Non-Abelian Stokes Theorems in Yang-Mills and Gravity Theories

TL;DR

This paper extends the non-Abelian Stokes theorem to gravitational Wilson loops, providing new formulas for holonomies in curved spaces, including constant-curvature backgrounds and small-area loops across dimensions.

Contribution

It introduces non-Abelian Stokes theorems for gravitational holonomies, generalizing previous Yang-Mills results to curved spacetime contexts.

Findings

Derived non-Abelian Stokes theorem for gravitational Wilson loops

Formulated holonomy formulas for constant-curvature backgrounds in 3D

Provided small-area loop formulas applicable in any dimension

Abstract

We recall the non-Abelian Stokes theorem for the Wilson loop in the Yang-Mills theory and discuss its meaning. Then we move to `gravitational Wilson loops', i.e. to holonomies in curved d=2,3,4 spaces and derive non-Abelian Stokes theorems for these quantities as well, which are similar to our formula in the Yang-Mills theory. In particular we derive an elegant formula for the holonomy in the case of a constant-curvature background in three dimensions and a formula for small-area loops in any number of dimensions.