cc} a^{-1}&{}0\\ 0&{}a\end{array}\right] \left[ \begin{array}{cc} 1&{}a\\ 0&{}1\end{array}\right] \end{aligned}$$

(13.4.7)

for $a\ne 0.$

Proposition 13.2

Formulas (13.4.1) define a representation of ${\widetilde{\mathrm {SL}}}_2$ in which an element of $\pi _1(\mathrm{{SL}}_2)$ acts by $e^{\frac{\pi i}{2}c}$ where c is its image in $\pi _1(\Lambda )\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\mathtt {\mathbb Z}}.$

Proof

All the Bruhat relations except (13.4.7) are true for operators T(g) defined in (13.4.1), whereas

Lemma 13.3

$$ T\left( {\left[ \begin{array}{cc} 1&{}0\\ a&{}1\end{array}\right] }\right) T\left( {\left[ \begin{array}{cc} 0&{}1\\ -1&{}0\end{array}\right] }\right) T\left( {\left[ \begin{array}{cc} 1&{}0\\ a^{-1}&{}1\end{array}\right] }\right) = $$

$$ =\frac{\sqrt{|a|}}{\sqrt{a}} e^{{\frac{\pi i}{2}} p(a)} T\left( {\left[ \begin{array}{cc} a^{-1}&{}0\\ 0&{}a\end{array}\right] } \right) T\left( {\left[ \begin{array}{cc} 1&{}a\\ 0&{}1\end{array}\right] }\right) $$

   $\square$

1.5 13.5 The Case of a General n

Now define

$$\begin{aligned} {\mathcal H}=\bigoplus _{I\subset \{1,\ldots , n\}}\bigoplus _a F_{I,J} {\mathcal H}^{\widehat{x}}/\sim \end{aligned}$$

(13.5.1)

where a runs through all symmetric $n\times n$ real matrices and the equivalence relation is defined as follows. For every subset K of $\{1,2,\ldots ,n\}$, define

$$\begin{aligned} F_K:\bigoplus _a F_I{\mathtt {\mathcal H}}^{\widehat{x}}\rightarrow \bigoplus _a F_{I\triangle K}{\mathtt {\mathcal H}}^{\widehat{x}} \end{aligned}$$

(13.5.2)

(where $\triangle$ stand for the symmetric difference) as follows: if L is the complement of $I\cap K$, then

$$\begin{aligned} (F_{K}F_I{\mathbf f})(\widehat{x}_{I\cap K}, \widehat{x}_L)=i^{-|I\cap K|}F_{I\triangle K}{} \mathbf{f}(-\widehat{x}_{I\cap K}, \widehat{x}_L) \end{aligned}$$

(13.5.3)

Let J be the complement of I.

$$\begin{aligned} \mathbf{f}(\widehat{x}_I ,\widehat{x}_J)=\exp \left( {\frac{a\widehat{x}_I^2+b\widehat{x}_I\widehat{x}_J+c\widehat{x}_J^2}{2i\hbar }}\right) f(\widehat{x}_{I}, \widehat{x}_J) \end{aligned}$$

(13.5.4)

such that $a_I$ is a nondegenerate symmetric matrix. Then

$$\begin{aligned} F_KF_{I}{\mathbf f}\sim F_K\frac{\exp \left( -\frac{\pi i}{2}p(a)\right) }{\sqrt{\det (|a|)}} f\left( i\hbar \frac{\partial }{\partial \widehat{x}_I}\right) \exp \left( \frac{-a^{-1}(\widehat{x}_I+b\widehat{x}_J)^2}{2i\hbar }\right) \end{aligned}$$

(13.5.5)

for all K.

1.5.1 13.5.1 Operators on ${\mathtt {\mathcal H}}$

Clearly, the operators $F_K$ (13.5.2) preserve the equivalence relation and therefore descend to ${\mathtt {\mathcal H}}.$

1.5.2 13.5.2 The Action of ${\widehat{{\widehat{\mathbb A}}}}$ on ${\mathcal H}$

The algebra ${\widehat{{\widehat{\mathbb A}}}}$ acts on the space ${\mathcal H}$ as follows. On the summand $F_I {\mathcal H}^{\widehat{x}}_a$,

$$\begin{aligned} \widehat{x}_j F_I {\mathbf f}=-F_I i\hbar \frac{\partial }{\partial \widehat{x}_j}{\mathbf f}, j\in I;\; \widehat{x}_j F_I {\mathbf f}=F_I \widehat{x}_j {\mathbf f},j\notin I; \end{aligned}$$

(13.5.6)

$$\begin{aligned} \widehat{\xi }_j F_I {\mathbf f}=F_I i\hbar \frac{\partial }{\partial \widehat{x}_j}{\mathbf f}, j\notin I;\; \widehat{\xi }_j F_I {\mathbf f}=F_I \widehat{x}_j {\mathbf f},j\in I. \end{aligned}$$

(13.5.7)

1.5.3 13.5.3 The Action of ${\text {Sp}}^4(2n)$ on ${\mathcal H}$

This action is exactly as described in Sect. 13.3.3. In particular, ${\text {Sp}}^4(2n,{\mathbb R})$ acts by the metaplectic representation as in Sect. 13.4:

$$\begin{aligned} T:\left[ \begin{array}{cc} 1&{}0\\ a&{}1\end{array}\right] \mapsto \exp \left( \frac{a\widehat{x}^2}{2i\hbar }\right) ; \; \left[ \begin{array}{cc} 0&{}1\\ -1&{}0\end{array}\right] \mapsto F; \end{aligned}$$

(13.5.8)

more generally, let ${\mathbf F}_I$ be the matrix that is the direct sum of $\left[ \begin{array}{cc} 0&{}1\\ -1&{}0\end{array}\right]$ in coordinates $\widehat{x}_I,\widehat{\xi }_I$ and the identity matrix in the rest of the Darboux coordinates maps to $F_I;$

$$\begin{aligned} \left[ \begin{array}{cc} b&{}0\\ 0&{}^tb^{-1}\end{array}\right] \mapsto T_b;\;(T_bf)(x)=\frac{1}{\sqrt{\det (b)}}f(b^{-1}x) \end{aligned}$$

(13.5.9)

Remark 13.4

The construction of ${\mathcal H}$ mimics very closely the construction of the orbit of 1 under the action of ${\text {Sp}}^4(2n)\ltimes {\mathbb C}[\widehat{x}, \widehat{\xi }]$ on the space of distributions via differential operators and the standard metaplectic representation.

Lemma 13.5

Assign to $F_I \exp ({\frac{ a\widehat{x}^2}{2i\hbar }})f(\widehat{x})\in {\mathtt {\mathcal H}}$ the Lagrangian subspace ${\mathbf F}_I (\{\widehat{\xi }=a\widehat{x}\})$ where ${\mathbf F}_I$ was defined after (13.5.8). This is a well-defined map ${\mathtt {\mathcal H}}\rightarrow \Lambda (n)$ where $\Lambda (n)$ is the Grassmannian of Lagrangian subspaces in ${\mathbb R}^{2n}.$ The space ${\mathcal H}$ is identified with the space of finitely supported sections of a ${\widetilde{\mathbf G}}$-equivariant vector bundle on $\Lambda (n).$

The Lagrangian Grassmannian is a homogeneous space of ${\widetilde{\mathbf G}}$ via the projection ${\widetilde{\mathbf G}}\rightarrow {\text {Sp}}^4\rightarrow {\text {Sp}}.$ In fact,

$$\Lambda (n)\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\widetilde{\mathbf G}}/{\widetilde{\mathbf P}}.$$

Lemma 13.6

The lines ${\mathbb C}F_I \exp (\frac{a\widehat{x}^2}{2i\hbar })$ where a runs through real symmetric $n\times n$ matrices and I through subsets of $\{1,\ldots ,n\}$ form a line subbundle of ${\mathcal H}$ which is isomorphic to the bundle on $\Lambda (n)$ determined by the Čech one-cohomology class $(-1)^{\mu _L}$ where $\mu _L$ is the generator of $H^1(\Lambda (n), {\mathbb Z})$ (the Maslov class).

Lemma 13.7

The actions described in Sects. 13.5.2 and 13.5.3 turn ${\mathcal H}$ into an ${\mathtt {\mathcal A}}$-module.

1.6 13.6 The Algebraic Metaplectic Representation as an Induced Module

Proposition 13.8

$${\mathtt {\mathcal H}}\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\widehat{{\mathtt {\mathcal V}}}}={\mathtt {\mathcal A}}{\widehat{\otimes }} _{{\mathcal B}} \widehat{\widehat{{\mathbb V}}}_{{\mathbb K}}$$

(cf. Sect. 9.2.1).

14 Appendix. Twisted Bundles and Groupoids

1.1 14.1 Charts and Cocycles

Suppose we have a manifold X with two sheaves of groups $\underline{ C}\subset \underline{ G}$ where $\underline{ C}$ is constant and central in $\underline{ G}.$ Consider a class $c\in H^2(X,\underline{ C}).$ A $\underline{ G}$-bundle on X twisted by c is given by an equivalence class of $g_{ij} \in \underline{ G}(U_i\cap U_j)$ for an open cover $X=\cup U_i$ such that

$$\begin{aligned} g_{ij}g_{jk}=c_{ijk} g_{ik} \end{aligned}$$

(14.1.1)

where $c_{ijk}$ is a Čech cocycle representing c. Two data $g_{ij}$ and $g'_{ij}$ are equivalent if

$$\begin{aligned} g_{ij}=h_i g'_{ij} h_j^{-1} b_{ij} \end{aligned}$$

(14.1.2)

for some common refinement of the two covers, where $h_i \in \underline{ G}(U_i)$ and $b_{ij}\in \underline{ C}(U_i \cap U_j).$ Note that this definition makes all $\underline{ C}$-bundles equivalent.

By a chart we mean a map $T\rightarrow X$ where T is any topological space. A good collection of charts on X is a collection of charts $T\rightarrow X$, $T\in {\mathtt {\mathcal T}}$, such that for every $T_0, \ldots , T_p$ in ${\mathtt {\mathcal T}}$, every one-cocycle on $T_0 \times _X \ldots \times _X T_p$ with values in the pullback of $\underline{ G}$, and every one- or two-cocycle with values in the pullback of $\underline{ C}$, can be trivialized.

Lemma 14.1

For any good collection of charts and any twisted bundle, one can define

$$\begin{aligned} c_{TT'T''} \in \underline{ C}(T\times _X T' \times _X T''); \; g_{TT'} \in \underline{ G}(T\times _X T') \end{aligned}$$

(14.1.3)

satisfying

$$\begin{aligned} c_{ TT'T''}c_{TT''T'''}=c_{TT'T'''}c_{T'T''T'''} \end{aligned}$$

(14.1.4)

and

$$\begin{aligned} g_{TT'}g_{T'T''}=c_{TT'T''}g_{TT''} \end{aligned}$$

(14.1.5)

in such a way that, if $T_i$ are a good open cover, then $c_{T_iT_jT_k}$ is cohomologous to $c_{ijk}$ and $g_{T_iT_j}$ is equivalent to $g_{ij}.$ The choice is unique up to equivalence in the following sense:

$$\begin{aligned} c_{TT'T''}=c'_{TT'T''}b_{TT'}b_{T'T''}b_{TT''}^{-1}; \; g_{TT'}=h_Tg'_{TT'} h_{T'}^{-1} b_{TT'} \end{aligned}$$

(14.1.6)

for some $b_{TT'}\in \underline{ C}(T\times T')$ and $h_T\in \underline{ G}(T).$

Proof

Consider inverse images on charts T of an open cover $\{U_i\}$ of X. Let

$$c_{ijk}=\alpha _{ij}(T)\alpha _{jk}(T)\alpha _{ik}(T)^{-1}$$

be a trivialization of c on T. Choose trivializations

$$g_{ij}\alpha _{ij}(T)^{-1}=h_i(T)h_j(T)^{-1}$$

on T and

$$\alpha _{ij}(T)\alpha _{ij}(T')=\beta _i(T,T')\beta _j(T,T')^{-1}$$

where $\alpha _{ij}, \,\beta _{ij}$ are sections of $\underline{ C}$ and $h_i$ are sections of $\underline{ G}.$ Put

$$\begin{aligned} c_{TT'T''}=\beta _i(T,T') \beta _i (T',T'') \beta _i (T,T'')^{-1} \end{aligned}$$

(14.1.7)

and

$$\begin{aligned} g_{TT'}=h_i(T)^{-1} h_i(T') \beta _i(T,T') \end{aligned}$$

(14.1.8)

The relations above show that these do not depend on i.   $\square$

1.2 14.2 The Groupoid of a Twisted G-Bundle

Let G be a Lie group and $\underline{ G}$ the sheaf of smooth G-valued functions. Let C be a central subgroup of G and $\underline{ C}$ the sheaf of locally constant C-valued functions. Consider a $\underline{ C}$-valued two-cohomology class represented by a cocycle $c_{ijk}$ and a twisted G-bundle represented by a $\underline{ G}$-valued cochain $g_{jk}.$

Define a groupoid on X as follows.

For $x_0$ and $x_1$ in X, define the set ${\widetilde{\mathbf G}}_{x_0,x_1}.$ Let $\gamma : [0,1]\rightarrow X$ be a smooth map. View it as a chart that we denote by T. An element of ${\widetilde{\mathbf G}}_{x_0,x_1}$ is a class of an element $g_T\in G$ with respect to the following equivalence relation. Consider two charts T and $T'$ representing two smooth maps $\gamma ,\,\gamma ': [0,1]\rightarrow X$ and a homotopy $\sigma : [0,1]^2\rightarrow X$ such that $\sigma (0, s)=x_0,\, \sigma (s,t)=x_1,\, \sigma (t,0)=\gamma (t)$, and $\sigma (t,1)=\gamma '(t).$ We will view $\sigma$ as a chart S. We call S a homotopy between S and $S'.$ Now generate the equivalence relation by the following.

$$\begin{aligned} g_T \sim (g_{TT'}c_{TT'S}^{-1} )(x_0) g_{T'} (g_{TT'}c_{TT'S}^{-1} )(x_1)^{-1} \end{aligned}$$

(14.2.1)

Lemma 14.2

Let S be a homotopy between T and $T'$, $S'$ a homotopy between $T'$ and $T''$, and $S''$ a homotopy between T and $T''.$ If we denote the right hand side of (14.2.1) by $a(S)g_T$, then $a(S)a(S')=\langle c, [\Sigma ] \rangle a(S'')$ where $\Sigma$ is the sphere formed by S, $S'$, and $S''.$

Proof

We have

$$a(S)a(S')g_T=$$

$$g_{TT'}g_{T'T''}c_{TT'S}^{-1}c_{T'T''S'}^{-1}(x_0) g_{T''} (g_{TT'}g_{T'T''}c_{TT'S}^{-1}c_{T'T''S'}^{-1}(x_1))^{-1} $$

The right hand side is equal to

$$(g_{TT''}c_{TT'T''}c_{TT'S}^{-1}c_{T'T''S'}^{-1})(x_0) g_{T''} (g_{TT''}c_{TT'T''}c_{TT'S}^{-1}c_{T'T''S'}^{-1})(x_1)^{-1} ;$$

therefore

$$a(S)a(S')= \frac{c_{TT'T''}c_{TT''S''}}{c_{TT'S}c_{T'T''S'}}(x_0) (\frac{c_{TT'T''}c_{TT''S''}}{c_{TT'S}c_{T'T''S'}}(x_1))^{-1}a(S'')$$

Applying the cocyclicity condition to the quadruple of charts $TT'T''S$, we get

$$\frac{c_{TT'T''}c_{TT''S''}}{c_{TT'S}c_{T'T''S'}}=\frac{c_{TT''S''}c_{T'T''S}}{c_{TT''S}c_{T'T''S'}}$$

Applying the same condition to $TT''SS''$ and then to $SS'S''T''$, we replace the right hand side with

$$\frac{c_{TSS''}c_{T''SS'}}{c_{T''S'S''}c_{T'SS'}}=\frac{c_{SS'S''}c_{TSS''}}{c_{T'SS'}c_{T''S'S''}}.$$

But $T\times _X S\times _X S''=T, \, T'\times _X S\times _X S'=T'$, and $T''\times _X S'\times _X S''=T''.$ Therefore the values of $c_{TSS''}$, etc. at $x_0$ and $x_1$ are the same. Therefore

$$a(S)a(S')= c_{SS'S''}(x_0) c_{SS'S''}(x_1) ^{-1} a(S'')$$

But

$$ c_{SS'S''}(x_0) c_{SS'S''}(x_1) ^{-1} = \langle c, [\Sigma ] \rangle $$

for any two-cocycle c. To see this, note that the left hand side is 1 for any coboundary c. On the other hand, if we enlarge $S,\,S',\, S''$ a little bit to make them an open cover of $\Sigma$, take an element a of C, and define $c_{SS'S''}(x_0)=a, \,c_{SS'S''}(x_1)=1$, the result will be $a=\langle c,[\Sigma ] \rangle$.   $\square$

Corollary 14.3

There is an epimorphism

$$\begin{aligned} {\widetilde{\mathbf G}}_{x_0,x_1}\rightarrow \pi _1(x_0,x_1) \end{aligned}$$

(14.2.2)

When $x_0=x_1=x$, the kernel of this epimorphism is isomorphic to $G/\langle c, \pi _2(X)\rangle .$

1.2.1 14.2.1 Example: The Holonomy Groupoid of a Vector Bundle

Let E be a real oriented vector bundle of rank N. Let $G=\mathrm{{SO}}_N({\mathtt {\mathbb R}})$ and ${\widetilde{G}}=\mathrm{{Spin}}_N ({\mathtt {\mathbb R}})$ its universal cover. Reduce the structure group of E to G using a Riemannian metric. Let ${\widetilde{{\text {Isom}}}}(E)_{x_0, x_1}$ be the set of equivalence classes of data $(\gamma , u_t)$ where $\gamma : [0,1]\rightarrow X$ is a smooth map, $\gamma (0)=x_0$, $\gamma (1)=x_1$, and $u_t : E_{\gamma (t)}\,{\mathop {\rightarrow }\limits ^{\sim }}\,E_{\gamma (0)}$ a metric-preserving linear isomorphism smoothly depending on t and satisfying $u_0={\mathtt {Id}}.$ An equivalence between $(\gamma , u_t)$ and $(\gamma ', u_t')$ is a smooth map $\sigma : [0,1]\times [0,1]\rightarrow X$ such that $\sigma (0,s)=x_0,\,\sigma (1,s)=x_1$, $\sigma (t,0)=\gamma (t),\,\sigma (t,1)=\gamma '(t)$, and a linear metric-preserving isomorphism $v_{t,s}: E_{\sigma (t,s)}\,{\mathop {\rightarrow }\limits ^{\sim }}\,E_{x_0}$ smooth in (t, s), such that $v_{0,s}={\mathtt {Id}}$, $v_{t,0}=u_t$, $v_{t,1}=u'_t$, and $v_{1,s}=u_1=u'_1.$

Lift the transition isomorphisms $g^E _{ij}$ of E to some ${\widetilde{g}}_{ij}.$ Put $c_{ijk}={\widetilde{g}}_{ij} {\widetilde{g}}_{jk} {\widetilde{g}}_{ik}^{-1}.$ This cocycle represents the second Stiefel–Whitney class $w_2(E).$ Note that the groupoid ${\widetilde{{\text {Isom}}}}(E)$ is isomorphic to the groupoid ${\widetilde{\mathbf G}}'$ constructed from the twisted bundle defined by ${\widetilde{g}}_{ij}, c_{ijk}.$ In fact, note that for the charts T and S defined by maps $\gamma$ and $\sigma$, there is a natural lifting $\widetilde{ g}_{TS}$ of $g_{TS}.$ Namely, $\widetilde{ g}_{TS} (\gamma (t))$ is the class of the path $g_{TS} (\gamma (\tau )),\,0\le \tau \le t.$ Similarly with $\widetilde{ g}_{ST'}.$ Identify with $\widetilde{G}$ the set of equivalence classes of $(\gamma , u_t)$ with fixed $\gamma$ (and with $\sigma (t,s)=\gamma (t)$ in the definition of the equivalence). Now, given an equivalence $\sigma ,v$ between $\gamma , u$ and $\gamma ',u'$, $g_T\in {\widetilde{G}}$ gets identified with $\widetilde{ g}_{TS} \widetilde{ g}_{ST'}=\widetilde{ g}_{TT'} c_{TST'}.$

Corollary 14.4

There is an epimorphism

$$\begin{aligned} {\widetilde{{\text {Isom}}}}(E)_{x_0,x_1}\rightarrow \pi _1(x_0,x_1) \end{aligned}$$

(14.2.3)

and every preimage is a homogeneous space $\mathrm{{Spin}}(N,{\mathtt {\mathbb R}})/\langle w_2 (E), \pi _2(X)\rangle .$ (We identify ${\mathbb Z}/2$ with the center of $\mathrm{{Spin}}(N,{\mathtt {\mathbb R}})$).

1.2.2 14.2.2 Connections on Twisted Bundles

As in Sect. 14.2, let G be a simply-connected (pro) Lie group and $\underline{ G}$ the sheaf of smooth G-valued functions. Let C be a central subgroup of G and $\underline{ C}$ the sheaf of smooth C-valued functions. In addition, fix some algebra ${\mathtt {\mathcal A}}$ on which G acts by automorphisms. Consider a twisted bundle defined by the data $(g_{ij}, c_{ijk}).$ A connection in this twisted bundle is a collection of ${\mathtt {\mathcal A}}$-valued forms on $U_i$ such that

$${\text {Ad}}_{g_{ij}}(d+A_j)=d+A_i$$

on every $U_{ij}.$ Here ${\text {Ad}}_g(d)=-dg\cdot g^{-1}.$ Note that, because $c_{ijk}$ are locally constant and central, ${\text {Ad}}_{g_{ij}}{\text {Ad}}_{g_{jk}}(d+A_k)={\text {Ad}}_{g_{ik}}(d+A_k)$, so the conditions above are consistent on $U_{ijk}.$ The curvature $R=dA_i+A_i^2$ is a well-defined ${\mathtt {\mathcal A}}$-valued two-form.

1.2.3 14.2.3 The Flat Connection up to Inner Derivations

Here we will construct a flat connection up to inner derivations on the associated bundle of algebras ${\mathtt {\mathcal A}}$ compatible with the action of the groupoid ${\widetilde{\mathbf G}}$ of a twisted bundle (cf. Sect. 14.2). We will start from a flat connection on the twisted bundle itself.

First define special coordinate charts on ${\widetilde{\mathbf G}}$ as follows. Fix:

• two open charts $U_0$ and $U_1$ of X; 

• two points $x_0^*\in U_0$ and $x_1^*\in U_1;$

• a path $\gamma$ from $x_0^*$ to $x_1^*$ in X;

• smooth maps $\tau _0: [0,1]\times U_0\rightarrow U_0$ and $\tau _1: [0,1] \times U_1\rightarrow U_1$, $\tau _0(0,x_0)=x_0$, $\tau _0(1,x_0)=x^*_0$, $\tau _1(0,x_1)=x_1$, $\tau _1(1,x_1)=x^*_1.$

For every $x_0\in U_0$ and $x_1\in U_1$, we will denote the path $t\mapsto \tau _0(t,x_0)$ by $\tau _{x_0}$ and the path $t\mapsto \tau _1(t,x_1)$ by $\tau _{x_1}.$ For the data as above, we construct a chart T in ${\widetilde{\mathbf G}}$ as a map

$$U_0\times U_1\rightarrow {\widetilde{\mathbf G}};\; (x_0,x_1)\mapsto \tau _{x_0} \circ \gamma \circ \tau _{x_1}: x_0\rightarrow x_1$$

(the composition of paths).

Now consider a flat connection in our twisted bundle. In a local trivialization, on any open chart W, we write $\nabla _{\mathtt {\mathcal V}}=d+A_W.$ We can identify a local section of ${\widetilde{\mathbf G}}$ on T with a ${\widetilde{G}}$-valued function $g_T(x_0,x_1)$ on $U_0\times U_1.$

Definition 14.5

$$\alpha (g_T)=-d g_T\cdot g_T^{-1}-A_0+{\text {Ad}}_{g_T}(A_1)$$

where $A_0=\pi _0 ^*(A_{U_0})$ and $A_1=\pi _1 ^*(A_{U_1});$

$$R=dA_0+A_0^2.$$

Lemma 14.6

The above formulas define a flat connection up to inner derivations on the associated bundle of algebras ${\mathtt {\mathcal A}}$ compatible with the action of ${\widetilde{\mathbf G}}.$

15 Appendix. Modules Associated to Lagrangian Submanifolds and Lagrangian Distributions

For any Lagrangian submanifold L of a symplectic manifold M with a given ${\text {Sp}}^4$ structure we constructed a bundle of modules ${\widehat{\widehat{{\mathbb V}}}}_L$ with a flat connection $\nabla _{\mathbb V}$ (cf. Sect. 9.2.2). This is a bundle of ${\widehat{\mathtt {\mathbb A}}}_M$-modules, and the connections $\nabla _{\mathbb V}$ and $\nabla _\mathtt {\mathbb A}$ are compatible. In particular, denote by $\mathtt {\mathbb A}_M$ the sheaf of algebras of horizontal sections of $\nabla _\mathtt {\mathbb A}$ and by ${\mathbb V}_L$ the sheaf of horizontal sections of $\nabla _{\mathbb V}.$ Then ${\mathbb V}_L$ is a sheaf of $\mathtt {\mathbb A}_M$-modules.

Now apply the same construction to L but instead of M take a tubular neighborhood of L and identify it with the tubular neighborhood of L in $T^*L$ by Darboux–Weinstein theorem. Use the ${\text {Sp}}^4$ structure provided by the Lagrangian polarization by fibers of $T^*L$ (cf. Lemma 12.8). We get another $\mathtt {\mathbb A}_M$-module that we denote by ${\mathbb V}^{(0)}_L.$

Lemma 15.1

${\mathbb V}_L$ is isomorphic to ${\mathbb V}^{(0)}_L$ twisted by the $\{\pm 1,\pm i\}$-valued Maslov class of L.

We denote this class by $\exp (\frac{\pi i}{2} \mu (L)).$ Note that $\mu (L)$ can be chosen as a ${\mathtt {\mathbb Z}}$-valued cocycle only if $2c_1(M)=0.$

1.1 15.1 The Asymptotic Construction of Hörmander and Maslov

As we have seen in Sect. 9.2.2, the oscillatory module ${\mathtt {\mathcal V}}^\bullet _L$ is induced from the module of forms with coefficients in ${\widehat{\widehat{{\mathbb V}}}}.$ But it is the twisted version of the latter module that serves as an asymptotic version of the classical construction of Lagrangian distributions with wave front L.

Put

$$\begin{aligned} {\mathbb V}_{L, {\mathbb K}} ={\mathbb K}{\widehat{\otimes }}{\mathbb V}_L= \left\{ \sum _{k=0}^\infty e^{\frac{1}{i\hbar }c_k} v_k |v_k \in {\mathbb V}_L ;\, c_k\in {\mathtt {\mathbb R}};\, c_k \rightarrow \infty \right\} \end{aligned}$$

(15.1.1)

Definition 15.2

Assume $M=T^*X.$ Let ${\mathbb V}_{L,{\mathbb K}}^{\eta }$ be the twist of the sheaf ${\mathbb V}_{L, {\mathbb K}}$ by the Čech cohomology class $\exp (-\frac{1}{i\hbar }\eta ) \in H^1(L,\exp (\frac{1}{i\hbar }{\mathtt {\mathbb R}}))$ where $\eta$ is the class of the form $\xi dx|L.$

Let $X=\cup U_\alpha$ is a small open cover. Let $L=\cup W_\gamma$ be a refinement of the cover $L=\cup (T^*U_\alpha \cap L).$ In particular, a choice is made of $\gamma \mapsto \alpha = \alpha (\gamma )$ such that $W_\gamma \subset T^*U_\alpha \cap L.$

1.1.1 15.1.1 Quantization Procedure

First let us review our deformation quantization picture in the case $M=T^*X.$ First, we have the sheaf of algebras $\mathtt {\mathbb A}_{T^*X}.$ It can be described by products $*_\alpha$ on $C^\infty (T^*U_\alpha )[[\hbar ]]$

$$\begin{aligned} a*_\alpha b=\sum _{k=0}^\infty (i\hbar )^k P_{\alpha ,k} (a,b) \end{aligned}$$

(15.1.2)

and by transition functions

$$\begin{aligned} G_{\alpha \beta }(a)=\sum _{k=0}^\infty (i\hbar )^k T_{\alpha \beta , k}(a) \end{aligned}$$

(15.1.3)

where $P_{\alpha ,k}$ are bilinear bidifferential expressions, $T_{\alpha \beta ,k}$ are differential operators, $P_{\alpha ,0}(f,g)=fg$, $P_{\alpha ,1}(f,g)=\frac{1}{2}\{f,g\}$, and $T_{\alpha \beta ,0}(f)=f.$ One has $G_{\alpha \beta }(a*_\beta b)=G_{\alpha \beta }(a)*_\alpha G_{\alpha \beta }(b).$ Actually in our $C^\infty$ case, unlike the complex analytic or algebraic case, $G_{\alpha \beta }$ can be made the identity automorphisms, but this is not necessarily the most natural choice.

The sheaf of modules ${\mathbb V}^\eta _L$ is described by the action

$$\begin{aligned} a*_\gamma f=\sum _{k=0}^\infty (i\hbar )^k Q_{\gamma ,k} (a,f) \end{aligned}$$

(15.1.4)

where $f\in |\Omega |^{\frac{1}{2}}(W_\gamma )$ and $a\in C^\infty (U_{\alpha (\gamma )})$, and by the transition functions

$$\begin{aligned} H_{\gamma \delta }(f)=\exp \left( -\frac{1}{i\hbar }\eta _{\gamma \delta }\right) \sum _{k=0}^\infty (i\hbar )^k S_{\gamma \delta , k}(f) \end{aligned}$$

(15.1.5)

where $Q_{\gamma ,k}$ are bidifferential and $S_{\gamma \delta ,k}$ are differential. Moreover, $Q_{\gamma \delta ,0}(a,f)=af$ and

$$\begin{aligned} S_{\gamma \delta ,0}(f)=\exp \left( \frac{\pi i}{2}\mu _{\gamma \delta }(L)\right) f. \end{aligned}$$

(15.1.6)

One has

$$a*_\gamma (b*_\gamma f)=(a*_{\alpha (\gamma )} b)*_\gamma f$$

and

$$S_{\gamma \delta }(a*_\delta f)=T_{\alpha (\gamma )\alpha (\delta )}(a)*_\gamma S_{\gamma \delta }(f)$$

Again, all higher $S_{\gamma \delta , k}$ can be made zero, but this is not the most natural choice.

Let $C^\infty _\mathrm{{poly}}$ denote functions on $T^*X$ that are polynomial on fibers. A quantization procedure is the following.

(1) For any $\alpha$, a map

$$\begin{aligned} \mathrm{{Op}}^\alpha _\hbar : C^\infty _\mathrm{{poly}} (T^*(U_\alpha ) )\rightarrow {\mathtt {\mathcal D}}(U_\alpha , |\Omega |_X^{\frac{1}{2}}) \end{aligned}$$

(15.1.7)

such that

$$\mathrm{{Op}}^\alpha _\hbar (a) \mathrm{{Op}}^\alpha _\hbar (b)= \mathrm{{Op}}^\alpha _\hbar (a*_\alpha b)$$

and

$$\mathrm{{Op}}^\alpha _\hbar (G_{\alpha \beta }(a))=\mathrm{{Op}}^\beta (a)$$

on $U_\alpha \cap U_\beta .$ (We can ask for exact equalities, not for asymptotic equalities like we use below, when a and b are polynomial).

(2) A map

$$\begin{aligned} u^\gamma _\hbar : |\Omega |^{\frac{1}{2}}_{c} (W_\gamma ) \rightarrow |\Omega |^{\frac{1}{2}}_{c} ( U_{\alpha (\gamma )}) \end{aligned}$$

(15.1.8)

for all $\hbar >0$, such that

$${\text {Op}}^{\alpha (\gamma )}_\hbar (a) u^\gamma _\hbar (f) - \sum _{k=0}^N (i\hbar )^k u^\gamma _\hbar (Q_{\gamma ,k} (a,f))=O(h^{N+1})$$

and

$$u^\gamma (f)-\sum _{k=0}^N (i\hbar )^k u^\delta _\hbar (S_{\gamma ,\delta ,k}(f)) = O(h^{N+1})$$

for all N.

Let us recall how a quantization procedure is carried out. For every $\gamma$ choose a phase function for $L\cap W_\gamma$ as follows. Let $\theta =(\theta _1,\ldots ,\theta _k)$ be a coordinate system on ${\mathtt {\mathbb R}}^k.$ Choose a coordinate system $x=(x_1,\ldots ,x_n)$ on $U_{\alpha (\gamma )}$. Choose a phase function for $L\cap W_\gamma$, i.e. a function $\varphi (x,\theta )$ such that

$$\begin{aligned} L\cap W_\gamma =\{(\xi ,x)|\exists \theta \; \mathrm{{ such}}\; \mathrm{{that}} \;\xi =\varphi _x (x,\theta )\; \mathrm{{and}}\; \varphi _\theta (x,\theta )=0\} \end{aligned}$$

(15.1.9)

Here $\varphi _x$ and $\varphi _\theta$ stand for partial derivatives. We assume that the $n\times (n+k)$ matrix $(\varphi _{xx}, \varphi _{x\theta })$ is nondegenerate.

Example 15.3

Let $n=1.$ Assume that $L=\{\xi =\varphi '(x)\}.$ Then we can choose $k=0$ and $\varphi =\varphi (x).$ Now let $L=\{x=-\psi '(\xi )\}.$ Then we can take $k=1$ and $\varphi (x,\theta )=x\theta +\psi (\theta ).$

Example 15.4

More generally, one can always subdivide the coordinates into two groups and write $x=(x_1,x_2);$ $\xi =(\xi _1,\xi _2)$ so that $L\cap W_\gamma$ will be of the form

$$\begin{aligned} \xi _1=F_{x_1}(x_1,\xi _2);\; x_2=-F_{\xi _1}(x_1,\xi _2) \end{aligned}$$

(15.1.10)

In this case one can take $\varphi (x_1,x_2,\theta )=x_2\theta +F(x_1,\theta ).$

Note that the condition that the matrix of second derivatives is nondegenerate means that $\theta$ in (15.1.9) is unique and therefore $L\cap W_\gamma$ can be identified with $\{(x,\theta )|\varphi _\theta (x,\theta )=0\}.$ (To do that, one may need to pass to a finer open cover). Moreover, we can choose n out of $n+k$ coordinates $x,\theta$ so that they will be coordinates on $\{\varphi _\theta =0\}.$ Namely, we can take any n coordinates such that the corresponding square submatrix of $(\varphi _{xx}, \varphi _{x\theta })$ is nondegenerate. Denote these coordinates by z and the other k coordinates by $\zeta .$ Choose a procedure for extending functions f(z) to functions on $\{(x,\theta )\}.$ Namely, extend f(z) to $f(z)\rho (z')$ where $\rho$ is a function with small support near zero and $\rho (z')=0.$

Given a phase function and a compactly supported half-form $f=f(z)|dz|^{\frac{1}{2}}$, define $u^\gamma _\hbar (f)$ as follows. Denote by $f(x,\theta )$ the extension of f(z) as above. Then define

$$\begin{aligned} u_\hbar (f) = \frac{e^{\frac{-\pi i k}{4}}}{(2\pi \hbar )^{\frac{k}{2}}} \int e^{-\frac{\varphi (x,\theta )}{i\hbar }} f(x,\theta ) d\theta |dx|^{\frac{1}{2}} \end{aligned}$$

(15.1.11)

For the sake of completeness let us outline the proof of the fact that this is indeed a quantization procedure as described above (it is contained essentially in [15, 16], as well as in [32]).

First, as proven in [16], any two local phase functions differ by a coordinate change

$$ \varphi (x,\theta )\mapsto \varphi (g(x), h(x,\theta )) $$

followed by iterated application of

$$ \varphi (x,\theta )\mapsto \varphi (x,\theta )\pm \theta _1^2 $$

to one or the other phase function. Here $\theta _1$ is an extra variable. So we can assume that our local phase functions are as in Example 15.4, possibly with some $\theta _1^2$ added or subtracted. We have two choices of subdivision $x=(x_1,x_2).$ Namely, for $W_\gamma$ we will have

$$x^\gamma _1=(x_1,x_2);\;x^\gamma _2=(x_3,x_4);$$

for $W_\delta$,

$$\;x^\delta _1=(x_1,x_3);\;x^\delta _2=(x_2,x_4).$$

Let $F_\gamma (x_1,x_2,\xi _3,\xi _4)$ and $F_\delta (x_1,x_3,\xi _2,\xi _4)$ be functions as in Example 15.4. Let us look for functions $f_\gamma$ and $f_\delta$ such that (15.1.11) will give the same answer for the charts $W_\gamma$ and $W_\delta .$

$$\begin{aligned} \frac{e^{-\frac{\pi i}{4} (k_3+k_4)}}{(2\pi \hbar )^{\frac{k_3+k_4}{2}}} \int e^{-\frac{1}{i\hbar }(x_3\xi _3+x_4\xi _4 +F_\gamma (x_1,x_2,\xi _3,\xi _4)} f_\gamma (x_1,x_2,\xi _3,\xi _4)d\xi _3d\xi _4= \end{aligned}$$

(15.1.12)

$$ =\frac{e^{-\frac{\pi i}{4} (k_2+k_4)}}{(2\pi \hbar )^{\frac{k_2+k_4}{2}}} \int e^{-\frac{1}{i\hbar }(x_2\xi _2+x_4\xi _4 +F_\delta (x_1,x_3,\xi _2,\xi _4)} f_\delta (x_1,x_3,\xi _2,\xi _4)d\xi _2d\xi _4 $$

Applying the inverse Fourier transform we get

$$\begin{aligned} e^{-F_\gamma }f_\gamma = \frac{e^{-\frac{\pi i}{4} (k_2-k_3)}}{(2\pi \hbar )^{\frac{k_2+k_3}{2}}} \int e^{\frac{1}{i\hbar }(-x_2\xi _2+x_3\xi _3-F_\delta })f_\delta d\xi _2 dx_3 \end{aligned}$$

(15.1.13)

Compute the right hand side by the stationary phase method. The critical points satisfy

$$\begin{aligned} x_2=-\frac{\partial F_\delta }{\partial \xi _2};\; \xi _3=\frac{\partial F_{\delta }}{\partial x_3} \end{aligned}$$

(15.1.14)

In other words, the critical point $(\xi _2,x_3)$ is such that $(x_1,x_2,\xi _1,\xi _2)$ is in L.

$$\begin{aligned} f_\gamma = \epsilon _{\gamma \delta } \exp (\frac{1}{i\hbar } ((x_3\xi _3-F_\delta )-(x_2\xi _2-F_\gamma ))) \mathrm{{mod}}\hbar \end{aligned}$$

(15.1.15)

or

$$\begin{aligned} f_\gamma = \epsilon _{\gamma \delta } \exp \left( \frac{1}{i\hbar } (\varphi _\delta -\varphi _\gamma )\right) \mathrm{{mod}}\hbar \end{aligned}$$

(15.1.16)

Here

$$\begin{aligned} \epsilon _{\gamma \delta } = e^{-\frac{\pi i}{4} (k_2-k_3)} e ^{-\frac{\pi i}{4} (n_-(\gamma ,\delta )-n_+(\gamma ,\delta ))} \end{aligned}$$

(15.1.17)

where $n_-(\gamma ,\delta )$, resp. $n_+(\gamma ,\delta )$, is the number of negative, resp. positive, eigenvalues of the matrix of second derivatives of $F_\delta$ with respect to variables $\xi _2$ and $x_3.$ We can re-write (15.1.17) as

$$\begin{aligned} \epsilon _{\gamma \delta }=\exp {\frac{\pi i}{2}(n_+-k_2)} \end{aligned}$$

(15.1.18)

where, as above, $n_+$ is the number of positive eigenvalues of the matrix of second derivatives of $F_\delta$ in variables $x_2,\xi _3.$

Example 15.5

Let $F_\gamma (x)=\varphi (x)$ and $F_\delta (x,\theta )=x\theta -\psi (\theta )$ as in Example 15.3. Let us compute $\epsilon _{\gamma _\delta }.$ One has $k_2=1.$ If $\varphi _{xx}>0$ then $n_2=0$. If $\varphi _{xx}<0$ then $n_2=1.$ Therefore

$$\epsilon _{\gamma _\delta }=-1\;\mathrm{for}\; \varphi _{xx}>0;\; \epsilon _{\gamma _\delta }=0 \;\mathrm{{for}}\; \varphi _{xx}<0.$$

Now compute $\epsilon _{\delta \gamma }.$ One has $k_2=0.$ If $\varphi _{xx}>0$ then $n_2=1$. If $\varphi _{xx}<0$ then $n_2=0.$ Therefore

$$\epsilon _{\delta \gamma }=1\;\mathrm{{for}}\;\varphi _{xx}>0;\;\epsilon _{\delta \gamma }=0\;\mathrm{{for}}\;\varphi _{xx}<0.$$

Now note that $d\varphi _\gamma = \xi dx|L$ on $L\cap W_\gamma$ and $d\varphi _\delta = \xi dx|L$ on $L\cap W_\delta .$ Therefore, if $\eta _{\gamma \delta }=\varphi _\gamma -\varphi _\delta$ on $L\cap W_\gamma \cap W_\delta$, then $(\eta _{\gamma \delta })$ represents the cohomology class $\eta$ corresponding to the De Rham class of $\xi dx|L.$

On the other hand, a choice of a local presentation (15.1.10) of L determines a choice of lifting of transition isomorphisms as in (12.1.4). Indeed, in a tangent space $T_{(x,\xi )} L$ to a point of $L\cap W_\gamma$, let $\widehat{x}$, $\widehat{\xi }$ be formal Darboux coordinates coming from some local coordinate system. Choose a presentation

$$\begin{aligned} \widehat{\xi }_1=A\widehat{x}_1+B\widehat{\xi }_2;\; \widehat{x}_2=-C\widehat{x}_1-D\widehat{\xi }_2 \end{aligned}$$

(15.1.19)

Construct a symplectic matrix sending $L_0=\{\widehat{\xi }_1=\widehat{\xi }_2=0\}$ to $T_{(x,\xi )} L$ as follows. Let

$$\begin{aligned} p(A,B,C,D): (\widehat{x}_1, \widehat{x}_2, \widehat{\xi }_1, \widehat{\xi }_2)\mapsto (\widehat{x}_1,\widehat{x}_2,A\widehat{x}_1+B\widehat{x}_2,C\widehat{x}_1+D\widehat{x}_2) \end{aligned}$$

(15.1.20)

and

$$\begin{aligned} F_{\widehat{x}_2} : (\widehat{x}_1, \widehat{x}_2, \widehat{\xi }_1, \widehat{\xi }_2)\mapsto (\widehat{x}_1, -\widehat{\xi }_2, \widehat{\xi }_1, \widehat{x}_2) \end{aligned}$$

(15.1.21)

One has

$$\begin{aligned} T_{(x,\xi )} L=F_{\widehat{x}_2} p(A,B,C,D) L_0 \end{aligned}$$

(15.1.22)

Note also that both factors of the right hand side extend automatically to elements in ${\text {Sp}}^4$. Indeed, one can replace p(A, B, C, D) by the homotopy class of the path p(tA, tB, tC, tD), $0\le t\le 1$, and $F_{\widehat{x}_2}$ by the homotopy class of the path

$$(\widehat{x}_1, \widehat{x}_2, \widehat{\xi }_1, \widehat{\xi }_2)\mapsto (\widehat{x}_1, \widehat{x}_2\cos t -\widehat{\xi }_2 \sin t, \widehat{\xi }_1, \widehat{x}_2 \sin t + \widehat{\xi }_2 \cos t),\, 0\le t\le \frac{\pi }{2}$$

It is easy to see that the Maslov class $\mu$ corresponding to the lifted transition functions thus defined is inverse to the one defined by (15.1.18).

16 Appendix. Twisted $A_\infty$ Modules and $A_\infty$ Functors

1.1 16.1 Differential Graded Categories of $A_\infty$ Functors

Our references for this Section are [23] and [24] (see also [8] and the survey [41]).

Let A and B be two differential graded (DG) categories. For two maps

$$\mathbf f,\mathbf g:{\text {Ob}}\,(A)\rightarrow {\text {Ob}}\,(B)$$

define

$${\overline{C}}^\bullet _{\mathbf f,\mathbf g} (A,B)= \prod _{{n\ge 1};x_0,\ldots ,x_n} {\text {Hom}}^\bullet (A(x_0,x_1)\otimes \ldots \otimes A(x_{n-1},x_n)[n], B(f(x_0), g(x_n)))$$

where the product is taken over all $x_0,\ldots , x_n\in {\text {Ob}}\,(A).$ Put

$$\begin{aligned} C^\bullet _{\mathbf f,\mathbf g}(A,B)=\prod _{x_0\in {\text {Ob}}\,(A)} B(f(x_0), g(x_0)) \times {\overline{C}}^\bullet _{\mathbf f,\mathbf g} (A,B) \end{aligned}$$

(16.1.1)

Define the differential d by

$$\begin{aligned} (d_1\varphi )(a_1,\ldots ,a_{n+1})=\sum _{j=1}^n (-1)^{\sum _{p\le j} (|a_p|+1)} \varphi (a_1,\ldots , a_ja_{j+1},\ldots , a_{n+1}) \end{aligned}$$

(16.1.2)

($d_1=0$ on the first factor of (16.1.1));

$$\begin{aligned} (d_2\varphi )(a_1,\ldots ,a_{n})=\sum _{j=1}^n (-1)^{\sum _{p< j} (|a_p|+1)} \varphi (a_1,\ldots , d_Aa_j,\ldots , a_{n+1})+d_B\varphi (a_1,\ldots ,a_n) \end{aligned}$$

(16.1.3)

Define

$$d=d_1+d_2$$

Also define the product

$${\overline{C}}^\bullet _{\mathbf f,\mathbf g} (A,B) \otimes {\overline{C}}^\bullet _{\mathbf g,\mathbf h} (A,B) \rightarrow {\overline{C}}^\bullet _{\mathbf f,\mathbf h} (A,B)$$

by

$$\begin{aligned} (\varphi \smile \psi )(a_1,\ldots ,a_{m+n})=(-1)^{|\psi | \sum _{j=1}^m (|a_j|+1)} \varphi (a_1,\ldots ,a_m)\psi (a_{m+1},\ldots , a_{m+n}) \end{aligned}$$

(16.1.4)

(Note that here m or n can be zero, which corresponds to the case of one or both factors lying in the first factor of (16.1.1)).

Definition 16.1

An $A_\infty$ functor $f:A\rightarrow B$ is a map $f:{\text {Ob}}\,(A)\rightarrow {\text {Ob}}\,(B)$ together with an element f of degree 1 in ${\overline{C}}^\bullet _{\mathbf f,\mathbf f} (A,B)$ such that

$$df+f\smile f=0$$

A curved $A_\infty$ functor is defined the same way but now the cochain f is allowed to be in ${C}^\bullet _{\mathbf f,\mathbf f} (A,B).$

Definition 16.2

Define the DG category $\mathbf{C}(A,B)$ as follows. Let objects be $A_\infty$ functors $\mathbf f:A\rightarrow B;$ set

$$\mathbf{C}^\bullet (A,B)(f,g) = C^\bullet _{\mathbf f,\mathbf g}(A,B)$$

with the differential

$$\delta \varphi = d\varphi +f\smile \varphi - (-1)^{|\varphi |} \varphi \smile f$$

We define the composition to be the cup product.

Also, define the DG category $\mathbf{C}_+(A,B)$ the same way as above but with objects being curved $A_\infty$ functors.

1.1.1 16.1.1 Equivalence of Objects in a DG Category

Let ${\mathtt {\mathbf C}}_1$ be the category with two objects 0 and 1 and two mutually inverse morphisms $g:0\rightarrow 1$ and $g^{-1}:1\rightarrow 0.$

Definition 16.3

Two objects $\mathtt {\mathbf x},\,\mathtt {\mathbf y}$ of a DG category C are equivalent if there is an $A_\infty$ functor $\mathtt {\mathbf C}_1\rightarrow C$ sending 0 to $\mathtt {\mathbf x}$ and 1 to $\mathtt {\mathbf y}.$

Lemma 16.4

The relation defined above is an equivalence relation.

Proof

Let $\mathtt {\mathbf C}_2$ be the category with three objects 0, 1, 2 and with unique morphism between any two objects. There are functors $i_{pq}:{\mathbf C}_1\rightarrow {\mathbf C}_2$ that send 0 to p and 1 to q, $0\le p<q\le 2.$ If we have one equivalence between $\mathtt {\mathbf x}$ and $\mathtt {\mathbf y}$ and another between $\mathtt {\mathbf y}$ and ${\mathbf z}$, then we have a functor (cf. Definitions and Lemma 16.7 below):

$$\begin{aligned} {{\text {Cobar}}}\,{{\text {Bar}}}\,k[i_{01}\mathtt {\mathbf C}_1] *_{k[1]} {{\text {Cobar}}}\,{{\text {Bar}}}\,k[i_{12}\mathtt {\mathbf C}_1] \rightarrow C \end{aligned}$$

(16.1.5)

that sends 0 to $\mathtt {\mathbf x}$, 1 to $\mathtt {\mathbf y}$, and 2 to ${\mathbf z}.$ Here $*$ stands for free product of categories; for any category $\mathtt {\mathbf C}$, $k[\mathtt {\mathbf C}]$ is its linearization, and k[1] is the category with one object 1 whose ring of endomorphisms is k. But the left hand side of (16.1.5) is quasi-isomorphic to $k[i_{01}\mathtt {\mathbf C}_1] *_{k[1]} k[i_{12}\mathtt {\mathbf C}_1]\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\mathtt {\mathbf C}}_2.$ By the standard transfer of structure [24, 25, 28], we get an $A_\infty$ morphism ${\mathbf C}_2\rightarrow C$ that sends 0 to $\mathtt {\mathbf x}$, 1 to $\mathtt {\mathbf y}$, and 2 to ${\mathbf z}.$ Composing it with $i_{02}$, we get an equivalence between $\mathtt {\mathbf x}$ and ${\mathbf z}.$   $\square$

Definition 16.5

Two $A_\infty$ functors $A\rightarrow B$ are equivalent if they are equivalent as objects in ${\mathbf C}(A,B).$

1.1.2 16.1.2 The Bar Construction

The bar construction of a DG category A is a DG cocategory ${\text {Bar}}(A)$ with the same objects where

$${{\text {Bar}}}(A)(x,y)=\bigoplus _{n\ge 0}\bigoplus _{x_1,\ldots , x_n}A(x,x_1)[1]\otimes A(x_1,x_2)[1]\otimes \cdots \otimes A(x_n,x)[1]$$

with the differential

$$d=d_1+d_2;$$

$$d_1(a_1|\cdots | a_{n+1})=\sum _{i=1}^{n+1}\pm (a_1|\cdots |da_i|\cdots |a_{n+1});$$

$$d_2(a_1|\cdots | a_{n+1})=\sum _{i=1}^{n}\pm (a_1|\cdots |a_ia_{i+1}|\cdots |a_{n+1})$$

The second sum is taken over n-tuples $x_1,\ldots ,x_n$ of objects of A. The signs are $(-1)^{\sum _{j<i}(|a_i|+1)+1}$ for the first sum and $(-1)^{\sum _{j\le i}(|a_i|+1)}$ for the second. The comultiplication is given by

$$\Delta (a_1|\cdots |a_{n})=\sum _{i=1}^{n-1}(a_1|\cdots |a_i)\otimes (a_{i+1}|\cdots |a_{n})$$

Dually, for a DG cocategory B one defines the DG category ${\text {Cobar}}(B)$. The DG category ${{\text {Cobar}}}\,{{\text {Bar}}}(A)$ is a cofibrant resolution of A.

It is convenient for us to work with DG (co)categories without (co)units. For example, this is the case for ${\text {Bar}}(A)$ and ${\text {Cobar}}(B)$ (we sum, by definition, over all tensor products with at least one factor). Let $A^+$ be the (co)category A with the (co)units added, i.e. $A^+(x,y)=A(x,y)$ for $x\ne y$ and $A^+(x,x)=A(x,x)\oplus k{\mathtt {Id}}_x.$ If A is a DG category then $A^+$ is an augmented DG category with units, i.e. there is a DG functor $\epsilon : A^+\rightarrow k_{{\text {Ob}}\,(A)}$. (For a set I, $k_I$ is the DG category with the set of objects I and with $k_I(x,y)=0$ for $x\ne y$, $k_I(x,x)=k$). Dually, one defines the DG cocategory $k^{{\text {Ob}}\,(B)}$ and the DG functor $\eta : k^{{\text {Ob}}\,(B)} \rightarrow B^+$ for a DG cocategory B.

For DG (co)categories with (co)units, define $A\otimes B$ as follows: $\mathrm{{Ob}}(A\otimes B)=\mathrm{{Ob}}(A)\times \mathrm{{Ob}}(B);$ $(A\otimes B)((x_1,y_1), (x_2, y_2))=A(x_1,y_1)\otimes B(x_2,y_2);$ the product is defined as $(a_1\otimes b_1)(a_2\otimes b_2)=(-1)^{|a_2||b_1|}a_1a_2\otimes b_1b_2$, and the coproduct in the dual way. This tensor product, when applied to two (co)augmented DG (co)categories with (co)units, is again a (co)augmented DG (co)category with (co)units: the (co)augmentation is given by $\epsilon \otimes \epsilon$, resp. $\eta \otimes \eta .$

Definition 16.6

For DG categories A and B without units, put

$$A\otimes B={\text {Ker}}(\epsilon \otimes \epsilon : A^+\otimes B^+\rightarrow k_{{\text {Ob}}\,(A)} \otimes k_{{\text {Ob}}\,(B)}).$$

Dually, for DG cocategories A and B without counits, put

$$A\otimes B={\text {Coker}}(\eta \otimes \eta :k^{{\text {Ob}}\,(A)}\otimes k^{{\text {Ob}}\,(B)} \rightarrow A^+\otimes B^+).$$

The following is standard (and straightforward).

Lemma 16.7

There are natural bijections

$${\text {Ob}}\,{\mathbf C}(A,B)\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {Hom}}({{\text {Cobar}}}\,{{\text {Bar}}}(A), B);$$

$${\text {Ob}}\,{\mathbf C}_+(A,B)\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {Hom}}({{\text {Cobar}}}\,{{\text {Bar}}}^+ (A), B)$$

In other words, an $A_\infty$ functor $A\rightarrow B$ is the same as a DG functor ${{\text {Cobar}}}\,{{\text {Bar}}}(A) \rightarrow B.$ A curved $A_\infty$ functor $A\rightarrow B$ is the same as a DG functor ${{\text {Cobar}}}\,{{\text {Bar}}}^+ (A) \rightarrow B.$

1.1.3 16.1.3 The Adjunction Formula

Lemma 16.8

There are natural bijections

$${\text {Ob}}\,{\mathbf C}(A, {\mathbf C}(B,C))\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {Hom}}_{\mathrm{{DGcat}}} ({\text {Cobar}}({\text {Bar}}^+(A)\otimes \mathrm {Bar}(B)), C)$$

$${\text {Ob}}\,{\mathbf C}_+(A, {\mathbf C}_+(B,C))\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {Hom}}_{\mathrm{{DGcat}}} ({\text {Cobar}}({\text {Bar}}^+(A)\otimes \mathrm {Bar}^+(B)), C)$$

This (as well as Lemma 16.7) follows from Lemmas 16.10, 16.11, 16.12 below.

1.1.4 16.1.4 Convolution Categories

Let $\mathbb B$ be a DG cocategory and C a DG category. For

$$\mathbf f,\mathbf g: {\text {Ob}}\,(\mathbb B)\rightarrow {\text {Ob}}\,(C),$$

put

$${\overline{{\text {Conv}}}}_{\mathbf f,\mathbf g} (\mathbb B,C)=\prod _{x,y\in {\text {Ob}}\,({\mathcal B})} {\text {Hom}}^\bullet (\mathbb B(x,y), C(fx,gy))$$

$${{{\text {Conv}}}}_{\mathbf f,\mathbf g} (\mathbb B,C)=\prod _{x,y\in {\text {Ob}}\,({\mathcal B})} {\text {Hom}}^\bullet (\mathbb B^+(x,y), C(fx,gy))$$

The differential d is the usual one (induced by the differentials on $\mathbb B$ and C). Define the product

$${\text {Conv}}_{\mathbf f,\mathbf g}(\mathbb B,C)\otimes {\text {Conv}}_{\mathbf g,\mathbf h}(\mathbb B,C)\rightarrow {\text {Conv}}_{\mathbf f,\mathbf h}(\mathbb B,C)$$

$$\overline{{\text {Conv}}}_{\mathbf f,\mathbf g}(\mathbb B,C)\otimes \overline{{\text {Conv}}}_{\mathbf g,\mathbf h}(\mathbb B,C)\rightarrow \overline{{\text {Conv}}}_{\mathbf f,\mathbf h}(\mathbb B,C)$$

as follows. If

$$\Delta b=\sum b^{(1)}\otimes b^{(2)}$$

then

$$\begin{aligned} (\varphi \smile \psi )(b)=\sum (-1)^{|\psi ||{b^{(1)}|}} \varphi (b^{(1)}) \psi (b^{(2)} ) \end{aligned}$$

(16.1.6)

Definition 16.9

Define DG categories $\mathbf {Conv}(\mathbb B,C)$ and $\mathbf {Conv}_+(\mathbb B,C)$ as follows. Their objects are maps $\mathbf f:{\text {Ob}}\,(\mathbb B)\rightarrow {\text {Ob}}\,(C)$ together with elements f of degree one in $\overline{{\text {Conv}}}_{\mathbf f,\mathbf f}(\mathbb B,C)$ (resp. in ${\text {Conv}}_{\mathbf f,\mathbf f}(\mathbb B,C)$) satisfying

$$df+f\smile f=0.$$

The complex of morphisms between f and g is ${\text {Conv}}_{\mathbf f,\mathbf g}(\mathbb B,C)$ with the differential

$$\delta \varphi =d\varphi +f\smile \varphi -(-1)^{|\varphi |} \varphi \smile f$$

The composition is the cup product (16.1.6).

Lemma 16.10

There are natural isomorphisms of DG categories

$${\mathtt {\mathbb C}}(A,B)\,{\mathop {\rightarrow }\limits ^{\sim }}\,\mathbf {Conv}({\text {Bar}}(A),B)$$

$${\mathtt {\mathbb C}}_+(A,B)\,{\mathop {\rightarrow }\limits ^{\sim }}\,\mathbf {Conv}_+({\text {Bar}}(A),B)$$

Lemma 16.11

There is a natural bijection

$${\text {Hom}}_\mathrm{{DGcat}}({\text {Cobar}}(\mathbb B), C)\,{\mathop {\rightarrow }\limits ^{\sim }}\,{\text {Ob}}\,(\mathbf {Conv}(\mathbb B, C))$$

Lemma 16.12

There is a natural isomorphism of DG categories

$$\mathbf {Conv}(\mathbb B_1,\mathbf {Conv}(\mathbb B_2, C))\,{\mathop {\rightarrow }\limits ^{\sim }}\,\mathbf {Conv}(\mathbb B_1\otimes \mathbb B_2,C)$$

This is a reformulation of a result in [23].

1.1.5 16.1.5 An $A_\infty$ Functor to $A_\infty$ Modules

Let k be a field. By ${\text {dgmod}}(k)$ we denote the differential graded category of complexes of modules over k. Let R be an associative algebra over k.

Definition 16.13

We denote the DG category $\mathtt {\mathbf C}({\text {Bar}}(R), {\text {dgmod}}(k))$ by ${\text {Mod}}_\infty (R)$ and call it the DG category of $A_\infty$ modules over R.

Let ${\mathfrak X}(R)$ be the category whose objects are pairs $({\mathcal B}{\mathop {\longrightarrow }\limits ^{\pi }}R, {\mathtt {\mathcal M}})$ where ${\mathcal B}$ is a differential graded algebra, $\pi$ a quasi-isomorphism of DGA, and ${\mathtt {\mathcal M}}$ a DG module over ${\mathcal B}.$ A morphism $({\mathcal B}{\mathop {\longrightarrow }\limits ^{\pi }}R, {\mathtt {\mathcal M}})\rightarrow ({\mathcal B}'{\mathop {\longrightarrow }\limits ^{\pi '}}R, {\mathtt {\mathcal M}}')$ is a morphism ${\mathcal B}\rightarrow {\mathcal B}'$ of DGA over R together with a compatible morphism ${\mathtt {\mathcal M}}\rightarrow {\mathtt {\mathcal M}}'.$

We will construct an $A_\infty$ functor

$$\begin{aligned} {\mathfrak X}(R)\rightarrow {\text {Mod}}_\infty (R) \end{aligned}$$

(16.1.7)

Remark 16.14

An $A_\infty$ functor from a category ${\mathfrak X}$ to a DG category ${\mathtt {\mathcal A}}$ is by definition an $A_\infty$ functor from the linearization of ${\mathfrak X}$ (viewed as a DG category with zero differential) to ${\mathtt {\mathcal A}}.$

Define the DG category ${\mathfrak B}$ as follows. Its objects are the same as objects of ${\mathfrak {X}}(R)$ but repeated countably many times, i.e. an object of ${\mathfrak B}$ is a pair $(\mathtt {\mathbf x}, n)$ where $\mathtt {\mathbf x}$ is an object of ${\mathfrak X}(R)$ and $n\in {\mathbb Z}.$ The spaces of morphisms are as follows.

$$\begin{aligned} {\mathfrak B}((\mathtt {\mathbf x}, m), (\mathtt {\mathbf y}, n))=0 \end{aligned}$$

(16.1.8)

if $m<n$ or $m=n$ but $\mathtt {\mathbf x}\ne \mathtt {\mathbf y}.$ If $m>n$ and

$$\begin{aligned} \mathtt {\mathbf x}=({\mathcal B}{\mathop {\longrightarrow }\limits ^{\pi }}R, {\mathtt {\mathcal M}}),\; \mathtt {\mathbf y}=({\mathcal B}'{\mathop {\longrightarrow }\limits ^{\pi '}}R, {\mathtt {\mathcal M}}'), \end{aligned}$$

(16.1.9)

then

$$\begin{aligned} {\mathfrak B}((\mathtt {\mathbf x}, m), (\mathtt {\mathbf y}, n))={\mathcal B}'\times {\mathfrak X}(R)(\mathtt {\mathbf x},\mathtt {\mathbf y}) \end{aligned}$$

(16.1.10)

By $a'{\mathbf b}$ we denote the pair $(a', {\mathbf b} )$ where $a'\in {\mathcal B}'$ and ${\mathbf b}: \mathtt {\mathbf x}\rightarrow \mathtt {\mathbf y}$ is a morphism in ${\mathfrak X}(R).$ We denote the underlying morphism ${\mathcal B}\rightarrow {\mathcal B}'$ also by ${\mathbf b}.$ Put

$$\begin{aligned} {\mathfrak B}((\mathtt {\mathbf x}, n), (\mathtt {\mathbf x}, n))={\mathcal B}\end{aligned}$$

(16.1.11)

We also denote the right hand side by ${\mathcal B}{\mathtt {Id}}_{\mathtt {\mathbf x}}.$ The composition is given by

$$\begin{aligned} (a'' {\mathbf b}')(a'{\mathbf b})=(a''{\mathbf b}'(a')) {\mathbf b}'{\mathbf b} \end{aligned}$$

(16.1.12)

Consider the following right DG module ${\mathtt {\mathbf M}}$ over ${\mathfrak B}.$ Define

$$\mathtt {\mathbf M}(\mathtt {\mathbf x}, n)={\mathtt {\mathcal M}}$$

where $\mathtt {\mathbf x}=({\mathcal B}\rightarrow R, {\mathtt {\mathcal M}}).$ Define the action

$$\mathtt {\mathbf M}(\mathtt {\mathbf x}, m) \otimes \mathfrak B((\mathtt {\mathbf x}, m), (\mathtt {\mathbf y}, n))\rightarrow \mathtt {\mathbf M}(\mathtt {\mathbf y}, n)$$

by

$$v\otimes (a'{\mathbf b})=a' {\mathbf b}(v)$$

Here we denote by ${\mathbf b}$ the underlying action of the morphism ${\mathbf b}:\mathtt {\mathbf x}\rightarrow \mathtt {\mathbf y}$ on the module, as well as on the algebra.

Define another DG category ${\mathtt {\mathcal R}}$ exactly like $\mathfrak B$ above with the only difference that we put

$$\begin{aligned} {{\mathtt {\mathcal R}}}((\mathtt {\mathbf x}, m), (\mathtt {\mathbf y}, n))=R\times {\mathfrak X}(R)(\mathtt {\mathbf x},\mathtt {\mathbf y}) \end{aligned}$$

(16.1.13)

instead of (16.1.10) and

$$\begin{aligned} {\mathfrak B}((\mathtt {\mathbf x}, n), (\mathtt {\mathbf x}, n))=R \end{aligned}$$

(16.1.14)

instead of (16.1.11). We also denote the right hand side by ${\mathtt {Id}}_{\mathtt {\mathbf x}} R.$ Instead of (16.1.12), the composition is given by

$$\begin{aligned} (a'' {\mathbf b}')(a'{\mathbf b})=(a''a') {\mathbf b}'{\mathbf b} \end{aligned}$$

(16.1.15)

The morphisms $\pi : {\mathcal B}\rightarrow R$ induce a quasi-isomorphism of DG categories $\mathfrak B{\mathop {\longrightarrow }\limits ^{\pi }}{\mathtt {\mathcal R}}.$ The transfer of structure argument makes $\mathtt {\mathbf M}$ a right $A_\infty$ module over ${\mathtt {\mathcal R}}$ as follows. Fix a linear map ${\mathtt {\mathcal R}}{{\mathop {\longrightarrow }\limits ^{i}}} {\mathcal B}$ that is inverse to $\pi$ at the level of cohomology. (This is where we use the assumption that k is a field). Fix also homotopies for ${\mathtt {Id}}_\mathfrak B-i\pi$ and for ${\mathtt {Id}}_{\mathtt {\mathcal R}}-\pi i.$ (By this we mean collections of maps ${\mathtt {\mathcal R}}(\mathtt {\mathbf x},\mathtt {\mathbf y})\rightarrow \mathfrak B(\mathtt {\mathbf x},\mathtt {\mathbf y})$, etc., for any objects $\mathtt {\mathbf x}$ and $\mathtt {\mathbf y}$). From his data one constructs an $A_\infty$ functor ${\mathtt {\mathcal R}}\rightarrow \mathfrak B$ which is inverse to $\pi$ up to equivalence (cf. [24, 25, 28]). Furthermore, the map i and the homotopies can be chosen to be invariant under the action of ${\mathtt {\mathbb Z}}$ on $\mathfrak B$ and on ${\mathtt {\mathcal R}}.$ Therefore the $A_\infty$ functor is also ${\mathtt {\mathbb Z}}$-invariant. We denote it by ${\mathbb T}$, and the corresponding twisting cochain $\rho$ by $\rho _{\mathbb T}.$

This, in turn, defines the desired $A_\infty$ functor (16.1.7). In fact, for any object $\mathtt {\mathbf x}=({\mathcal B}\rightarrow R, {\mathtt {\mathcal M}})$, the value of this $A_\infty$ functor on $\mathtt {\mathbf x}$ is the underlying complex ${\mathtt {\mathcal M}}.$ For $g_1,\ldots ,g_p\in R$, put

$$\begin{aligned} \rho (g_1,\ldots ,g_p)=\rho _{\mathbb T}(g_1{\mathtt {Id}}_\mathtt {\mathbf x}, \ldots ,g_p{\mathtt {Id}}_\mathtt {\mathbf x}) \end{aligned}$$

(16.1.16)

where we view $\rho _j{\mathtt {Id}}_\mathtt {\mathbf x}$ as morphisms $(\mathtt {\mathbf x}, 0)\rightarrow (\mathtt {\mathbf x}, 0)$ in $\mathfrak B.$ This makes each ${\mathtt {\mathcal M}}$ an $A_\infty$ module over R. Now consider morphisms

$$\begin{aligned} \mathtt {\mathbf x}_0{\mathop {\longleftarrow }\limits ^{{\mathbf b}_1}}\mathtt {\mathbf x}_1{\mathop {\longleftarrow }\limits ^{{\mathbf b}_2}} \cdots {\mathop {\longleftarrow }\limits ^{{\mathbf b}_n}}\mathtt {\mathbf x}_n \end{aligned}$$

(16.1.17)

in $\mathfrak X(R)$, as well as corresponding morphisms

$$\begin{aligned} (\mathtt {\mathbf x}_0, 0){\mathop {\longleftarrow }\limits ^{{\mathbf b}_1}}(\mathtt {\mathbf x}_1, 1) {\mathop {\longleftarrow }\limits ^{{\mathbf b}_2}} \cdots {\mathop {\longleftarrow }\limits ^{{\mathbf b}_n}}(\mathtt {\mathbf x}_n, n) \end{aligned}$$

(16.1.18)

in ${\mathtt {\mathcal R}}.$ Now put

$$\rho (g_1,\ldots ,g_p)= \sum \pm \rho _{\mathbb T}(g_1{\mathtt {Id}}_{\mathtt {\mathbf x}_0}, \ldots ,g_{p_1}{\mathtt {Id}}_{\mathtt {\mathbf x}_0}, {\mathbf b}_1, $$

$$g_{p_1+1}{\mathtt {Id}}_{\mathtt {\mathbf x}_1}, \ldots ,g_{p_2}{\mathtt {Id}}_{\mathtt {\mathbf x}_1}, \ldots , {\mathbf b}_n, g_{p_n+1}{\mathtt {Id}}_{\mathtt {\mathbf x}_n}, \ldots ,g_{p}{\mathtt {Id}}_{\mathtt {\mathbf x}_n }) $$

where the sum is taken over all $0\le p_1\le \ldots \le p_{n}\le n.$ The sign rule: both $g_j{\mathtt {Id}}_{\mathtt {\mathbf x}_k}$ and ${\mathbf b}_i$ are treated as odd (the former has degree $(-1)^{|g_j|+1}$ if R is graded).

It is straightforward to check that thus defined $\rho$, when viewed as a cochain

$$\rho ({\mathbf b}_1,\ldots ,{\mathbf b_n}) \in \mathrm{{Mod}}_\infty (R)({\mathtt {\mathcal M}}_0,{\mathtt {\mathcal M}}_n),$$

is an $A_\infty$ functor $\mathfrak X(R)\rightarrow \mathrm{{Mod}}_\infty (R).$ (Here ${\mathtt {\mathcal M}}_j$ is the underlying DG module of $\mathtt {\mathbf x}_j$, viewed as a complex).

1.2 16.2 Twisted $A_\infty$ Modules on a Space

Let ${\mathtt {\mathcal R}}$ be a sheaf of algebras on a topological space X. Fix an open cover ${\mathfrak U}$ of X. For two collections $\mathtt {\mathbf M}=\{{\mathtt {\mathcal M}}_U|U\in {\mathfrak U}\}$ and $\mathtt {\mathbf N}=\{{\mathtt {\mathcal N}}_U|U\in {\mathfrak U}\}$ of sheaves of ${\mathtt {\mathcal R}}_U$-modules, define the complex $C^\bullet _{\mathtt {\mathbf M}, \mathtt {\mathbf N}}({\mathfrak U})$ as follows. Put

$$\begin{aligned} C^\bullet _{\mathtt {\mathbf M}, \mathtt {\mathbf N}}({\mathfrak U})=\prod _{p,q=0}^\infty \prod _{U_0,\ldots ,U_p\in \mathfrak U} \underline{{\text {Hom}}}^{\bullet -p-q}({\mathtt {\mathcal R}}^{\otimes q}, \underline{{\text {Hom}}}^{\bullet } ({\mathtt {\mathcal N}}_{U_p}, {\mathtt {\mathcal M}}_{U_0}))(U_0\cap \ldots \cap U_p) \end{aligned}$$

(16.2.1)

Define the differentials

$$\begin{aligned} ({\check{\partial }} \varphi )_{U_0\ldots U_{p+1} }=\sum _{j=1}^{p} (-1)^j \varphi _{U_0\ldots {\widehat{U_j}}\ldots U_{p+1}}; \end{aligned}$$

(16.2.2)

$$\begin{aligned} (\partial \varphi )(g_1,\ldots ,g_{q+1})=(-1)^{p|\varphi |} \sum _{j=1}^q\varphi (g_1,\ldots ,g_jg_{j+1},\ldots ,g_{q+1}) \end{aligned}$$

(16.2.3)

for local sections $g_1,\ldots$ of ${\mathtt {\mathcal R}};$

$$\begin{aligned} d\varphi ={\check{\partial }}\varphi +\partial \varphi +d_{\mathtt {\mathcal M}}\varphi -(-1)^{|\varphi |} \varphi d_{\mathtt {\mathcal N}}\end{aligned}$$

(16.2.4)

Define also the product

$$\begin{aligned} C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}} (\mathfrak U) \otimes C^\bullet _{\mathtt {\mathbf N},\mathbf P} (\mathfrak U) \rightarrow C^\bullet _{\mathtt {\mathbf M},\mathbf P} (\mathfrak U) \end{aligned}$$

(16.2.5)

by

$$ (\varphi \smile \psi )_{U_0\ldots U_{p_1+p_2}}(g_1,\ldots ,g_{q_1+q_2})= $$

$$ (-1)^{{|\varphi |p_2+(|\psi |+p_2)q_1}} \varphi _{U_0\ldots U_{p_1}}(g_1,\ldots ,g_{q_1}) \psi _{U_{p_1},\ldots ,U_{p_1+p_2}}(g_{q_1+1},\ldots ,g_{q_1+q_2}) $$

Set

$$\begin{aligned} C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}} (X)=\varinjlim _{\mathfrak U} C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}} (\mathfrak U) \end{aligned}$$

(16.2.6)

The differential and the cup product are well defined on the above complexes.

Definition 16.15

A twisted $A_\infty$ module ${\mathtt {\mathcal M}}$ over ${\mathtt {\mathcal R}}$ is a collection $\mathtt {\mathbf M}=\{{\mathtt {\mathcal M}}_U|U\in {\mathfrak U}\}$, of sheaves of ${\mathtt {\mathcal R}}_U$-modules together with a cochain $\rho$ of degree one in $C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf M}}(X)$ such that

$$d\rho +\rho \smile \rho =0.$$

The DG category $\mathrm{{Tw}}{\text {Mod}}_\infty ({\mathtt {\mathcal R}})$ has twisted $A_\infty$ modules as objects. The complex of morphisms between ${\mathtt {\mathcal M}}=(\mathtt {\mathbf M},\rho )$ and ${\mathtt {\mathcal N}}=(\mathtt {\mathbf N}, \sigma )$ is the complex $C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}}(X)$ with the differential $\delta \varphi =d\varphi +\rho \smile \varphi -(-1)^{|\varphi |} \varphi \smile \sigma .$

The above definition is an extension of the definition of twisted cochains from [39]. Cf. also [5, 33, 42].

Remark 16.16

The DG category of twisted $A_\infty$ modules is obtained almost verbatim as a partial case of the left hand side of Lemma 16.8. Formally, one could choose B to be the category with one object whose complex of morphisms is ${\mathtt {\mathcal R}}$, and $A=\mathrm{{Op}}_X$ to be the category of open subsets of X. More precisely, we perform all the computations as if A were the category whose objects are open subsets $U_\alpha$, and there is one morphism $U_\alpha \rightarrow U_\beta$ for any two intersecting open subsets. This is not literally true (there may be nonempty intersections $U_\alpha \cap U_\beta$ and $U_\beta \cap U_\gamma$ but not $U_\alpha \cap U_\gamma$), but all the formulas work. The above motivation may be given rigorous meaning using the techniques of [13] or [5].

1.3 16.3 Twisted $A_\infty$ Modules over Groupoids

For $q\ge 0$, we use notation ${\mathbb U}=(U^{(0)}, \ldots , U^{(q)}).$ We denote by $\mathfrak U_q$ the set of all such ${\mathbb U}$ where $U_j$ is in a given open cover $\mathfrak U.$ For $p+1$ such q-tuples ${\mathbb U}_{j_0}, \ldots , {\mathbb U}_{j_p}$, denote

$$\begin{aligned} U^{(k)}_{j_0\ldots j_p}=U^{(k)}_{j_0}\cap \cdots \cap U^{(k)}_{j_p} \end{aligned}$$

(16.3.1)

for all $0\le k \le q.$ Denote also

$$\begin{aligned} {\mathbb U}_{j_0\ldots j_p}=(U^{(0)}_{j_0\ldots j_p},\ldots ,U^{(q)}_{j_0\ldots j_p}). \end{aligned}$$

(16.3.2)

Let $\Gamma$ be an étale groupoid on a manifold X (in our applications, $\Gamma =\pi _1(X)$). For $\mathtt {\mathbf M}=\{{\mathtt {\mathcal M}}_U|U\in \mathfrak U\}$ and $\mathtt {\mathbf N}=\{{\mathtt {\mathcal N}}_U|U\in \mathfrak U\}$ as in the beginning of Sect. 16.2, put

$$C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}}(\mathfrak U,\Gamma )=\prod _{p,q\ge 0} \prod _{{\mathbb U}_0,\ldots ,{\mathbb U}_p \in \mathfrak U_q} \underline{{\text {Hom}}}^{\bullet -p-q}({\underline{\Gamma }}^{(q)}, \underline{{\text {Hom}}}^\bullet ({\mathtt {\mathcal N}}_{U_p^{(q)}},{\mathtt {\mathcal M}}_{U^{(0)}_0})(\prod _{k=0}^q U^{(k)}_{01\ldots p})$$

Here ${\mathtt {\mathcal M}}_{U^{(0)}_0}$ stands for its inverse image under the map

$$\prod _k \cap _j U_j^{(k)} \rightarrow \prod _k U_0^{(k)}\rightarrow U^{(0)}_0$$

The differential and the cup product are defined exactly as in (16.2.5), (16.2.3), (16.2.2) (with $U_j$ replaced by ${\mathbb U}_j$). Define

$$\begin{aligned} C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}}(X,\Gamma ) =\varinjlim _{\mathfrak U} C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}}(\mathfrak U,\Gamma ) \end{aligned}$$

(16.3.3)

Definition 16.17

(a) Define the DG category $\mathrm{Tw}{\text {Mod}}_\infty (\Gamma )$ exactly as in Definition 16.15 using complexes $C^\bullet _{\mathtt {\mathbf M},\mathtt {\mathbf N}}(X,\Gamma ).$

(b) The DG category

$$\mathrm{Tw}{\text {Mod}}_\infty (\Gamma , \Omega ^\bullet _{{\mathbb K}, X})$$

is defined the same way but with ${\mathtt {\mathcal M}}_U$ being $\Omega ^\bullet _{{\mathbb K}, U}$-modules as in Sect. 8.1.

Remark 16.18

By

$$\mathrm{{Loc}}_{\infty ,{\mathbb K}}(X)$$

we denote the DG category of $A_\infty$ representations of the fundamental groupoid $\pi _1(X).$ This is the partial case of the above Definition 16.17, (a) when $\Gamma =\pi _1(X)$, the topology on X is discrete, and the ground ring is ${\mathbb K}.$ Objects of this DG category are infinity local systems as in Sect. 8.4.

1.3.1 16.3.1 From ${\mathtt {\mathcal A}}_M^\bullet$-Modules with an Action of $\pi _1(M)$ up to Inner Automorphisms to Twisted $(\Omega _{{\mathbb K},M}^\bullet ,\pi _1(M))$-Modules

Given two ${\mathtt {\mathcal A}}_M^\bullet$-modules ${\mathtt {\mathcal V}}^\bullet$ and ${\mathtt {\mathcal W}}^\bullet$ with an action of $\pi _1(M)$ up to inner automorphisms, consider the standard complex

$${\mathtt {\mathcal M}}={\mathtt {\mathcal C}}^\bullet ({\mathtt {\mathcal V}}^\bullet , {\mathtt {\mathcal A}}^\bullet , {\mathtt {\mathcal W}}^\bullet ).$$

As it is shown in Sect. 6.2.5, ${\mathtt {\mathcal M}}$ has the following structure.

For a number of open subsets $U^{(j)}$ indexed by $j\in J$, write ${\mathbb U}_{ij}=(U^{(i)},U^{(j)}).$ We have constructed:

(a) For every $U^{(0)}$ and $U^{(1)}$, an $\Omega ^\bullet _{{\mathbb K}, U^{(0)}\times U^{(1)}}$-module ${\mathcal B}_{\mathbb U_{01}}$ together with a quasi-isomorphism

$$\begin{aligned} {\mathcal B}_{\mathbb U_{01}}\rightarrow {\mathbb K}{\underline{\pi _1}}(M)|(U^{(0)}\times U^{(1)}); \end{aligned}$$

(16.3.4)

(b) a morphism

$$\begin{aligned} p_{01}^*{\mathcal B}_{{\mathbb U}_{01}}\otimes p_{12}^*{\mathcal B}_{{\mathbb U}_{12}} \rightarrow p_{02}^*{\mathcal B}_{{\mathbb U}_{02}} \end{aligned}$$

(16.3.5)

which commutes with the composition on ${\underline{\pi _1}}(M)$ under (16.3.4);

(c) for any $U_0^{(j)}$ and $U^{(j)}_1$, an isomorphism

$$\begin{aligned} {\mathbf b}_{01}: {\mathcal B}_{{\mathbb U}_0}{\mathop {\leftarrow }\limits ^{\sim }}{\mathcal B}_{{\mathbb U}_1} \end{aligned}$$

(16.3.6)

that commutes with (16.3.4) and (16.3.5) and satisfies

$${\mathbf b}_{01} {\mathbf b}_{12} = {\mathbf b}_{02} $$

on the intersections.

Now repeat the procedure from Sect. 16.1.5, together with Remark 16.16, in the above context. First note that the constructions of Sect. 16.1.5 can be carried out in the case when R is a category (and all ${\mathcal B}$ are DG categories with the same objects). Now act as if R were the category with objects $U^{(j)}$, with

$$R(i,j)={\underline{\pi _1}}(M)|(U^{(i)}\times U^{(j)})$$

and the composition being the one on ${\underline{\pi _1}}.$ Now, let $\mathrm{{Op}}_M$ be the category whose objects are open subsets $U_j$, exactly as discussed in Remark 16.16. View the data (a), (b), (c) above as a DG functor $\mathrm{{Op}}_M \rightarrow {\mathfrak X}(R).$ Applying formulas from Sect. 16.1.5, we get an $A_\infty$ functor $\mathrm{{Op}}_M\rightarrow \mathtt {\mathbf C}(R,\mathrm{{dgmod}}({\mathbb K}))$, which is the same as an $\Omega ^\bullet _{{\mathbb K},M}$-module with a twisted action of $\pi _1(M).$

1.3.2 16.3.2 From Twisted $( \Omega ^\bullet _{{\mathbb K}, X}, \pi _1(X))$ Modules to Infinity Local Systems

Here we extend the construction from Sect. 8.4.1. Consider all open covers of the type $\mathfrak U=\{U_x|x\in X\}.$ For an object ${\mathtt {\mathcal M}}$ of $\mathrm{Tw}{\text {Mod}}_\infty (\pi _1(X), \Omega ^\bullet _{{\mathbb K}, X})$ choose a cover $\mathfrak U$ as above and define

$$\begin{aligned} {{\mathtt {\mathcal M}}_x}=\varinjlim _{U\subset U_x} C^\bullet (U, {\mathtt {\mathcal M}}_{U_x}) \end{aligned}$$

(16.3.7)

The $A_\infty$ operators $T(g_1,\ldots ,g_n)$ are by definition $\rho _{\mathbb U}(g_1,\ldots ,g_n)$ where $g_j\in \pi _1(X)_{x_{j-1},x_{j}}$ and ${\mathbb U}=(U_{x_0},\ldots ,U_{x_n})$. Let us show that different choices of $\mathfrak U$ lead to equivalent infinity local systems (in the sense of Definition 16.5). Choose two covers $\mathfrak U'$ and $\mathfrak U''.$ Apply (16.3.7) to all covers of the form $\mathfrak U=\{U_x| x\in X\}$ where for any x either $U_x=U'_x$ or $U_x=U_x''.$ This data defines an $A_\infty$ functor ${\mathbb K}\mathtt {\mathbf C}_1\otimes {\mathbb K}\pi _1(X)\rightarrow \mathrm{{dgmod}}({\mathbb K})$ (cf. Sect. 16.1.1). Let ${\mathbb K}(0)$, resp. ${\mathbb K}(1)$, be the full subcategory of $\mathtt {\mathbf C}_1$ with one object 0, resp. 1. When restricted to ${\mathbb K}(0)$, resp. to ${\mathbb K}(1)$, our $A_\infty$ functor coincides with the infinity local system obtained from $\mathfrak U'$, resp. from $\mathfrak U''.$ By the adjunction formula (Lemma 16.8), the two infinity local systems are equivalent.

Remark 16.19

It is easy to modify the above construction and obtain an $A_\infty$ functor

$$\mathrm{{TwMod}}(\Omega ^\bullet _{{\mathbb K},X}, \pi _1(X))\rightarrow \mathrm{{Loc}}_{\infty ,{\mathbb K}}(X).$$

Moreover, the right hand side is a monoidal category up to homotopy, and the assignment ${\mathtt {\mathcal M}}, {\mathtt {\mathcal N}}\mapsto {\underline{{\mathtt {\mathbb R}}{\text {HOM}}}}({\mathtt {\mathcal M}},{\mathtt {\mathcal N}})$ turns oscillatory modules, as well as $\Omega ^\bullet _{{\mathbb K},M}$-modules with an action of $\pi _1(M)$ up to inner automorphisms, into a category enriched over it. The main reason for this is Lemma 6.17. We will provide the details in a subsequent work.

17 Appendix. Jets and Twisted Bundles

Here we will describe the deformation quantization and the twisted bundle ${\mathtt {\mathcal H}}_M$ in terms of bundles of jets.

1.1 17.1 Jet Bundles

Let M be any manifold and let ${\mathtt {\mathcal E}}$ be a complex vector bundle of rank N on M. Here we recall the construction of the bundle whose fiber at a point x is the space of jets of sections of ${\mathtt {\mathcal E}}$ at x. This bundle has the canonical connection; its horizontal sections are determined by sections s of ${\mathtt {\mathcal E}}$. The value of such a section at any x is the jet of s at x.

Let $\{U_\alpha \}$ is an open cover and $x_\alpha =(x_{\alpha , 1}, \ldots , x_{\alpha , n})$ a local coordinate system on $U_\alpha .$ For $x\in U_{\alpha }\cap U_\beta$, we denote by $x_\alpha$, resp. $x_\beta$, its coordinates in the corresponding coordinate system and write

$$\begin{aligned} x_\alpha =g_{\alpha \beta }(x_\beta ) \end{aligned}$$

(17.1.1)

Let $h_{\alpha \beta }: U_{\alpha }\cap U_\beta \rightarrow {\text {GL}}_N$ be the transition isomorphisms of ${\mathtt {\mathcal E}}.$ We identify a local section of ${\mathtt {\mathcal E}}$ on $U_{\alpha }\cap U_{\beta }$ with a ${\mathtt {\mathbb C}}^N$-valued function in the coordinates $x_\beta .$

Let ${\mathbb C}^N[[\widehat{x}]]={\mathbb C}^N[[\widehat{x}_1,\ldots ,\widehat{x}_n]].$ For $x\in U_\alpha$ define $G_{\beta \alpha }(x): {\mathbb C}^N[[\widehat{x}]]\rightarrow {\mathbb C}^N[[\widehat{x}]]$ by $G_{\beta \alpha }(x):f_\alpha \mapsto f_\beta$ where

$$\begin{aligned} f_\beta (\widehat{x})=h_{\alpha \beta }(x_\beta +\widehat{x}) f_\alpha (g_{\alpha \beta }(x_\beta +\widehat{x})-x_\alpha ) \end{aligned}$$

(17.1.2)

It is easy to see that different choices of covers and of local trivializations lead to isomorphic bundles. We denote the bundle defined in (17.1.2) by ${\text {Jets}}(\Gamma ({\mathtt {\mathcal E}})).$

The canonical flat connection is given in any local coordinate system by

$$\begin{aligned} \nabla _\mathrm{{can}}=\left( \frac{\partial }{\partial x_\alpha }-\frac{\partial }{\partial \widehat{x}}\right) dx_\alpha \end{aligned}$$

(17.1.3)

If a local section of ${\mathtt {\mathcal E}}$ is represented by a vector-valued function $f(x_\alpha )$, it defines a horizontal section which is given in local coordinates by $f(x_\alpha +\widehat{x}).$

1.2 17.2 Real Polarization

Recall that a real polarization is an integrable distribution of Lagrangian subspaces. Let ${\mathcal P}$ be a real polarization on M. In this case, automatically $2c_1(TM)=0$ modulo 4 (cf. [36]).

1.2.1 17.2.1 The Line Bundle ${\mathtt {\mathcal L}}$

Assume that $\omega$ admits a real polarization ${\mathtt {\mathcal P}}$ (i.e. a foliation by Lagrangian submanifolds). By $T_{\mathtt {\mathcal P}}$ we denote the quotient of TM by the subbundle of vectors tangent to the leaves. Choose local Darboux coordinates $\xi _\alpha ,x_\alpha$ such that $x_{j,\alpha }$ are constant along the leaves. Then the transition coordinate changes are of the form

$$\begin{aligned} x_\alpha =g_{\alpha \beta }(x_\beta );\; \xi _\alpha =({{g'_{\alpha \beta }}(x_\beta )}^{t})^{-1}(\xi _\beta +\varphi _{\alpha \beta }(x_\beta )) \end{aligned}$$

(17.2.1)

Assume that $i\omega$ is a $2\pi i\mathbb Z$-valued cohomology class. Construct explicitly the line bundle ${\mathtt {\mathcal L}}$ such that $c_1({\mathtt {\mathcal L}})=i\omega .$ Adding some constants to $\varphi _{\alpha \beta }$, we may assume that $i\varphi _{\alpha \beta }-i \varphi _{\alpha \gamma }+i\varphi _{\beta \gamma }\in 2\pi i{\mathbb Z};$ define ${\mathtt {\mathcal L}}$ to be the line bundle with transition isomorphisms $\exp (i\varphi _{\alpha \beta }).$ Formulas

$$\begin{aligned} A_\alpha =-i\xi _\alpha dx_\alpha \end{aligned}$$

(17.2.2)

define a connection in this bundle, since

$$\xi _\alpha dx_\alpha =\xi _\beta dx_\beta +d\varphi _{\alpha \beta };$$

the curvature of this connection is $-i\omega .$

1.2.2 17.2.2 The Jet Bundle ${\text {Jets}}(\Gamma _{{\text {hor}}}({\Omega ^{\frac{1}{2}}}\otimes {\mathtt {\mathcal L}}^k))$

Define for $x\in U_\alpha \cap U_\beta$

$$G_{\beta \alpha }(x): {\mathtt {\mathbb C}}[[\widehat{x}]]\rightarrow {\mathtt {\mathbb C}}[[\widehat{x},\hbar ]]$$

by $(G_{\beta \alpha } f_\alpha )(\widehat{x})=f_\beta (\widehat{x})$ where

$$\begin{aligned} f_\beta (\widehat{x})=\det g'_{\alpha \beta }(x_\beta +\widehat{x})^{\frac{1}{2}} e^{ik\varphi _{\alpha \beta }(x_\beta +\widehat{x})} f_\alpha (g_{\alpha \beta }(x_\beta +\widehat{x})-x_\alpha ) \end{aligned}$$

(17.2.3)

The square root of the determinant comes from the metalinear structure. The above formula defines the transition functions for the bundle of jets of ${\mathtt {\mathcal P}}$-horizontal sections of the bundle $(\wedge ^{\max }T^*_{\mathtt {\mathcal P}})^{\frac{1}{2}} \otimes {\mathtt {\mathcal L}}^k.$

1.2.3 17.2.3 The Jet Bundle $\mathrm{{Rees}}\, {\text {Jets}}\,D(\Gamma _{{\text {hor}}}({\Omega ^{\frac{1}{2}}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}))$

Recall the construction of the Rees ring and the Rees module [2] of a filtered ring and a filtered module. If A is a ring with an increasing filtration $F_p A$, $p\ge 0$, and V an A-module with a compatible filtration $F_p$, $p\ge 0$, we put

$$\begin{aligned} {\text {Rees}}\,A=\oplus _{p\ge 0} \hbar ^p F_p A ; \;{\text {Rees}}\,V=\oplus _{p\ge 0} \hbar ^p F_p V . \end{aligned}$$

(17.2.4)

$$\begin{aligned} \mathrm{Rees}_f\,A=\prod _{p\ge 0} \hbar ^p F_p A ; \;\mathrm{Rees}_f\,V=\prod _{p\ge 0} \hbar ^p F_p V . \end{aligned}$$

(17.2.5)

When applied to the ring of formal differential operators with its filtration by order, (17.2.4) produces the ring ${\mathbb C}[[\widehat{x}]][\widehat{\xi },\hbar ]$ with the usual Heisenberg relations ($\widehat{\xi }_j=i\hbar \frac{\partial }{\partial \widehat{x}_j}$). When applied to the module of formal functions $V={\mathbb C}[[\widehat{x}]]$ whose filtration is given by $F_0 V=V$, it gives ${\mathbb C}[[\widehat{x}]][ \hbar ].$ The completed version (17.2.5) produces the complete Weyl algebra ${\mathbb C}[[\widehat{x},\widehat{\xi },\hbar ]]$ and the complete module ${\mathbb C}[[\widehat{x}, \hbar ]].$

Observe that in the expression $G_{\beta \alpha } (i\hbar \frac{\partial }{\partial \widehat{x}}) G_{\alpha \beta }$ one can substitute $\frac{1}{i\hbar }$ for k. The result will be given (in vector/matrix notation) by the following:

$$\frac{1}{2}\left( i\hbar \frac{\partial }{\partial \widehat{x}}\right) ({{g'_{\alpha \beta }}(x_\beta +\widehat{x})}^{t})+\left( {{g'_{\alpha \beta }}(x_\beta +\widehat{x})}^{t}\right) \left( i\hbar \frac{\partial }{\partial \widehat{x}}\right) -\varphi '_{\alpha \beta }(x_\beta +\widehat{x})$$

Define the bundle of algebras $\mathrm{{Rees}}\, {\text {Jets}}\,D(\Gamma _{{\text {hor}}}({\Omega ^{\frac{1}{2}}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}))$ whose fiber is ${\mathbb C}[[\widehat{x}, \hbar ]][\widehat{\xi }]$ and whose transition isomorphisms are

$$\begin{aligned} G_{\beta \alpha }(\widehat{x})=g_{\beta \alpha }(x_\alpha +\widehat{x})-x_\beta ; \end{aligned}$$

(17.2.6)

$$\begin{aligned} G_{\beta \alpha }(\widehat{\xi })= {{g'_{\alpha \beta }}(x_\beta +\widehat{x})}^t * \widehat{\xi }- \varphi '_{\alpha \beta }(x_\beta +\widehat{x}) \end{aligned}$$

(17.2.7)

(the multiplication in the left hand side is the (matrix) Moyal–Weyl multiplication). We see that our bundle is the result of formally substituting $\frac{1}{\hbar }$ for k in the bundle of jets of Rees rings of ${\mathtt {\mathcal P}}$-horizontal differential operators on $(\wedge ^{\max }T^*_{\mathtt {\mathcal P}})^{\frac{1}{2}} \otimes {\mathtt {\mathcal L}}^k.$

The above formula is the result of formally substituting k by $\frac{1}{\hbar }$ into the transition functions for the bundle

$$\mathrm{{Rees}}\,{\text {Jets}}\,D(\Gamma _{{\text {hor}}} ((\wedge ^{\max }T^*_{\mathtt {\mathcal P}})^{\frac{1}{2}} \otimes {\mathtt {\mathcal L}}^k)).$$

1.2.4 17.2.4 The Bundle of Algebras ${\widehat{\mathtt {\mathbb A}}}_M$ and the Twisted Bundle of Modules ${\mathtt {\mathcal H}}_M$

Now apply to the bundle above the gauge transformation [32]

$$\begin{aligned} {\text {Ad}}\exp \left( \frac{1}{i\hbar }\xi _\alpha \widehat{x}\right) \end{aligned}$$

(17.2.8)

We get transition isomorphisms

$$\begin{aligned} G_{\beta \alpha }(\widehat{x})=g_{\beta \alpha }(x_\alpha +\widehat{x})-x_\beta ; \end{aligned}$$

(17.2.9)

$$\begin{aligned} G_{\beta \alpha }(\widehat{\xi })= {{g'_{\alpha \beta }}(x_\beta +\widehat{x})}^t *( \widehat{\xi }+\xi _\alpha ) - \varphi '_{\alpha \beta }(x_\beta +\widehat{x})-\xi _\beta \end{aligned}$$

(17.2.10)

Unlike in (17.2.6) and (17.2.7), these transition isomorphisms preserve the maximal ideal $\langle \widehat{x},\,\widehat{\xi }, \hbar \rangle$ and therefore extend to the complete Weyl algebra ${\widehat{\mathtt {\mathbb A}}}={\mathbb C}[[\widehat{x},\widehat{\xi },\hbar ]]$, cf. Sect. 2.1. We use them to construct a bundle of algebras $\mathtt {\mathbb A}_M$ whose fiber is the Weyl algebra ${\widehat{\mathtt {\mathbb A}}}.$ We see immediately that the bundle of algebras ${\widehat{\mathtt {\mathbb A}}}_M$ is a deformation of the bundle of jets of functions on M.

Moreover, after we apply the gauge transformation (17.2.8), the formula (17.2.11) allows to replace k by $\frac{1}{\hbar }$. We get new transition isomorphisms

$$\begin{aligned} f_\beta (\widehat{x})=\det g'_{\alpha \beta }(x_\beta +\widehat{x})^{\frac{1}{2}} e^{-\frac{1}{i\hbar }(\varphi _{\alpha \beta }(x_\beta +\widehat{x})-\varphi '_{\alpha \beta }(x_\beta )\widehat{x})} f_\alpha (g_{\alpha \beta }(x_\beta +\widehat{x})-x_\alpha ) \end{aligned}$$

(17.2.11)

that define a twisted bundle of modules ${\mathtt {\mathcal H}}_M$ whose fiber is the space ${\mathtt {\mathcal H}}$ of the formal metaplectic representation (cf. (13.5.1)). The cocycle c from the definition of a twisted module (14.1.1) is $\exp (\frac{1}{i\hbar }(\varphi _{\alpha \beta }-\varphi _{\alpha \gamma }+\varphi _{\beta \gamma }))$. (The summand $-\varphi '_{\alpha \beta }(x_\beta )\widehat{x}$ in the exponent comes from the difference of $\xi _\alpha \widehat{x}$ and $\xi _\beta \widehat{x}$ that figure in the gauge transformation).

In other words, the bundle of algebras ${\widehat{\mathtt {\mathbb A}}}_M$ can be formally described as

$$\begin{aligned} {\widehat{\mathtt {\mathbb A}}}_M=\mathrm{{Rees}}_f\, \mathrm{{Jets}}\, D _\mathrm{{hor}}((\wedge ^{\frac{1}{2} }T_{\mathtt {\mathcal P}}^*)^{\frac{1}{2}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}) \end{aligned}$$

(17.2.12)

$$\begin{aligned} {\mathtt {\mathcal H}}_M=\mathrm{{Rees}}_f\, \mathrm{{Jets}}\, \Gamma _\mathrm{{hor}}((\wedge ^{\frac{1}{2} }T_{\mathtt {\mathcal P}}^*)^{\frac{1}{2}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}) \end{aligned}$$

(17.2.13)

(cf. (17.2.5) for the meaning of $\mathbf{{Rees}}_f$). The latter is only a twisted bundle because the transition functions of ${\mathtt {\mathcal L}}$ stop being a one-cocycle when elevated to the power $\frac{1}{\hbar }.$

1.2.5 17.2.5 The Canonical Connections

The bundle of horizontal sections of ${\Omega ^{\frac{1}{2}}}\otimes {\mathtt {\mathcal L}}^k$ has a canonical connection that is given by the formula

$$\nabla =\left( \frac{\partial }{\partial x}-\frac{\partial }{\partial \widehat{x}}\right) dx+\frac{\partial }{\partial \xi }d\xi $$

in all local coordinate systems.

This connection induces a connection in $\text {Rees Jets } D (\Gamma _{{\text {hor}}}({\Omega ^{\frac{1}{2}}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{i\hbar }}))$ that is given by the same formula. After the gauge transformation from Sect. 17.2.4 we get flat connections

$$\begin{aligned} \nabla _{\mathtt {\mathbb A}}=\left( \frac{\partial }{\partial x}-\frac{\partial }{\partial \widehat{x}}\right) dx+\left( \frac{\partial }{\partial \xi }-\frac{\partial }{\partial \widehat{\xi }}\right) d\xi \end{aligned}$$

(17.2.14)

in ${\mathtt {\mathcal A}}_M$ and

$$\begin{aligned} \nabla _{{\mathtt {\mathcal H}}}=-\frac{1}{i\hbar }\xi dx+\left( \frac{\partial }{\partial x}-\frac{\partial }{\partial \widehat{x}}\right) dx+\left( \frac{\partial }{\partial \xi }+\frac{1}{i\hbar }\widehat{x}\right) d\xi \end{aligned}$$

(17.2.15)

1.3 17.3 Complex Polarization

The following is largely based on the approach to deformation quantization from [19].

1.3.1 17.3.1 Kähler Potentials

Let M be a Kähler manifold. We can locally choose a Kähler potential, i.e. a real-valued function $\Phi$ such that the symplectic form is given by

$$\omega =-i \partial {\overline{\partial }}\Phi $$

A Kähler potential is unique up to a change $\Phi \mapsto \Phi +\varphi +{\overline{\varphi }}$ where $\varphi$ is holomorphic.

Lemma 17.1

Put $\zeta _j=i\frac{\partial \Phi }{\partial z_j}.$ Then

$$\{z_j,z_k\}=0;\; \{\zeta _k, z_j\}=\delta _{jk};\; \{\zeta _j,\zeta _k\}=0.$$

Proof

Choose local holomorphic coordinates and put

$$A_{jk}=\frac{\partial }{\partial z_j}\frac{\partial }{\partial \overline{ z}_k}\Phi (z,\overline{ z})$$

We have

$$\{z_j,\overline{ z}_k\}=i (A^{-1})_{kj};$$

$$\{z_j, \zeta _k\}=i\sum \frac{\partial \zeta _k}{\partial \overline{ z}_l} \{z_j,\overline{ z}_l\}=\sum A_{kl}(A^{-1})_{lj}= \delta _{jk};$$

$$-\{\zeta _j,\zeta _k\}=\sum \left( \frac{\partial \zeta _j}{\partial z_p} \frac{\partial \zeta _k}{\partial \overline{ z}_q}-\frac{\partial \zeta _k}{\partial z_p} \frac{\partial \zeta _j}{\partial \overline{ z}_q}\right) \{z_p,\overline{ z}_q\}=$$

$$i\sum \left( \frac{\partial ^2\Phi }{\partial z_j \partial z_p} A_{kq}- \frac{\partial ^2\Phi }{\partial z_k \partial z_p} A_{jq}\right) (A^{-1})_{qp}=i\left( \frac{\partial ^2\Phi }{\partial z_j \partial z_k}-\frac{\partial ^2\Phi }{\partial z_k\partial z_j}\right) =0$$

   $\square$

1.3.2 17.3.2 The Line Bundle ${\mathtt {\mathcal L}}$

Choose an open cover $\{U_{\alpha }\}$ of M and a holomorphic coordinate system $z_\alpha =(z_{\alpha , 1},\ldots ,z_{\alpha ,n})$ on every $U_{\alpha }.$ We write

$$\begin{aligned} z_\alpha =g_{\alpha \beta }(z_\beta ). \end{aligned}$$

(17.3.1)

Choose local Kähler potentials $\Phi _\alpha .$ We have

$$\begin{aligned} i\Phi _\alpha -i \Phi _\beta = \varphi _{\alpha \beta }+\overline{\varphi _{\alpha \beta }} \end{aligned}$$

(17.3.2)

where $\varphi _{\alpha \beta }$ are holomorphic.

Let us start with rewriting the transition isomorphisms in terms of the new complex Darboux coordinates $z,\zeta .$ We have

$$i\Phi _\alpha (z_\alpha )-i \Phi _\beta (z_\beta ) = \varphi _{\alpha \beta }+\overline{\varphi _{\alpha \beta }(z_\beta )}$$

Applying $\frac{\partial }{\partial z_\beta }$, we get

$$\frac{\partial z_\alpha }{\partial z_\beta }i\frac{\partial \Phi }{\partial z_\alpha }(z_\alpha )-i\frac{\partial \Phi }{\partial z_\beta }(z_\beta )=\frac{\partial \varphi _{\alpha \beta }}{\partial z_\beta }(z_\beta )$$

or

$$\begin{aligned} \zeta _\alpha =(g'_{\alpha \beta }(z_\beta )^{-1})^t\left( \zeta _\beta +\frac{\partial \varphi _{\alpha \beta }}{\partial z_\beta }(z_\beta )\right) \end{aligned}$$

(17.3.3)

Together with (17.3.1), this describes the rule for the change of new variables.

Assume that $i(\varphi _{\alpha \beta }+\varphi _{\beta \gamma }-\varphi _{\alpha \gamma })$ is a $2\pi i{\mathbb Z}$-valued two-cocycle. the line bundle ${\mathtt {\mathcal L}}$ with transition functions $\exp (\varphi _{\alpha \beta }).$ The curvature of this connection is $-i\omega .$

1.3.3 17.3.3 The Jet Bundles

Assume that the canonical sheaf has a square root $\Omega ^{\frac{1}{2}}.$ We call this line bundle the bundle of holomorphic half-forms on M. The transition isomorphisms of this line bundle are denoted by $\det {g_{\alpha \beta }'}^\frac{1}{2}.$ For any integer k, consider the bundle $\mathrm{{Jets}}(\Gamma _\mathrm{{hol}}({\mathtt {\mathcal L}}^k\otimes \Omega ^{\frac{1}{2}}))$ of jets of holomorphic sections of ${\mathtt {\mathcal L}}^k\otimes \Omega ^{\frac{1}{2}}.$ The fiber of this bundle is ${\mathtt {\mathbb C}}[[\widehat{z}]]$ where $\widehat{z}=(\widehat{z}_1,\ldots ,\widehat{z}_n).$ The transition isomorphisms of the jet bundle take a power series $f_\alpha (\widehat{z})$ to a power series $f_\beta (\widehat{z})$ according to the following formula.

$$\begin{aligned} f_\beta (\widehat{z})=f_\alpha (g_{\alpha \beta }(z_\beta +\widehat{z})-z_\alpha ) \det g'_{\alpha \beta }(z_\beta +\widehat{z})^{\frac{1}{2}} \exp (k\varphi _{\alpha \beta }(z_\beta +\widehat{z})) \end{aligned}$$

(17.3.4)

Exactly as in Sect. 17.2.3, we can define the bundle of algebras whose fiber is ${\mathbb C}[[\widehat{z}, \hbar ]][\widehat{\zeta }]$ by transition isomorphisms

$$\begin{aligned} G_{\beta \alpha }(\widehat{z})=g_{\beta \alpha }(z_\alpha +\widehat{z})-z_\beta ; \end{aligned}$$

(17.3.5)

$$\begin{aligned} G_{\beta \alpha }(\widehat{\zeta })= {{g'_{\alpha \beta }}(z_\beta +\widehat{z})}^t * \widehat{\zeta }- \partial _{z_\beta } \varphi _{\alpha \beta }(z_\beta +\widehat{\zeta }) \end{aligned}$$

(17.3.6)

We see that our bundle is the result of formally substituting $\frac{1}{\hbar }$ for k in the bundle of jets of Rees rings of holomorphic differential operators on $\Omega ^{\frac{1}{2}} \otimes {\mathtt {\mathcal L}}^k$ (if we map $\widehat{\zeta }_i$ to $i\hbar \partial _{\widehat{z}}$). On the other hand, because of (17.3.3), this bundle of algebras is a deformation of the bundle of jets of $C^\infty$ functions on M. The gauge transformation

$$\begin{aligned} {\text {Ad}}\exp \left( \frac{1}{i\hbar }\widehat{\zeta }_\alpha \widehat{z}\right) \end{aligned}$$

(17.3.7)

produces new transition functions

$$\begin{aligned} G_{\beta \alpha }(\widehat{z})=g_{\beta \alpha }(z_\alpha +\widehat{z})-z_\beta ; \end{aligned}$$

(17.3.8)

$$\begin{aligned} G_{\beta \alpha }(\widehat{\zeta })= {{g'_{\alpha \beta }}(z_\beta +\widehat{z})}^t * (\widehat{\zeta }+\zeta _\alpha )- \partial _{z_\beta } \varphi _{\alpha \beta }(z_\beta +\widehat{\zeta })-\zeta _\beta \end{aligned}$$

(17.3.9)

that extend to ${\widehat{\mathtt {\mathbb A}}}_M={\mathbb C}[[\widehat{z}, \widehat{\zeta }, \hbar ]].$ The transition isomorphisms for the module of jets (17.3.4) are now as follows (when we replace k by $\frac{1}{\hbar }$) which now define only a twisted module that we denote by ${\mathtt {\mathcal H}}_M.$

$$ f_\beta (\widehat{z})=f_\alpha (g_{\alpha \beta }(z_\beta +\widehat{z})-z_\alpha ) \det g'_{\alpha \beta }(z_\beta +\widehat{z})^{\frac{1}{2}} \exp \left( \frac{1}{i\hbar }\varphi _{\alpha \beta }(z_\beta +\widehat{z})-\partial _{z_\beta } \varphi _{\alpha \beta }(z_\beta )\widehat{z}\right) $$

As in the case of a real polarization, the canonical connections become

$$\begin{aligned} \nabla _{\mathtt {\mathbb A}}=\left( \frac{\partial }{\partial z}-\frac{\partial }{\partial \widehat{z}}\right) dz+\left( \frac{\partial }{\partial \zeta }-\frac{\partial }{\partial \widehat{\zeta }}\right) d\zeta \end{aligned}$$

(17.3.10)

on ${\mathtt {\mathcal A}}_M$ and

$$\begin{aligned} \nabla _{{\mathtt {\mathcal H}}}=-\frac{1}{i\hbar }\zeta dz+\left( \frac{\partial }{\partial z}-\frac{\partial }{\partial \widehat{z}}\right) dz+\left( \frac{\partial }{\partial \zeta }+\frac{1}{i\hbar }\widehat{z}\right) d\zeta \end{aligned}$$

(17.3.11)

on ${\mathtt {\mathcal H}}_M.$ We conclude that

$$\begin{aligned} {\widehat{\mathtt {\mathbb A}}}_M\,{\mathop {\rightarrow }\limits ^{\sim }}\,\mathrm{{Rees}}_f\, \mathrm{{Jets}}\,D_\mathrm{{hol}} (\Omega ^{\frac{1}{2}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}) \end{aligned}$$

(17.3.12)

$$\begin{aligned} {\mathtt {\mathcal H}}_M\,{\mathop {\rightarrow }\limits ^{\sim }}\,\mathrm{{Rees}}_f\,\mathrm{{Jets}}\,\Gamma _\mathrm{{hol}} (\Omega ^{\frac{1}{2}}\otimes {\mathtt {\mathcal L}}^{\frac{1}{\hbar }}) \end{aligned}$$

(17.3.13)