<p>A derived algebraic geometric study of classical <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{GL}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>GL</mtext> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>-Yang-Mills theory on the 2-dimensional square lattice <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {Z}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is presented. The derived critical locus of the Wilson action is described and its local data supported in rectangular subsets <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V =[a,b]\times [c,d]\subseteq {\mathbb {Z}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>=</mo> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo>×</mo> <mrow> <mo stretchy="false">[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mrow> <mo>⊆</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with both sides of length <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is extracted. A locally constant dg-category-valued prefactorization algebra on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathbb {Z}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> is constructed from the dg-categories of quasi-coherent complexes on the derived stacks of local data.</p>

Derived algebraic geometry of 2d lattice Yang-Mills theory

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2026年6月29日

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Marco Benini,
Tomás Fernández,
Alexander Schenkel

摘要

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