Axial anomaly in the reduced model: Higher representations

TL;DR

This paper investigates the axial anomaly in reduced $ ext{U}(N)$ and $ ext{SU}(N)$ models, revealing how the anomaly depends on fermion representations and analyzing its behavior in the large $N$ limit.

Contribution

It provides a detailed analysis of the axial anomaly for various fermion representations in reduced gauge models using the overlap-Dirac operator.

Findings

Anomaly vanishes for certain irreducible representations with Young tableaux having boxes multiple of $L^2$.

The anomaly exhibits expected algebraic properties in the large $N$ limit.

For $ ext{SU}(N)$, the anomaly lacks the structure of trace of products of traceless generators.

Abstract

The axial anomaly arising from the fermion sector of $\U(N)$ or $\SU(N)$ reduced model is studied under a certain restriction of gauge field configurations (the ``$\U(1)$ embedding'' with $N=L^d$). We use the overlap-Dirac operator and consider how the anomaly changes as a function of a gauge-group representation of the fermion. A simple argument shows that the anomaly vanishes for an irreducible representation expressed by a Young tableau whose number of boxes is a multiple of $L^2$ (such as the adjoint representation) and for a tensor-product of them. We also evaluate the anomaly for general gauge-group representations in the large $N$ limit. The large $N$ limit exhibits expected algebraic properties as the axial anomaly. Nevertheless, when the gauge group is $\SU(N)$, it does not have a structure such as the trace of a product of traceless gauge-group generators which is expected from the corresponding gauge field theory.