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Nonabelian Surface Holonomy from Multiplicative Integration

Hollis Williams

TL;DR

This work gives an analytic realization of nonabelian surface holonomy by deriving a 3D multiplicative Stokes theorem for Yekutieli’s multiplicative integration, bridging MI with Schreiber–Waldorf transport 2-functors. It shows that the MI of a 2-connection on a Lie crossed module satisfies the axioms of a transport 2-functor and that the variation of surface holonomy is governed by a 3-curvature obstruction, yielding a nonabelian generalization of the Wess–Zumino phase. The results unify higher gauge theory with concrete differential geometry, clarifying boundary phases in Chern–Simons theory and nonabelian B-field backgrounds while avoiding heavy categorical machinery. Applications to nonabelian string backgrounds, gauge anomalies, and CS boundary terms demonstrate the utility of an analytic framework for higher parallel transport in physics. The work also points to future extensions to higher n-categories, connections with differential cohomology, and potential topological field theory implications.

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 $(α,β)$ 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.

Nonabelian Surface Holonomy from Multiplicative Integration

TL;DR

This work gives an analytic realization of nonabelian surface holonomy by deriving a 3D multiplicative Stokes theorem for Yekutieli’s multiplicative integration, bridging MI with Schreiber–Waldorf transport 2-functors. It shows that the MI of a 2-connection on a Lie crossed module satisfies the axioms of a transport 2-functor and that the variation of surface holonomy is governed by a 3-curvature obstruction, yielding a nonabelian generalization of the Wess–Zumino phase. The results unify higher gauge theory with concrete differential geometry, clarifying boundary phases in Chern–Simons theory and nonabelian B-field backgrounds while avoiding heavy categorical machinery. Applications to nonabelian string backgrounds, gauge anomalies, and CS boundary terms demonstrate the utility of an analytic framework for higher parallel transport in physics. The work also points to future extensions to higher n-categories, connections with differential cohomology, and potential topological field theory implications.

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 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 -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.

Paper Structure

This paper contains 16 sections, 44 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and conventions
  4. Multiplicative integration on surfaces
  5. MI is a transport 2-functor
  6. Transport 2-Functors
  7. Verification of the functor axioms
  8. The Wess–Zumino Term and 3-Form Curvature Obstruction
  9. Curvature obstruction as a phase difference
  10. Abelian case and physical interpretation
  11. Relation to previous work
  12. Applications
  13. Nonabelian string backgrounds
  14. Gauge anomalies
  15. Chern-Simons boundary phases
  16. ...and 1 more sections

Theorems & Definitions (2)

  • proof
  • proof