Anomalies and Bosonization

TL;DR

This paper reviews advanced bosonization methods beyond 1+1 dimensions, extends them to systems with boundaries, and explores their implications for understanding anomalies in fermionic and topological phases.

Contribution

It introduces extensions of bosonization to boundary conditions and anomalous theories, providing new tools for analyzing fermionic system anomalies and topological phases.

Findings

Bosonization methods extended to boundary conditions.

Modified symmetry algebra in anomalous bosonized theories.

New constraints in SPT phases with domain defects.

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.