Global Anomalies on the Hilbert Space

TL;DR

This paper presents a method to detect global anomalies through the analysis of symmetry algebra representations on the Hilbert space, revealing their structure via cobordism classifications and implications for supersymmetry in quantum field theories.

Contribution

It introduces an elementary approach to identify global anomalies in the Hilbert space and explores their implications, including supersymmetry and detailed constructions in spin TQFTs.

Findings

Anomalies correspond to distinct cohomology layers in the Hilbert space.

Anomalies can imply an exact bose-fermi degeneracy, indicating supersymmetry.

Develops Hilbert space and modular data constructions for spin TQFTs.

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.