On some cocycles which represent the Dixmier-Douady class in simplicial de Rham complexes

TL;DR

This paper constructs explicit cocycles in simplicial de Rham complexes representing the Dixmier-Douady class for central U(1)-extensions of loop groups, extending previous work on special cases to the unitary group.

Contribution

It introduces a new construction of a central U(1)-extension of LU(2) and explicit cocycles representing the Dixmier-Douady class in this context.

Findings

Constructed a central U(1)-extension of LU(2).

Developed a cocycle in a triple complex representing the Dixmier-Douady class.

Extended previous transgression results to the unitary group case.

Abstract

When a Lie group $G$ has a central $U(1)$-extension, there is a cocycle in the simplicial de Rham complex $\Omega^3(NG)$ which represents the Dixmier-Douady class. Mickelsson and Brylinski, McLaughlin constructed a central $U(1)$-extension $\widehat{LSU(2)} \rightarrow LSU(2)$ whose Dixmier-Douady class in $\Omega^3(NLSU(2))$ is a kind of transgression of the second Chern class. In this paper, we consider the case of unitary group and construct a central $U(1)$-extension of $LU(2)$. After that we construct also a cocycle in a certain triple complex.