Towards 2-dimensional non-commutative integrals

TL;DR

This paper extends the concept of non-abelian integrals from one dimension to two dimensions, introducing a new framework for surface parallel transport and integration that generalizes existing formulas and concepts.

Contribution

It proposes a novel notion of non-abelian 2-form integrals based on Lie algebroid theory, broadening the scope of non-abelian integration to higher dimensions.

Findings

Includes abelian integrals as special cases

Generalizes Baker-Campbell-Hausdorff formula to 2D

Provides a framework for non-abelian surface transport

Abstract

We collect evidence that the notion of path-ordered non-abelian integration admits an extension to two dimensions. We propose the corresponding notion of non-abelian 2-form along the lines of Lie algebroid theory and argue it is an appropriate one. The processes of parallel transport and integration turn out to be subtly different in the 2-dimensional case; we discuss parallel transport along surfaces and present an indirect definition of a non-abelian integral. This integral includes, for specific choices of 2-forms, both abelian integrals and the continuous limit of Baker-Campbell-Hausdorff formula as special cases; it interpolates between those cases and broadly generalizes them, allowing, for example, an analog of path-exponential with local, point-depending commutators to be spoken about. We comment on all these objects, their relations, gauge symmetries and geometrical meaning, and roughly sketch a plausible order-by-order procedure for obtaining formulas for non-abelian integrals. The exposition is reasonably concrete, relying on no notions more abstract than sections of vector bundles and homotopies.