Hamiltonian anomalies from extended field theories

TL;DR

This paper develops a framework to understand Hamiltonian anomalies in quantum field theories through extended field theories and anomaly field theories, including non-invertible cases relevant to six-dimensional superconformal theories.

Contribution

It proposes a new perspective on anomalous field theories as relative theories valued in higher-dimensional anomaly theories, extending to non-invertible cases and constructing explicit examples.

Findings

Hamiltonian anomalies correspond to projective representations of symmetry groups.

Anomalous state spaces are described by abelian or non-abelian gerbes.

Constructed Dai-Freed and Wess-Zumino theories as extended field theories.

Abstract

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the state space, or that the latter is really an abelian gerbe rather than an ordinary Hilbert space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2,0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms, as extended field theories down to codimension 2.