New insights in brane and Kaluza--Klein theory through almost product structures

TL;DR

This paper explores the geometric structures underlying gauge and Kaluza-Klein theories using almost product structures, linking gauge fields to Nijenhuis tensors and classifying M-brane solutions within this framework.

Contribution

It introduces a novel geometric approach to gauge and Kaluza-Klein theories via almost product structures, providing a classification scheme for M-brane solutions.

Findings

Gauge theory described by almost product structures

Gauge field strength as Nijenhuis tensor

Classification of M-brane solutions within this geometric framework

Abstract

We will show that gauge theory can be described by an almost product structure, which is a certain type of endomorphism of the tangent bundle. We will recover the gauge field strength as the Nijenhuis tensor of this endomorphism. We discuss a generalization to the case of a general Kaluza-Klein theory. Furthermore, we will look at the classification of these almost product structures in the case where we have a manifold with metric, and fit the M-brane solutions into this classification scheme. In this analysis certain algebraic properties of the space of differential forms and multivectors are obtained. All analysis is global but we will give local expressions where we find it suitable.