Derived algebraic geometry of 2d lattice Yang-Mills theory

TL;DR

This paper applies derived algebraic geometry to analyze 2D lattice Yang-Mills theory, describing the derived critical locus of the Wilson action and constructing a local dg-category-valued algebra on the lattice.

Contribution

It introduces a derived geometric framework for 2D lattice Yang-Mills theory and constructs a local prefactorization algebra from derived stacks of local data.

Findings

Derived critical locus of Wilson action characterized

Local data described on rectangular subsets of the lattice

Constructed a dg-category-valued prefactorization algebra

Abstract

A derived algebraic geometric study of classical $\mathrm{GL}_n$-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 $\geq 2$ 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.