Nonabelian Surface Holonomy from Multiplicative Integration

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.