Contents

Idea

There are at least two things that are called quantum anomalies in the context of quantum field theory

• anomalous action functional: the action functional (in path integral quantization) is not a globally well defined function, but instead a section of a line bundle on configuration space?;

• anomalous symmetry: a symmetry of the action functional does not extend to a symmetry of the exponentiated action times the path integral measure; or equivalently the action of a group on classical phase space is not preserved by deformation quantization.

Definition

Anomalous action functional

There are two major kinds of action functionals that may be anomalous in that they are not actually functions/functionals on the configuration space of fields, but just sections of some line bundle:

• theories with fermions (see e.g. spinors in Yang-Mills theory)

• gauge theories with higher degree gauge fields (differential cocycles of higher degree.)

Fermionic anomalies

The path integral for a quantum field theory with fermions can be decomposed into the integral over the fermionic fields follows by that over the bosonic fields. The former, a Berezin integral? is typically well defined for a fixed configuration of the bosonic fields, but does not produce a well defined function on the space of all bosonic fields: but a twisted function , a section of some line bundle called a determinant line bundle or, in $8k+2$ dimensions, its square root, the Pfaffian line bundle.

So to even start making sense of the remaining path integral over the bosonic degree of freedom, this determinant line bundle or the corresponding Pfaffian line bundle has to be trivializale. Its non-trivializability is the fermionic anomaly .

Higher gauge-theoreric anomalies

For the moment see Green-Schwarz mechanism for more.

Anomalous symmetry

…

Examples

Anomalous action functional

Spinning particles and super-branes

The sigma-model for a supersymmetric fundamental brane on a target space $X$ has an anomaly coming from the nontriviality of Pfaffian line bundles associated with the fermioninc fields on the worldvolume. These anomalies disappear (i.e. these bundles are trivializable) when the structure group of the tangent bundle of $X$ has a sufficiently high lift through the Whitehead tower of $O(n)$.

• Spin structure the worldline anomaly for the spinning particle/superparticle vanishes when $X$ has Spin structure

  This is a classical result. A concrete derivation is in

  • Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

• String structure the worldsheet anomaly for the spinning string/superstring in heterotic string theory? vanishes (essentially) when $X$ has String structure

  This is originally due to Killingback and Witten. A commented list of literature is here. Recently Ulrich Bunke gave the rigorous proof

  • Ulrich Bunke, String structures and trivialisations of a Pfaffian line bundle (arXiv)

• Fivebrane structure the worldvolume anomaly for the super-5-brane in dual heterotic string theory? vanishes (essentially) when $X$ has Fivebrane structure. See there.

Anomalous symmetry

Conformal anomaly

For the moment see Liouville cocycle.

References

The role of spin structures as the qnomaly cancellation condition for the spinning particle is discussed in

• Edward Witten, Global anomalies in String theory in Symposium on anomalies, geometry, topology , World Scientific Publishing, Singapore (1985)

The fundamental article for the role of the determinant line bundle in understanding the anomalies is

• M. F. Atiyah, I. M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984)

A physicists’ monograph:

• Reinhold A. Bertlmann, Anomalies in quantum field theory, Oxford Science Publ., 1996, 2000

A reference that very clearly identifies the mathematical nature of quantum anomalies for higher gauge theories is

• Dan Freed, Dirac charge quantization and generalized differential cohomology (arXiv)