Derived algebraic geometry of 2d lattice Yang-Mills theory

A derived algebraic geometric study of classical ${\mathrm{G}\mathrm{L}}_n$-Yang-Mills theory on the $2$-dimensional square lattice ${\mathbb{Z}}^2$ is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets $V=[a,b]\times [c,d]\subseteq {\mathbb{Z}}^2$ with both sides of length $\ge 2$ is extracted. A locally constant dg-category-valued prefactorization algebra on ${\mathbb{Z}}^2$ is constructed from the dg-categories of perfect complexes on the derived stacks of local data.