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Anomalies and Bosonization

Ryan Thorngren

TL;DR

The paper develops a unified framework to extend bosonization beyond 1+1 dimensions, incorporating boundaries, twisted spin structures, and higher fermionic probes (Kitaev strings, $p+ip$ membranes) to study fermionic anomalies via bosonized descriptions. It connects anomaly inflow, spin cobordism, and the AHSS to derive differential constraints that govern decorated domain-wall constructions and the resulting symmetry algebras, revealing how fermionic anomalies can modify the bosonic symmetry structure (via 2-group and higher-group extensions). Through detailed case studies of 1+1D chiral $U(1)$ and 2+1D time-reversal anomalies, it demonstrates how 0th/1st/2nd (and higher) bosonizations reproduce known fermionic SPT phenomena and predict nontrivial bulk-boundary interplays, including Ising-like nonabelian behavior and duality-based symmetry extensions. The framework provides tools for deriving new constraints in SPT phases, connecting decorated-domain-wall pictures with AHSS differentials, and clarifying how fermionic and bosonic anomalies relate under bosonization, with implications for understanding boundary gauge variations and higher symmetry structures.

Abstract

Recently, general methods of bosonization beyond 1+1 dimensions have been developed. In this article, we review these bosonizations and extend them to the case with boundary conditions. In particular, we study the case when the bulk theory is a $G$-symmetry protected topological phase and the boundary theory has a $G$ 't Hooft anomaly. We discuss how, when the anomaly is not realizable in a bosonic system, the $G$ symmetry algebra becomes modified in the bosonization of the anomalous theory. This gives us a useful tool for understanding anomalies of fermionic systems, since there is no way to compute a boundary gauge variation of the anomaly polynomial, as one does for anomalies of bosonic systems. We take the chiral anomalies in 1+1D and the parity/time reversal anomalies in 2+1D as case studies. We also provide a derivation of new constraints in SPT phases with domain defects decorated by $p+ip$ superconductors and Kitaev strings.

Anomalies and Bosonization

TL;DR

The paper develops a unified framework to extend bosonization beyond 1+1 dimensions, incorporating boundaries, twisted spin structures, and higher fermionic probes (Kitaev strings, membranes) to study fermionic anomalies via bosonized descriptions. It connects anomaly inflow, spin cobordism, and the AHSS to derive differential constraints that govern decorated domain-wall constructions and the resulting symmetry algebras, revealing how fermionic anomalies can modify the bosonic symmetry structure (via 2-group and higher-group extensions). Through detailed case studies of 1+1D chiral and 2+1D time-reversal anomalies, it demonstrates how 0th/1st/2nd (and higher) bosonizations reproduce known fermionic SPT phenomena and predict nontrivial bulk-boundary interplays, including Ising-like nonabelian behavior and duality-based symmetry extensions. The framework provides tools for deriving new constraints in SPT phases, connecting decorated-domain-wall pictures with AHSS differentials, and clarifying how fermionic and bosonic anomalies relate under bosonization, with implications for understanding boundary gauge variations and higher symmetry structures.

Abstract

Recently, general methods of bosonization beyond 1+1 dimensions have been developed. In this article, we review these bosonizations and extend them to the case with boundary conditions. In particular, we study the case when the bulk theory is a -symmetry protected topological phase and the boundary theory has a 't Hooft anomaly. We discuss how, when the anomaly is not realizable in a bosonic system, the symmetry algebra becomes modified in the bosonization of the anomalous theory. This gives us a useful tool for understanding anomalies of fermionic systems, since there is no way to compute a boundary gauge variation of the anomaly polynomial, as one does for anomalies of bosonic systems. We take the chiral anomalies in 1+1D and the parity/time reversal anomalies in 2+1D as case studies. We also provide a derivation of new constraints in SPT phases with domain defects decorated by superconductors and Kitaev strings.

Paper Structure

This paper contains 30 sections, 253 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of Bosonization
  3. 0th and 1st Bosonization
  4. Some Gapped 1+1D Examples and Convolution
  5. 1st Bosonization of the Trivial Fermionic Phase
  6. Bosonization with Global Symmetries and Twisted Spin Structures
  7. SPT Phases
  8. Higher Bosonization
  9. SPTs Beyond Gu-Wen-Freed
  10. 1st Bosonization with Boundary
  11. 1st bosonization of Gu-Wen-Freed anomalies
  12. 1+1D Chiral $U(1)$ Anomaly
  13. 2nd bosonization and more general anomalies
  14. 1+1D Chiral $\mathbb{Z}_2$ Anomaly
  15. Overview
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The $G = \mathbb{Z}_2$ symmetry action of the chiral fermion parity of the free massless Majorana fermion in 1+1D when bosonized is extended nontrivially by the canonical $E_{2,2}$ groupoid symmetry of 1+1D bosonizations, such that the generator of $G$ acts as a duality transformation between two a priori inequivalent bosonizations.