Anomalies in gauge theory and gerbes over quotient stacks

In Yang–Mills theory one is interested in lifting the action of the gauge transformation group G=G(P)G=G(P) on the space of connection one-forms A=A(P)A=A(P), where P⟶MP⟶M is a principal GG-bundle over a compact Riemannian spin manifold MM, to the total space of the Fock bundle F⟶AF⟶A in a consistent way with the second quantized Dirac operators D/Aˆ,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 Gˆ, not necessarily central, of GG that acts in the Fock bundle. The Faddeev–Mickelsson anomaly is then essentially the class of the Lie group extension Gˆ.When M=S1M=S1 and PP is the trivial GG-bundle, we are dealing with S1S1-central extensions of loop groups LGLG 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 dimM>1dimM>1 the group multiplication in Gˆ depends also on the elements A∈AA∈A and hence is no longer an S1S1-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 Gˆ can be replaced with a Lie groupoid   extension of the action Lie groupoid A⋊GA⋊G, where AA 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 S1S1-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 S1S1-gerbe over the quotient stack [A/G][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 dimM=3dimM=3.