Towards deformation quantization over a Z-graded base

TL;DR

This paper explores a new class of formal deformations of symplectic manifolds over a Z-graded base, involving A-infinity structures and cohomology classes of degree ≥ 2, expanding deformation quantization theory.

Contribution

It introduces a framework for deformation quantization over a Z-graded base, incorporating A-infinity deformations and higher-degree cohomology classes.

Findings

Describes formal deformations with non-positive degree parameters.

Highlights the role of all cohomology classes of degree ≥ 2 in deformations.

Provides a new perspective on deformation quantization in graded settings.

Abstract

The goal of this note is to describe a class of formal deformations of a symplectic manifold $M$ in the case when the base ring of the deformation problem involves parameters of non-positive degrees. The interesting feature of such deformations is that these are deformations "in $A_{\infty}$-direction" and, in general, their description involves all cohomology classes of $M$ of degrees $\ge 2$.