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Vanishing of the $H^3$ obstruction for time-reversal symmetry in (2+1)D abelian bosonic TQFTs

Ippo Orii

Abstract

In $(2+1)$-dimensional topological quantum field theories (TQFTs), the action of a global symmetry group on the anyon system is one of the central topics of research. Owing to the subtle categorical nature of anyons, a global symmetry acting on them is generally realized in a projective manner. Symmetry fractionalization encodes this projective realization. The obstruction to defining symmetry fractionalization is captured by a cohomology class, known as the $H^3$ obstruction, whose nontriviality signals a failure to define symmetry fractionalization consistently. In this short note, we prove that the $H^3$ obstruction for time-reversal symmetry always vanishes in abelian bosonic TQFTs.

Vanishing of the $H^3$ obstruction for time-reversal symmetry in (2+1)D abelian bosonic TQFTs

Abstract

In -dimensional topological quantum field theories (TQFTs), the action of a global symmetry group on the anyon system is one of the central topics of research. Owing to the subtle categorical nature of anyons, a global symmetry acting on them is generally realized in a projective manner. Symmetry fractionalization encodes this projective realization. The obstruction to defining symmetry fractionalization is captured by a cohomology class, known as the obstruction, whose nontriviality signals a failure to define symmetry fractionalization consistently. In this short note, we prove that the obstruction for time-reversal symmetry always vanishes in abelian bosonic TQFTs.

Paper Structure

This paper contains 25 sections, 73 equations.

Table of Contents

  1. Introduction and Summary
  2. Organization of the paper :
  3. Basics of abelian TQFTs
  4. $U(1)^N$ Chern--Simons Theory
  5. Defining data of abelian systems
  6. Relation to $U(1)^N$ Chern--Simons theory :
  7. Moore--Seiberg Data
  8. Comment on three constructions :
  9. What is the obstruction
  10. Time Reversal on Moore--Seiberg Data
  11. Time reversal on anyon :
  12. Time reversal on $(F,R)$ :
  13. Redundancy for $U$ :
  14. Redundancy for $\beta$ :
  15. Definition of the $H^3$ obstruction
  16. ...and 10 more sections