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Hamiltonian anomalies from extended field theories

Samuel Monnier

TL;DR

This work reframes anomalous quantum field theories as relative theories valued in an anomaly field theory of one higher dimension, following Freed. By extending the anomaly theory down to codimension two, it naturally recovers the Hamiltonian anomaly as a projective representation of the symmetry group and reveals the state space as an abelian gerbe, with a non-invertible generalization governed by a degree-2 non-abelian cohomology class. The paper then develops explicit extended-field-theory models for anomalies: Wess-Zumino terms via extended Chern–Simons constructions, and Dai–Freed theories for chiral fermions, each defined down to codimension two and compatible with gluing, dagger, and monoidal structures. These constructions illuminate the role of higher-categorical structures (2-Hilbert spaces, 2-Hilbert lines, non-abelian gerbes) in encoding anomalies and provide a unified framework for anomalies in 2D rational CFTs and 6D $(2,0)$ theories, as well as the Green–Schwarz-type mechanisms via Wess–Zumino terms. The approach offers a principled path to analyze Hamiltonian anomalies and their symmetry actions in both invertible and non-invertible settings, with significant implications for understanding the geometric and categorical underpinnings of anomalous 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.

Hamiltonian anomalies from extended field theories

TL;DR

This work reframes anomalous quantum field theories as relative theories valued in an anomaly field theory of one higher dimension, following Freed. By extending the anomaly theory down to codimension two, it naturally recovers the Hamiltonian anomaly as a projective representation of the symmetry group and reveals the state space as an abelian gerbe, with a non-invertible generalization governed by a degree-2 non-abelian cohomology class. The paper then develops explicit extended-field-theory models for anomalies: Wess-Zumino terms via extended Chern–Simons constructions, and Dai–Freed theories for chiral fermions, each defined down to codimension two and compatible with gluing, dagger, and monoidal structures. These constructions illuminate the role of higher-categorical structures (2-Hilbert spaces, 2-Hilbert lines, non-abelian gerbes) in encoding anomalies and provide a unified framework for anomalies in 2D rational CFTs and 6D theories, as well as the Green–Schwarz-type mechanisms via Wess–Zumino terms. The approach offers a principled path to analyze Hamiltonian anomalies and their symmetry actions in both invertible and non-invertible settings, with significant implications for understanding the geometric and categorical underpinnings of anomalous 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.

Paper Structure

This paper contains 61 sections, 6 theorems, 46 equations.

Table of Contents

  1. Introduction and summary
  2. Notation
  3. Field theories
  4. The functorial picture of field theories
  5. Extended field theories
  6. Anomalous field theories
  7. Definition
  8. Properties of anomalous theories
  9. Anomalous field theories with infinite dimensional state space
  10. The anomaly of the partition function
  11. The symmetry group
  12. The partition function anomaly
  13. The Hamiltonian anomaly in the invertible case
  14. The Hamiltonian anomaly in the general case
  15. Unpacking the definitions
  16. ...and 46 more sections

Key Result

Proposition 3.1

The vector of partition functions $\textit{\large f}\:(M^{d-1})$ vanishes outside the space of invariants of the action of ${\rm Aut}_{\mathsf{E},\mathsf{F}}(M^{d-1})$ on $\mathcal{A}(M^{d-1})$.

Theorems & Definitions (8)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Lemma 5.1
  • proof