Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On anomalies and fermionic unitary operators

Masaki Okada, Shutaro Shimamura, Yuji Tachikawa, Yi Zhang

TL;DR

The paper develops a Hamiltonian perspective on anomalies by focusing on localized symmetry operations realized as unitary operators, revealing that fermionic unitary operators can anticommute when anomalies are present. Through two complementary derivations in 2D, the authors derive the commutator map $\gamma(f,g)$ and the 2-cocycle $\beta(f,g)$ for position-dependent $U(1)$ actions, showing that odd level $k$ and odd windings yield fermionic behavior. They extend the analysis to finite subgroups $\mathbb{Z}_n \subset U(1)$ and to 4D SU(2) with the Witten anomaly, where anticommutation of disjointly supported, winding-one transformations is established via higher-dimensional invertible phases and $\eta$-invariants, highlighting the unifying role of differential cohomology, quadratic refinements, and anomaly data. The work connects bosonic and fermionic anomaly classifications and provides concrete, computable expressions for commutator maps that encode locality obstructions in the presence of anomalies. The results offer a coherent framework linking Hamiltonian locality, topological phases, and symmetry anomalies across dimensions, with potential implications for understanding gauging obstructions and edge-bulk correspondences in QFTs.

Abstract

We point out that fermionic unitary operators which anticommute among themselves appear in various situations in quantum field theories with anomalies in the Hamiltonian formalism. To illustrate, we give multiple derivations of the fact that position-dependent $U(1)$ transformations of two-dimensional theories with $U(1)$ symmetry of odd level are fermionic when the winding number is odd. We then relate this mechanism to the anomalies of the discrete $\mathbb{Z}_N \subset U(1)$ symmetry, whose description also crucially uses unitary operators which are fermionic. We also show that position-dependent $SU(2)$ transformations of four-dimensional theories with $SU(2)$ symmetry with Witten anomaly are fermionic and anticommute among themselves when the winding number is odd.

On anomalies and fermionic unitary operators

TL;DR

The paper develops a Hamiltonian perspective on anomalies by focusing on localized symmetry operations realized as unitary operators, revealing that fermionic unitary operators can anticommute when anomalies are present. Through two complementary derivations in 2D, the authors derive the commutator map and the 2-cocycle for position-dependent actions, showing that odd level and odd windings yield fermionic behavior. They extend the analysis to finite subgroups and to 4D SU(2) with the Witten anomaly, where anticommutation of disjointly supported, winding-one transformations is established via higher-dimensional invertible phases and -invariants, highlighting the unifying role of differential cohomology, quadratic refinements, and anomaly data. The work connects bosonic and fermionic anomaly classifications and provides concrete, computable expressions for commutator maps that encode locality obstructions in the presence of anomalies. The results offer a coherent framework linking Hamiltonian locality, topological phases, and symmetry anomalies across dimensions, with potential implications for understanding gauging obstructions and edge-bulk correspondences in QFTs.

Abstract

We point out that fermionic unitary operators which anticommute among themselves appear in various situations in quantum field theories with anomalies in the Hamiltonian formalism. To illustrate, we give multiple derivations of the fact that position-dependent transformations of two-dimensional theories with symmetry of odd level are fermionic when the winding number is odd. We then relate this mechanism to the anomalies of the discrete symmetry, whose description also crucially uses unitary operators which are fermionic. We also show that position-dependent transformations of four-dimensional theories with symmetry with Witten anomaly are fermionic and anticommute among themselves when the winding number is odd.

Paper Structure

This paper contains 37 sections, 3 theorems, 203 equations, 9 figures.

Table of Contents

  1. Introduction and summary
  2. U(1) symmetry in two dimensions
  3. The setup
  4. Comments
  5. First derivation
  6. Mode expansion.
  7. The regularized operator.
  8. The 2-cocycle and the commutator.
  9. Second derivation
  10. The commutator map $\gamma$.
  11. The 2-cocycle $\beta$.
  12. Zn symmetry in two dimensions
  13. Anomaly of finite group G in 2d bosonic theories
  14. Anomaly of finite group G in 2d fermionic theories
  15. The data.
  16. ...and 22 more sections

Key Result

Theorem 1

If there is a commutator map $\gamma$ satisfying the consistency conditions ($\gamma$-0)--($\gamma$-3), then $k$ is an integer. For any integer $k$, there is a unique such commutator map $\gamma$, and its explicit formula can be given as

Figures (9)

  • Figure 1: The patching of $S^1$ with multiple intervals $\sigma_u$ and points $\sigma_{u,u+1}$.
  • Figure 2: Unitary fusion operators at the vicinities of boundary of supports.
  • Figure 3: The profile and auxiliary functions used in the analysis of the anomaly of $\mathbb{Z}_n\subset U(1)$.
  • Figure 4: The 5d configuration we employ. The light-red shaded region is where the $SU(2)$ field is nontrivial due to the gauge transformation by $f$. (This entire gauge configuration on $M_3\times S^1_X$ is pulled back rather trivially to $S^1_Y$, and therefore it might be better to fill by light-red along the $S^1_Y$ direction. That would make the figure more complicated, so we decided not to do so.) Similarly, the light-blue shaded region is from $g$.
  • Figure 5: The $n=1$ patching. where the winding points of $f$ and $g$ are $0$ and $0$ respectively.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof