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Global Anomalies on the Hilbert Space

Diego Delmastro, Davide Gaiotto, Jaume Gomis

TL;DR

The paper develops a systematic, elementary approach to detecting global anomalies by examining how symmetry algebras are realized on torus Hilbert spaces, linking Hilbert-space fingerprints to cobordism-layer anomaly classifications. It analyzes both free-fermion systems and spin TQFTs across dimensions, identifying concrete signatures for Arf, psi, and higher layers, including time-reversal and projective rotation effects. A key result is that certain anomalies force bose-fermi degeneracies or projective symmetry actions in the Hilbert space, with explicit examples in 0+1d, 1+1d, 2+1d, and spin TQFTs, and a general construction for spin-TQFT Hilbert spaces from bosonic shadows. The work provides practical Hilbert-space tools and modular data constructions (S/T/Wilson lines) for diagnosing anomalies and predicts nontrivial spectral/symmetry structures in strongly coupled QFTs and TQFTs, including supersymmetric-like bose-fermi degeneracies arising from anomalies.

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.

Global Anomalies on the Hilbert Space

TL;DR

The paper develops a systematic, elementary approach to detecting global anomalies by examining how symmetry algebras are realized on torus Hilbert spaces, linking Hilbert-space fingerprints to cobordism-layer anomaly classifications. It analyzes both free-fermion systems and spin TQFTs across dimensions, identifying concrete signatures for Arf, psi, and higher layers, including time-reversal and projective rotation effects. A key result is that certain anomalies force bose-fermi degeneracies or projective symmetry actions in the Hilbert space, with explicit examples in 0+1d, 1+1d, 2+1d, and spin TQFTs, and a general construction for spin-TQFT Hilbert spaces from bosonic shadows. The work provides practical Hilbert-space tools and modular data constructions (S/T/Wilson lines) for diagnosing anomalies and predicts nontrivial spectral/symmetry structures in strongly coupled QFTs and TQFTs, including supersymmetric-like bose-fermi degeneracies arising from anomalies.

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.

Paper Structure

This paper contains 32 sections, 178 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Summary
  2. Anomalies from Layers
  3. Anomalies in free fermion Hilbert space
  4. Anomalous $\mathbb Z^{\mathsf T}_2$ in 0+1 dimensions
  5. SYK model.
  6. Anomalous $\mathbb Z^{\mathsf T}_4$ in 2+1 dimensions
  7. Anomalous $\mathbb Z_2$ in 1+1 dimensions
  8. Projective rotations.
  9. Anomalies in spin TQFT Hilbert space
  10. $\nu=2\mod4$: Arf layer
  11. Hilbert space.
  12. Modularity.
  13. Wilson lines.
  14. Time-reversal.
  15. $\nu$ odd: fermion layer
  16. ...and 17 more sections

Figures (1)

  • Figure 1: Schematic notation for an arbitrary configuration of anyons on the torus, in the presence of a puncture $\mathsf g$. The green line represents the vertical (time) direction, orthogonal to the torus. We insert an anyon $\mathsf g$ along this direction, which from the point of view of the torus becomes a puncture (a marked point). The red line represents the $\mathsf a$-cycle, and the blue one the $\mathsf b$-cycle. We insert Wilson lines with anyons $\alpha,\beta$ along these cycles, respectively. The cross $\times$ represents the hole. The states in the Hilbert space $\mathcal{H}$ are created by wrapping anyons around the $\mathsf b$-cycle. The states in the twisted Hilbert space $\mathcal{H}^{\mathsf g}$ are created by wrapping anyons around the $\mathsf b$-cycle, in presence of a vertical anyon $\mathsf g$.