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

TL;DR

The paper addresses the problem of identifying the leading-order term in the free energy for Euclidean lattice Yang-Mills theories with gauge group U(N) in large-volume limits. It builds on Cha1 by aligning the leading Gaussian fluctuations with lattice Maxwell theory in the axial gauge, recasting the problem in terms of a lattice differential operator Q_d and its spectral data. The main contribution is an explicit, gauge-group–independent formula for the constant K_d in the leading term, expressed through integrals that arise from the spectrum of Q_d under periodic boundary conditions. This advances understanding of the continuum-limit behavior of lattice YM measures and provides a closed-form constant that governs the dominant free energy density at small coupling. The results have potential implications for rigorous constructions of continuum Yang-Mills measures and for connecting lattice fluctuations to Gaussian field limits.

Abstract

In \cite{Cha1}, the leading order term of the free energy of $\text{U(N)}$ lattice Yang-Mills theory in $Λ_n=\{0,\ldots,n\}^d\subset \mathbb{Z}^d$ was determined, for every $N\geq 1$ and $d\geq 2$. The formula is explicit apart from a contribution $K_d$ 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 $K_d$ that admits its explicit computation.

Theorems & Definitions (13)

• Theorem 1: Cha1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof