Classification of Group Actions Extended to Symplectic Deformation Quantizations

TL;DR

This paper studies how group actions by symplectomorphisms can be extended to formal deformation quantizations of symplectic manifolds, providing a classification framework using non-Abelian group cohomology and examples.

Contribution

It introduces a new notion of extension for group actions on deformation quantizations, reformulates existence conditions, and classifies extensions via group cohomology with computational tools.

Findings

Extension conditions are not necessary for all cases.

Group cohomology H^1 classifies extensions up to equivalence.

Tools for computing H^1 are developed and applied to examples.

Abstract

Consider a group $\Gamma$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,\omega)$. This canonically induces an action of $\Gamma$ by symplectomorphisms on $M$. We examine the reverse problem of extending group actions by symplectomorphisms to the deformation quantization. To do this we first define a notion of extension that does not impose restrictions on the Fedosov connection realizing $\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition of pull-back with certain inner automorphisms of sections of the Weyl bundle. We reformulate the well known sufficient conditions for existence of such extensions and show by way of example that they are not necessary. We then turn to the corresponding classification problem of extensions of a given action by symplectomorphisms. To this end we associate a group $\bar{\mathcal{G}_\nabla}$ to each Fedosov connection $\nabla$ and show that the (non-Abelian) group cohomology $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ classifies extensions up to a natural notion of equivalence. Finally we develop several tools that allow for computation of $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ and apply these to compute $\mbox{H}^1(\Gamma,\bar{\mathcal{G}_\nabla})$ in a range of examples.