Anomalies in gauge theory and gerbes over quotient stacks

Abstract

In Yang–Mills theory one is interested in lifting the action of the gauge transformation group G = G ( P ) on the space of connection one-forms A = A ( P ) , where P ⟶ M is a principal G -bundle over a compact Riemannian spin manifold M , to the total space of the Fock bundle F ⟶ A in a consistent way with the second quantized Dirac operators D / A ˆ , A ∈ A . In general, there is an obstruction to this called the Faddeev–Mickelsson anomaly, and to overcome this one has to introduce a Lie group extension G ˆ , not necessarily central, of G that acts in the Fock bundle. The Faddeev–Mickelsson anomaly is then essentially the class of the Lie group extension G ˆ . When M = S 1 and P is the trivial G -bundle, we are dealing with S 1 -central extensions of loop groups L G as in [A. Pressley, G. Segal, Loop groups, in: Oxford Mathematical Monographs, Clarendon Press, 1986]. However, it was first noticed in the pioneering works of Mickelsson [J. Mickelsson, Chiral anomalies in even and odd dimensions, Comm. Math. Phys. 97 (1985)] and Faddeev, [L. Faddeev, Operator anomaly for the Gauss law, Phys. Lett. 145B (1984)] that when dim M > 1 the group multiplication in G ˆ depends also on the elements A ∈ A and hence is no longer an S 1 -central extension of Lie groups. We give a new interpretation of certain noncommutative versions of the Faddeev–Mickelsson anomaly (see for example [S.G. Rajeev, Universal gauge theory, Phys. Rev. D, 42 (8) (1990); E. Langmann, J. Mickelsson, S. Rydh, Anomalies and Schwinger terms in NCG field theory models, J. Math. Phys. 42 (10) (2001) 4779–4801; J. Arnlind, J. Mickelsson, Trace extensions, determinant bundles, and gauge group cocycles, Lett. Math. Phys. 62 (2002) 101–110]) and show that the analogous Lie group extensions G ˆ can be replaced with a Lie groupoid extension of the action Lie groupoid A ⋊ G , where A is now some relevant abstract analog of the space of connection one-forms. Then at the level of Lie groupoids, this extension proves out to be an S 1 - central extension and hence one may apply the general theory of these extensions developed by Behrend and Xu in [K. Behrend, P. Xu, Differentiable stacks and gerbes. arXiv:math.DG/0605694]. This makes it possible to consider the Faddeev–Mickelsson anomaly as the class of this Lie groupoid extension or equivalently as the class of a certain differentiable S 1 -gerbe over the quotient stack [ A / G ] . We also give examples from noncommutative gauge theory where our construction can be applied. The construction may also be used to give a geometric interpretation of the (classical) Faddeev–Mickelsson anomaly in Yang–Mills theory when dim M = 3 .