Derived algebraic geometry of 2d lattice Yang-Mills theory

Abstract A derived algebraic geometric study of classical $$\textrm{GL}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> -Yang-Mills theory on the 2-dimensional square lattice $${\mathbb {Z}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets $$V =[a,b]\times [c,d]\subseteq {\mathbb {Z}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> with both sides of length $$\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> is extracted. A locally constant dg-category-valued prefactorization algebra on $${\mathbb {Z}}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is constructed from the dg-categories of quasi-coherent complexes on the derived stacks of local data.