Quantization of derived cotangent stacks and gauge theory on directed graphs

We study the quantization of the canonical unshifted Poisson structure on the derived cotangent stack T∗[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 étale resolution of T∗[X/G] by stacky CDGAs that allows for an explicit description of the canonical Poisson structure on T∗[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.