### Gerbes and Quantum Field Theory

#### Posted by Urs Schreiber

Today one can find this nice article on the arXiv:

J. Mickelsson
*Gerbes and Quantum Field Theory*

math-ph/0603031.

This is an article for the Encyclopedia of Mathematical Physics. While in terms of its content a review of the role played by gerbes in the understanding of anolamies in quantum field theory, it is necessarily a terse accumulation of very rather deep ideas.

There are many ways to think about gerbes. This article emphasizes the notion of *lifting gerbes*, measuring the obstruction for the lift of $P U (H)$-bundles to $U(H)$-bundles. Here $H$ is a finite or infinite dimensional Hilbert space and $P U(H)$ is the group of projective unitary operators on $H$.

This is naturally the most efficient description for the examples of interest in this article, which are concerned with families of Dirac operators parameterized by moduli spaces $X$ of “background fields”, in particular moduli spaces of Riemannian metrics and gauge potentials.

(I had recently talked vaguely about a closely related issue here.)

## Re: Gerbes and Quantum Field Theory

For the past 15 years, I have on and off tried to understand what Jouko is doing. When chiral fermions are coupled to gauge fields, the algebra of gauge transformations acquires a Mickelsson-Faddeev (MF) cocycle (eq. 22) - a gauge anomaly. Mickelsson, together with Rajeev and maybe others, originally tried to construct representations of the this algebra. However, it was shown in

D. Pickrell, On the Mickelsson-Faddeev extensions and unitary representations, Comm. Math. Phys. 123 (1989) 617.

that the MF algebra does not possess any faithful unitary representation on a separable Hilbert space (or something like that). As a response to this disappointing result, Mickelsson developed a theory where cocycles depending on an external gauge potential are regarded as generalized representations. It is this theory which evidently is naturally formulated in terms of gerbes.

AFAIU, physically this describes quantized chiral fermions coupled to a classical background gauge field. To understand the chiral anomaly (which I wished that I did), this is apparently sufficient, because the anomaly arises in the fermion sector. However, I am unhappy with the presence of the classical background gauge field. Fundamentally, the gauge field should be quantized, just like all other fields, but if you do that, Pickrell probably comes back to haunt you.

In math-ph/0501023 I gave a simple plausibility argument why the MF algebra lacks unitary representations. My argument is not watertight and never pretended to be so, but what it lacks in rigor it may make up in simplicity. Anyway, I also contrast the MF cocycle to the Kassel (multi-dimensional affine) cocycle, to which the no-go theorem does not apply.