Global anomalies on the Hilbert space

We show that certain global anomalies can be detected in an elementary fashion by analyzing the way the symmetry algebra is realized on the torus Hilbert space of the anomalous theory. Distinct anomalous behaviours imprinted in the Hilbert space are identified with the distinct cohomology “layers” that appear in the classification of anomalies in terms of cobordism groups. We illustrate the manifestation of the layers in the Hilbert for a variety of anomalous symmetries and spacetime dimensions, including time-reversal symmetry, and both in systems of fermions and in anomalous topological quantum field theories (TQFTs) in 2 + 1d. We argue that anomalies can imply an exact bose-fermi degeneracy in the Hilbert space, thus revealing a supersymmetric spectrum of states; we provide a sharp characterization of when this phenomenon occurs and give nontrivial examples in various dimensions, including in strongly coupled QFTs. Unraveling the anomalies of TQFTs leads us to develop the construction of the Hilbert spaces, the action of operators and the modular data in spin TQFTs, material that can be read on its own.

ANOMALOUS CHIRAL ACTION FROM THE PATH INTEGRAL

By generalizing the Fujikawa approach, we show in the path integral formalism: (1) how the infinitesimal variation of the fermion measure can be integrated to obtain the full anomalous chiral action; (2) how the action derived in this way can be identified as the Chern–Simons term in five dimensions, if the anomaly is consistent; (3) how the regularization can be carried out, so as to lead to the consistent anomaly and not to the covariant anomaly. We consider a massless left-handed fermion interacting with a non-Abelian gauge field. The gauge field also interacts with a set of Goldstone bosons, so that a gauge-invariant configuration of the gauge field exists. We use Schwinger's "proper time" representation of the Green's function and the guage-invariant point-splitting technique, and find that the consistency requirement and the point-splitting technique allow both an anomalous and a nonanomalous action. In the end, the nature of the vacuum determines whether we have an anomalous theory or a nonanomalous theory.