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March 14, 2006

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 PU(H)P U (H)-bundles to U(H)U(H)-bundles. Here HH is a finite or infinite dimensional Hilbert space and PU(H)P U(H) is the group of projective unitary operators on HH.

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 XX of “background fields”, in particular moduli spaces of Riemannian metrics and gauge potentials.

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

Posted at March 14, 2006 9:33 AM UTC

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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.

Posted by: Thomas Larsson on March 15, 2006 2:32 AM | Permalink | Reply to this

Re: Gerbes and Quantum Field Theory

Hi Thomas,

I am currently at the airport, without much time and leisure, so the following remark may be affected by that.

It seems to me that what Jouko Mickelsson is secretly getting at is that the Mickelsson-Fadeev cocycle is not related to gerbes and loop groups as (16) and (17) are, but to 3-gerbes and “3-sphere groups”.

The construction of the centrally extended loop group of GG with the central extension induced by the 3-form (16) on GG is closely related to the construction of a gerbe on GG.

Now, he seems to be saying that the Mickelsson-Fadeev cocycle (21) really comes from a 5-form instead of a 3-form (he remarks on p. 11 that the classes [c][c] is the same as that of the transgression [c 1][c_1] of the 5-form ω 5\omega_5). Consequently, if I understand correctly, he says (in the paragrph on p. 12 starting with “This construction generalizes to higher loop groups […]”) that this gives not the central extension of a loop group but of a 3-sphere group.

contrast the MF cocycle to the Kassel (multi-dimensional affine) cocycle, to which the no-go theorem does not apply.

This is probably using higher tori instead of higher spheres, right?

coupled to a classical background gauge field

I would think so.

Posted by: urs on March 15, 2006 11:01 AM | Permalink | Reply to this

Re: Gerbes and Quantum Field Theory

contrast the MF cocycle to the Kassel (multi-dimensional affine) cocycle, to which the no-go theorem does not apply.
This is probably using higher tori instead of higher spheres, right?

No, the extensions differ already locally (or for Laurent polynomials). In my paper, I contrast the two cocycles in the same Fourier basis on the 3-torus (eqs (2) and (5)). MF has two derivatives and is proportional to the third Casimir, KM has one derivative and is proportional to the second Casimir.

Posted by: Thomas Larsson on March 15, 2006 11:22 AM | Permalink | Reply to this

Re: Gerbes and Quantum Field Theory

I have now taken a look at your simple plausibility argument in math-ph/0501023. The argument is the stronger the more noncontractible 1-cycles there are. In particular, if there is no such cycle, then the argument does not tell us much.

Ok, so the MF algebra does not have physical representations in the ordinary sense. The anomaly simply has to vanish. Still, when it does not vanish, then the information about how precisely it does not vanish is apparently encoded in some gerbe structure. I think that’s Mickelsson’s point in the review that we talked about.

Posted by: urs on March 15, 2006 3:21 PM | Permalink | Reply to this

Re: Gerbes and Quantum Field Theory

The argument is the stronger the more noncontractible 1-cycles there are. In particular, if there is no such cycle, then the argument does not tell us much.

It works for Laurent polynomials, too. We don’t really expect physics to depend to much on what happens at the end of the universe. At least my intuition from CFT applied to statphys says that the conformal anomaly is real, even if we choose boundary conditions such that our system is topologically a sphere.

A no-go theorem is one thing, but for the KM-like extension we have a go theorem. Unitary reps have been constructed (take J_X = int dt X_a(q(t)) J^a(t), where J^a(t) satisfy a unitary rep of an affine algebra and q(t) is a curve), and reps (not unitary) which are close to being faithful (which depend on X_a(x) at x=q(t)and its derivatives up to some finite but arbitrary high order p). Note that these reps do not depend on some background field, so in that sense they are fully quantum.

Posted by: Thomas Larsson on March 15, 2006 5:14 PM | Permalink | Reply to this