Anomaly (physics)

A global anomaly is the quantum violation of a global symmetry current conservation. A global anomaly can also mean that a non-perturbative global anomaly cannot be captured by one loop or any loop perturbative Feynman diagram calculations—examples include the Witten anomaly and Wang–Wen–Witten anomaly.

As these symmetries vanish at infinity, they cannot be constrained by boundary conditions and so must be summed over in the path integral. The sum of the gauge orbit of a state is a sum of phases which form a subgroup of U(1). As there is an anomaly, not all of these phases are the same, therefore it is not the identity subgroup. The sum of the phases in every other subgroup of U(1) is equal to zero, and so all path integrals are equal to zero when there is such an anomaly and a theory does not exist.

An exception may occur when the space of configurations is itself disconnected, in which case one may have the freedom to choose to integrate over any subset of the components. If the disconnected gauge symmetries map the system between disconnected configurations, then there is in general a consistent truncation of a theory in which one integrates only over those connected components that are not related by large gauge transformations. In this case the large gauge transformations do not act on the system and do not cause the path integral to vanish.

However, if we are only interested in the subgroup of gauge transformations that vanish at infinity, we may consider the 3-sphere at infinity to be a single point, as the gauge transformations vanish there anyway. If the 3-sphere at infinity is identified with a point, our Minkowski space is identified with the 4-sphere. Thus we see that the group of gauge transformations vanishing at infinity in Minkowski 4-space is isomorphic to the group of all gauge transformations on the 4-sphere.

When a theory contains an odd number of flavors of chiral fermions, the actions of gauge symmetries in the identity component and the disconnected component of the gauge group on a physical state differ by a sign. Thus when one sums over all physical configurations in the path integral, one finds that contributions come in pairs with opposite signs. As a result, all path integrals vanish and a theory does not exist.

Higher anomalies involving higher global symmetries: Pure Yang–Mills gauge theory as an example

Since cancelling anomalies is necessary for the consistency of gauge theories, such cancellations are of central importance in constraining the fermion content of the standard model, which is a chiral gauge theory.