Quantization of derived cotangent stacks and gauge theory on directed graphs

TL;DR

This paper develops a method to quantize the canonical Poisson structure on derived cotangent stacks of quotient stacks and applies it to construct a dg-category-valued algebra for gauge theories on directed graphs.

Contribution

It introduces an étale resolution approach for quantizing derived cotangent stacks and constructs a dg-category-valued algebra for gauge theories on directed graphs.

Findings

Explicit description of the Poisson structure on $T^*[X/G]$

Construction of a dg-category quantization of the gauge theory

Application to gauge theories on directed graphs

Abstract

We study the quantization of the canonical unshifted Poisson structure on the derived cotangent stack $T^\ast[X/G]$ of a quotient stack, where $X$ is a smooth affine scheme with an action of a (reductive) smooth affine group scheme $G$. This is achieved through an {\'e}tale resolution of $T^\ast[X/G]$ by stacky CDGAs that allows for an explicit description of the canonical Poisson structure on $T^\ast[X/G]$ and of the dg-category of modules quantizing it. These techniques are applied to construct a dg-category-valued prefactorization algebra that quantizes a gauge theory on directed graphs.