Nonabelian Surface Holonomy from Multiplicative Integration

TL;DR

This paper develops an explicit analytic framework for nonabelian surface holonomy using multiplicative integration, extending gauge theories and providing concrete models for higher parallel transport without complex categorical tools.

Contribution

It introduces a novel analytic formulation of nonabelian surface holonomy via multiplicative integration, connecting it to higher parallel transport and the Wess-Zumino phase.

Findings

Provides explicit models for nonabelian surface holonomy

Extends holonomy concepts to arbitrary gauge 2-bundles

Derives a 3D Stokes theorem linking to Chern-Simons theory

Abstract

Surface holonomy and the Wess-Zumino phase play a central role in string theory and Chern-Simons models, yet a completely analytic formulation of their nonabelian counterparts has remained elusive. In this work, we show that Yekutieli's theory of multiplicative integration provides such a formulation and realizes explicitly the higher parallel transport structure of Schreiber and Waldorf. Starting from a smooth 2-connection $(\alpha,\beta)$ on a Lie crossed module, we prove that the corresponding multiplicative integrals satisfy the axioms of a transport 2-functor, thereby providing an explicit model for nonabelian surface holonomy. This framework extends the familiar holonomy on $U(1)$-bundle gerbes to arbitrary gauge 2-bundles whilst avoiding abstract categorical machinery. The resulting three-dimensional Stokes theorem yields the Wess-Zumino phase law and gives an analytic counterpart of the boundary phase relation underlying the Chern-Simons functional.