Anomaly constraint on chiral central charge of (2+1)d topological order

Ryohei Kobayashi

doi:10.1103/physrevresearch.3.023107

Abstract

In this short paper, we argue that the chiral central charge c− of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of antiunitary global symmetry. For example, if a (2+1)d fermionic topological quantum field theory (TQFT) has a time-reversal anomaly with T2=(−1)F labeled as ν∈Z16, the TQFT must have c−=1/4 mod 1/2 for odd ν, while c−=0 mod 1/2 for even ν. This generalizes the fact that the bosonic TQFT with T anomaly in a particular class must carry c−=4 mod 8 to fermionic cases. We also study such a constraint for fermionic TQFT with U(1)×CT symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII.Received 5 January 2021Revised 16 March 2021Accepted 12 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023107Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSymmetry protected topological statesTopological field theoriesTopological superconductorsCondensed Matter, Materials & Applied Physics

Highlights

The ’t Hooft anomaly in quantum field theory is a mild violation of the conservation law due to quantum effects
In this paper we found the anomaly constraint on chiral central charge of (2 + 1)d topological order with T symmetry
The constraint comes from a chiral state localized on the T domain wall in the bulk Symmetry Protected Topological (SPT) phase

Summary

INTRODUCTION

The ’t Hooft anomaly in quantum field theory is a mild violation of the conservation law due to quantum effects. The ’t Hooft anomaly is typically matched by a symmetrybroken or gapless phase (e.g., perturbative anomaly), but in some cases the anomaly is known to be matched by a symmetry-preserving gapped phase, realized by topological quantum field theory (TQFT) enriched by the global symmetry [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] This implies that an anomaly in some particular class can be carried by topological degrees of freedom, not by gapless particles, and in particular, the system with an anomaly can have an energy gap.

RYOHEI KOBAYASHI

Therefore we have the Smith map for cobordism groups

DISCUSSION