Commuting elements in central products of special unitary groups

TL;DR

This paper investigates the structure of commuting elements in the central product of multiple special unitary groups, providing detailed descriptions of their connected components and the geometry of associated moduli spaces.

Contribution

It offers a comprehensive computation of the number of connected components and describes the geometry of the moduli space of flat bundles for these groups.

Findings

Number of path connected components computed

Geometry of the moduli space fully described

Results applicable for all primes p, and all n, m values

Abstract

In this paper the space of commuting elements in the central product $G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where $p$ is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles over the $n$--torus is completely described for all values of $n$, $m$ and $p$.