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

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 H gamma,O is proportional to R sigma. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop gamma embedded in a manifold M, H gamma,O is an element of a Lie group G; the curvature R sigma is an element of g is an element of the Lie algebra of G. However, it turns out that the curvature form R sigma obtained from the small loop approximation is ambiguous, as the information of gamma and H gamma,O are insufficient for determining a specific plane sigma responsible for R sigma. To resolve this ambiguity, it is necessary to specify the surface S enclosed by the loop gamma; hence, sigma is defined as the limit of S when gamma 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.

Automated summary

This article addresses the relationship between the holonomy along a loop and the integral of the curvature form over the surface it encloses. By specifying the surface enclosed by the loop, the ambiguity is resolved. The derived result can be viewed as an alternative proof of the non-Abelian Stokes theorem.