Abstract
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.
Highlights
Manifold and reproduces the anomaly on a manifold with a boundary
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
Summary
The classification of SPT phases in terms of generalized cohomology/cobordism and associated “layers” is somewhat forbidding, but has a rather transparent physical meaning. We can take the cell decomposition C dual to the triangulation, and place on the facets of C some collection of invertible TFTs (meaning here SPTs with no symmetries) of appropriate dimension, following some rules which take into account the discrete data we put on the manifold. The cocycle condition ensures that the partition function defined as the product of all the phases is independent of the choice of triangulation and gauge. Applying the rules above to M × S1 and reducing them to some evaluation on the triangulation of M one can figure out the resulting SPT theory in one dimension lower This was done for the Gu-Wen layer in [62], but has not been done in full generality
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