Finite Yang-Mills Integrals

TL;DR

This paper evaluates zero-dimensional SU(N) Yang-Mills integrals using Monte Carlo methods, establishing their existence under various conditions and confirming recent exact formulas for supersymmetric cases up to SU(5).

Contribution

It provides the first numerical verification of the existence of these integrals for certain dimensions and gauge groups, and confirms recent exact formulas in supersymmetric cases.

Findings

Non-supersymmetric integrals exist for D=3, N>3; D=4, N>2; D>=5, N>=2.

Integrals are well-defined in the large N limit for D=3 and D=4.

Confirmed recent exact formulas for supersymmetric D-instanton integrals up to SU(5).

Abstract

We use Monte Carlo methods to directly evaluate D-dimensional SU(N) Yang-Mills partition functions reduced to zero Euclidean dimensions, with and without supersymmetry. In the non-supersymmetric case, we find that the integrals exist for D=3, N>3 and D=4, N>2 and, lastly, D >= 5, N >= 2. We conclude that the D=3 and D=4 integrals exist in the large N limit, and therefore lead to a well-defined, new type of Eguchi-Kawai reduced gauge theory. For the supersymmetric case, we check, up to SU(5), recently proposed exact formulas for the D=4 and D=6 D-instanton integrals, including the explicit form of the normalization factor needed to interpret the integrals as the bulk contribution to the Witten index.