  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.
