Faddeev's anomaly and bundle gerbes
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 First, Segal's above mentioned work remains unpublished.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.
scientific article; zbMATH DE number 888951