The Convergence of Yang-Mills Integrals

TL;DR

This paper establishes the precise dimensions at which SU(N) bosonic Yang-Mills matrix integrals converge, confirming and extending previous numerical findings.

Contribution

It proves the critical dimensions for convergence of SU(N) bosonic Yang-Mills integrals for various N, refining the understanding of their mathematical behavior.

Findings

Convergence for N=2 at D≥5

Convergence for N=3 at D≥4

Convergence for N≥4 at D≥3

Abstract

We prove that SU(N) bosonic Yang-Mills matrix integrals are convergent for dimension (number of matrices) $D\ge D_c$. It is already known that $D_c=5$ for N=2; we prove that $D_c=4$ for N=3 and that $D_c=3$ for $N\ge 4$. These results are consistent with the numerical evaluations of the integrals by Krauth and Staudacher.