Abelian representation for nonabelian Wilson loops and the Non - Abelian Stokes theorem on the lattice

TL;DR

This paper derives an Abelian-like expression for lattice SU(N) Wilson loops in any representation, connecting continuum and lattice formulations, and presents a lattice version of the non-Abelian Stokes theorem.

Contribution

It introduces a novel Abelian-like lattice expression for SU(N) Wilson loops and formulates a lattice non-Abelian Stokes theorem, bridging continuum and lattice approaches.

Findings

Derived Abelian-like lattice Wilson loop expression for SU(N).

Connected continuum Abelian representation with lattice formulation.

Presented explicit lattice non-Abelian Stokes theorem.

Abstract

We derive the Abelian - like expression for the lattice SU(N) Wilson loop in arbitrary irreducible representation. The continuum Abelian representation of the SU(N) Wilson loop (for the loop without selfintersections) that has been obtained by Diakonov and Petrov appears to be a continuum limit of this expression. We also obtain the lattice variant of a non - Abelian Stokes theorem and present the explicit expression for the matrix $\cal H$ used in the Diakonov - Petrov approach.