Lorentz Symmetry Fractionalization and Dualities in (2+1)d

TL;DR

This paper explores how Lorentz symmetry fractionalization affects dualities in (2+1)d non-spin quantum field theories, revealing conditions under which different theories are dual when considering spin structures and anomalies.

Contribution

It establishes a precise criterion linking Lorentz symmetry fractionalization to dualities in non-spin QFTs, including the role of framing anomalies and applications to Chern-Simons matter dualities.

Findings

Duality as spin QFT depends on Lorentz symmetry fractionalization.

Framing anomalies differing by multiples of 8 influence duality equivalences.

Lorentz fractionalization is key in Chern-Simons matter dualities.

Abstract

We discuss symmetry fractionalization of the Lorentz group in (2+1)$d$ non-spin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent non-spin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous $\mathbb{Z}_2$ one-form symmetry. Moreover, if the framing anomalies of two non-spin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as non-spin QFTs. Applications to summing over the spin structures, time-reversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in Chern-Simons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as non-spin theories.