Towards the Theory of Non--Abelian Tensor Fields I

TL;DR

This paper introduces a triangulation-independent area-ordering method for non-Abelian tensor fields, generalizing path ordering and involving a three-index connection with a novel exponentiation definition.

Contribution

It proposes a new area-ordering prescription for non-Abelian tensor fields and defines how to exponentiate three-index matrices using fusion rule structure constants.

Findings

Provides a consistent area-ordering prescription

Generalizes path ordering to higher-dimensional objects

Defines exponentiation for three-index matrices

Abstract

We present a triangulation--independent area--ordering prescription which naturally generalizes the well known path ordering one. For such a prescription it is natural that the two--form ``connection'' should carry three ``color'' indices rather than two as it is in the case of the ordinary one--form gauge connection. To define the prescription in question we have to define how to {\it exponentiate} a matrix with three indices. The definition uses the fusion rule structure constants.