Loop Groups, Kaluza-Klein Reduction and M-Theory

TL;DR

This paper explores the mathematical structure of principal bundles and loop groups to understand the Kaluza-Klein reduction of M-theory, revealing how characteristic classes encode supergravity field identities and supporting existing conjectures.

Contribution

It establishes an equivalence between principal G-bundles over circle bundles and loop group bundles, linking topological invariants to supergravity Bianchi identities.

Findings

Loop group characteristic classes encode IIA supergravity Bianchi identities.

Central extension classes relate to massive IIA Bianchi identities.

Supports conjectures connecting loop groups to M-theory reductions.

Abstract

We show that the data of a principal G-bundle over a principal circle bundle is equivalent to that of a \hat{LG} = U(1) |x LG bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of hep-th/0203218.