The Atiyah algebroid of the path fibration over a Lie group

TL;DR

This paper explores the differential geometry of the Atiyah algebroid associated with a principal loop group bundle over a Lie group, revealing how the Cartan 3-form's cohomology class acts as an obstruction.

Contribution

It introduces a new geometric framework for the Atiyah algebroid of the path fibration over a Lie group, linking it to the Cartan 3-form and higher-degree analogues.

Findings

Cohomology class of the Cartan 3-form arises as an obstruction in the lifting problem.

Constructs a 2-form on the path space with differential equal to the pull-back of the Cartan form.

Develops G-equivariant primitives for higher-degree analogues of the Cartan form.

Abstract

Let G be a connected Lie group, LG its loop group, and PG->G the principal LG-bundle defined by quasi-periodic paths in G. This paper is devoted to differential geometry of the Atiyah algebroid A=T(PG)/LG of this bundle. Given a symmetric bilinear form on the Lie algebra g and the corresponding central extension of Lg, we consider the lifting problem for A, and show how the cohomology class of the Cartan 3-form on G arises as an obstruction. This involves the construction of a 2-form on PG with differential the pull-back of the Cartan form. In the second part of this paper we obtain similar LG-invariant primitives for the higher degree analogues of the Cartan form, and for their G-equivariant extensions.