<p>In [26], the leading order term of the free energy of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\text{ U(N) }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.333333em" /> <mtext>U(N)</mtext> <mspace width="0.333333em" /> </mrow> </math></EquationSource> </InlineEquation> lattice Yang-Mills theory in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda _n=\{0,\ldots ,n\}^d\subset \mathbb {Z}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Λ</mi> <mi>n</mi> </msub> <mo>=</mo> <msup> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> <mi>d</mi> </msup> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> was determined, for every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$N\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. The formula is explicit apart from a contribution <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> which corresponds to the limiting free energy of lattice Maxwell theory with boundary conditions induced by the axial gauge. By suitably adjusting the boundary conditions, we provide an equivalent characterization of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$K_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> that admits its explicit computation.</p>

On the Leading Order Term of the Lattice Yang-Mills Free Energy

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2026年6月24日

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Christian Brennecke

摘要

In [26], the leading order term of the free energy of $\text{ U(N) }$ $U(N)$ lattice Yang-Mills theory in $\Lambda _n=\{0,\ldots ,n\}^d\subset \mathbb {Z}^d$ $Λ_{n} = {0, \dots, n}^{d} \subset Z^{d}$ was determined, for every $N\ge 1$ $N \geq 1$ and $d\ge 2$ $d \geq 2$ . The formula is explicit apart from a contribution $K_d$ $K_{d}$ which corresponds to the limiting free energy of lattice Maxwell theory with boundary conditions induced by the axial gauge. By suitably adjusting the boundary conditions, we provide an equivalent characterization of $K_d$ $K_{d}$ that admits its explicit computation.