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

Abstract In [26], the leading order term of the free energy of $$\text{ U(N) }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mspace/> <mml:mtext>U(N)</mml:mtext> <mml:mspace/> </mml:mrow> </mml:math> lattice Yang-Mills theory in $$\Lambda _n=\{0,\ldots ,n\}^d\subset \mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> was determined, for every $$N\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The formula is explicit apart from a contribution $$K_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> 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 $$K_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> that admits its explicit computation.