Three-dimensional QCD in the adjoint representation and random matrix theory

TL;DR

This paper derives the effective low-energy partition function for three-dimensional adjoint QCD using random matrix theory, confirming symmetry breaking patterns and deriving sum rules.

Contribution

It extends random matrix theory methods to three-dimensional adjoint QCD, establishing symmetry breaking patterns and deriving the first Leutwyler--Smilga sum rule.

Findings

Symmetry breaking pattern: O(2N_f) to O(N_f) x O(N_f)

Derived the first Leutwyler--Smilga sum rule

Confirmed Dyson index β=4 for this case

Abstract

In this paper we complete the derivations of finite volume partition functions for QCD using random matrix theories by calculating the effective low-energy partition function for three-dimensional QCD in the adjoint representation from a random matrix theory with the same global symmetries. As expected, this case corresponds to Dyson index $\beta =4$, that is, the Dirac operator can be written in terms of real quaternions. After discussing the issue of defining Majorana fermions in Euclidean space, the actual matrix model calculation turns out to be simple. We find that the symmetry breaking pattern is $O(2N_f) \to O(N_f) \times O(N_f)$, as expected from the correspondence between symmetric (super)spaces and random matrix universality classes found by Zirnbauer. We also derive the first Leutwyler--Smilga sum rule.