Equivariant cohomology and current algebras

TL;DR

This paper introduces a new differential graded Lie algebra framework where contractions form a free Lie superalgebra, providing insights into twisted equivariant cohomology and higher-dimensional current algebra cocycles.

Contribution

It defines a novel dg Lie algebra with non-commuting contractions, classifies its central extensions via invariant polynomials, and links this to twisted equivariant cohomology and topological lifting problems.

Findings

Central extensions classified by invariant polynomials

Framework for twisted equivariant cohomology

New interpretation of higher-dimensional current algebra cocycles

Abstract

Let M be a manifold and g a Lie algebra acting on M. Differential forms Omega(M) carry a natural action of Lie derivatives L(x) and contractions I(x) of fundamental vector fields for x \in g. Contractions (anti-) commute with each other, [I(x), I(y)]=0. Together with the de Rham differential, they satisfy the Cartan's magic formula [d, I(x)]=L(x). In this paper, we define a differential graded Lie algebra Dg, where instead of commuting with each other, contractions form a free Lie superalgebra. It turns out that central extensions of Dg are classified (under certain assumptions) by invariant homogeneous polynomials p on g. This construction gives a natural framework for the theory of twisted equivariant cohomology and a new interpretation of Mickelsson-Faddeev-Shatashvili cocycles of higher dimensional current algebras. As a topological application, we consider principal G-bundles (with G a Lie group integrating g), and for every homogeneous polynomial p on g we define a lifting problem with the only obstruction the corresponding Chern-Weil class cw(p).