Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions

Abstract

The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy Hγ,O is proportional to Rσ. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop γ embedded in a manifold M, Hγ,O is an element of a Lie group G; the curvature Rσ∈g is an element of the Lie algebra of G. However, it turns out that the curvature form Rσ obtained from the small loop approximation is ambiguous, as the information of γ and Hγ,O are insufficient for determining a specific plane σ responsible for Rσ. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop γ; hence, σ is defined as the limit of S when γ shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.