Towards deformation quantization over a \(\mathbb Z\)-graded base (Q2418821)

If \(M\) is a real manifold, \(\mathcal{O}(M)\) is the algebra of smooth complex-valued functions on \(M\), and \(\varepsilon,\varepsilon_1,\dots,\varepsilon_g\) are formal variables of degrees \(\text{deg}(\varepsilon)=0\), \(\text{deg}(\varepsilon_1)=d_1,\dots,\text{deg}(\varepsilon_g)=d_g\), where \(d_i\) are non-positive integers, then a formal deformation of \(\mathcal{O}(M)\) is a \(\mathbb{C}[\![\varepsilon,\varepsilon_1,\dots,\varepsilon_g]\!]\)-linear \(A_\infty\)-structure on \(\mathcal{O}(M)[\![\varepsilon,\varepsilon_1,\dots,\varepsilon_g]\!]\) with the multiplications \(\{\textsf{m}_n\}_{n\ge2}\) of the form \[ (*)\,\mathsf{m}_n(a_1,\dots,a_n)=\begin{cases} a_1a_2+\sum\limits_{k_0d_0+\dots+k_gd_g=0}\varepsilon^{k_0}\varepsilon_1^{k_1}\cdots\varepsilon_g^{k_g}\mu_{k_0,\dots,k_g}(a_1,a_2) & \text{ if } n=2\\ \sum\limits_{k_0d_0+\dots+k_gd_g=2-n}\varepsilon^{k_0}\varepsilon_1^{k_1}\cdots\varepsilon_g^{k_g}\mu_{k_0,\dots,k_g}(a_1,\dots,a_n) & \text{ if } n>2 \end{cases}, \] where each \(\mu_{k_0,\dots,k_g}\) is a polydifferential operator on $M$ acting on \(2-(k_0d_0+\cdots+k_gd_g)\) arguments. Such \(A_\infty\)-structures are in bijection with Maurer-Cartan (MC) elements of the \(\mathsf{dg}\) Lie algebra \((**)\,(\varepsilon,\varepsilon_1,\dots,\varepsilon_g)\,\mathsf{PD}^\bullet(M)[\![\varepsilon,\varepsilon_1,\dots,\varepsilon_g]\!]\), where \(\mathsf{PD}^\bullet(M)\) denotes the algebra of polydifferential operators on $M$. The cohomology space of \(\mathsf{PD}^\bullet(M)\) is isomorphic to the space \(\mathsf{PV}^\bullet(M)\) of polyvector fields on $M$. So the Kodaira-Spencer class of every MC element of \((**)\) can be identified with a degree 1 vector in the graded space \(\varepsilon\,\mathsf{PV}^\bullet(M)\oplus\cdots\oplus\varepsilon_g\,\mathsf{PV}^\bullet(M)\). \par In this paper, the authors consider formal deformations \((*)\) of \(\mathcal{O}(M)\) such that \(M\) has a symplectic structure \(\omega\) and \(\alpha\in\mathsf{PV}^1(M)\) is the Poisson structure corresponding to \(\omega\), and the deformations satify \begin{itemize}\item[(i)] the Kodaira-Spencer class of this deformation is \(\varepsilon\alpha\), \item[(ii)] \( {\mathsf{m}_n}_{|\varepsilon=0}=\begin{cases} a_1a_2 & \text{ if } n=2\\ 0 & \text{ if } n>2\end{cases}. \) \end{itemize} If \(\mathsf{TL}\) is the set of equivalence classes of formal deformations \((*)\) satisfying the above conditions, then \(\mathsf{TL}\) is called the topological locus of the triple \((M,\omega,\{\varepsilon,\varepsilon_1,\dots,\varepsilon_g\})\). The authors give a description of the topological locus \(\mathsf{TL}\) in terms of the singular cohomology of \(M\), that is, they show that for every symplectic manifold \((M,\omega)\), the equivalence classes of formal deformations \((*)\) of \(\mathcal{O}(M)\) satisfying the above conditions are in bijections with degree 2 vectors of the graded vector space \(\bigoplus\limits_{q\ge 0}(\varepsilon,\varepsilon_1,\dots,\varepsilon_g)H^q(M,\mathbb{C})[\![\varepsilon,\varepsilon_1,\dots,\varepsilon_g]\!]\), where \(H^\bullet(M,\mathbb{C})\) is the singular cohomology of $M$ with coefficients in \(\mathbb{C}\) and every vector of \(H^q(M,\mathbb{C})\) carries degree \(q\).