Faddeev's anomaly and bundle gerbes
type: Item
instance of: scholarly article
title: Faddeev's anomaly and bundle gerbes
zbMATH Open document ID: 0848.22024
DOI: 10.1007/BF00400136
author: Michael K. Murray
author: A. L. Carey
published in: Letters in Mathematical Physics
publication date: 1996-10-21T00:00:00Z
review text: In [the authors, Mathematical remarks on the cohomology of gauge groups and anomalies, Intern. J. Modern Phys. (to appear)] we indicated how Segal's discussion of Faddeev's anomaly [\textit{G. B. Segal}, Faddeev's anomaly in Gauss' law (Preprint)] could be phrased in the language of bundle gerbes and conjectured that the Faddeev anomaly was the transgression of the Dixmier-Douady class of the corresponding bundle gerbe. In this letter, we prove this conjecture. We utilise the results of \textit{J. Mickelsson} and \textit{S. G. Rajeev} [Commun. Math. Phys. 116, 365-400 (1988; Zbl 0648.22013)] on the Faddeev-Mickelsson 2-cocycle on the gauge group to determine the Dixmier-Douady class via our notion of transgression (defined in Section 2). This proof is of interest for several reasons. First, Segal's above mentioned work remains unpublished. Second, the discussion of this issue by \textit{J. Mickelsson} [Current algebras and groups, New York-London (1989; Zbl 0726.22015)] gives the impression that there is no connection between Segal's approach and that of \textit{J. Mickelsson} and \textit{S. G. Rajeev} [op. cit.]. The reason for this is that the notion of bundle gerbe was invented recently and is essential for the method we adopt here of relating the two. Third, the notion of bundle gerbe provides a way of realising degree three integral cohomology of a manifold geometrically that is naturally adapted to the classes arising in the study of anomalies [see \textit{B. Wang}, \textit{A. L. Carrey} and \textit{M. K. Murray}, Higher bundle gerbes, descent equations and 3-cocycles, Preprint (1995)]. The example discussed in this letter lends further weight to this last assertion.
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 81T50
zbMATH DE Number: 888951
zbMATH Keywords: infinite-dimensional Lie groups
zbMATH Keywords: bundle gerbes
zbMATH Keywords: Faddeev anomaly
zbMATH Keywords: Dixmier-Douady class
zbMATH Keywords: Faddeev-Mickelsson 2-cocycle
zbMATH Keywords: gauge group
zbMATH Keywords: integral cohomology
zbMATH Keywords: manifold
MaRDI profile type: Publication
cites work: Q3999168
cites work: Current algebras in \(d+1\)-dimensions and determinant bundles over infinite-dimensional Grassmannians
description: scientific article; zbMATH DE number 888951