Properties of the Abelian Projection Fields in $SU(N)$ Lattice Gluodynamics

TL;DR

This paper investigates the properties of abelian projection fields in $SU(N)$ lattice gluodynamics, revealing a symmetry among species and demonstrating that inter-species interactions are suppressed at large N, supported by numerical simulations.

Contribution

It introduces the concept of species permutation symmetry in abelian projection fields and analyzes its implications for interactions in $SU(N)$ gauge theory.

Findings

Cross-species interactions are ${1\over N}$ suppressed at large N.

Numerical simulations at N=3 support symmetry predictions.

Inter-species interactions are of order ${\cal O}(1/(N-1))$.

Abstract

't~Hooft's abelian projection of $SU(N)$ gauge theory yields $N$ mutually constrained, compact abelian fields which are permutationally equivalent. We formulate the notion of ``species permutation'' symmetry of the $N$ abelian projection fields and discuss its consequences for cross-species correlators. We show that at large $N$ cross-species interactions are ${1\over N}$ suppressed relative to same-species interactions. Numerical simulations at $N=3$ support our symmetry arguments and reveal the existence of inter-species interactions of size ${\cal O\/}\bigl({1\over N-1}\bigr)$ as analytically predicted.