Gauge Theory on Fuzzy S^2 x S^2 and Regularization on Noncommutative R^4

TL;DR

This paper constructs a fuzzy S^2 x S^2 gauge theory as a matrix model, serving as a regularization for noncommutative R^4, and explores its solutions, quantization, and fermionic extensions.

Contribution

It introduces a new multi-matrix model for gauge theory on fuzzy S^2 x S^2 that regularizes noncommutative R^4 and includes topologically non-trivial solutions and fermionic structures.

Findings

Model reduces to Yang-Mills on S^2 x S^2 in the commutative limit

Identifies fluxon solutions and brane interpretations

Provides a finite, non-perturbative path integral formulation

Abstract

We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model, which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative limit N -> infinity. The model can be used as a regularization of gauge theory on noncommutative R^4_\theta in a particular scaling limit, which is studied in detail. We also find topologically non-trivial U(1) solutions, which reduce to the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full moduli space. Other solutions which can be interpreted as 2-dimensional branes are also found. The quantization of the model is defined non-perturbatively in terms of a path integral which is finite. A gauge-fixed BRST-invariant action is given as well. Fermions in the fundamental representation of the gauge group are included using a formulation based on SO(6), by defining a fuzzy Dirac operator which reduces to the standard Dirac operator on S^2 x S^2 in the commutative limit. The chirality operator and Weyl spinors are also introduced.