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Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs

Ippo Orii

TL;DR

The work extends the established time-reversal anomaly framework for (2+1)D abelian bosonic TQFTs by incorporating higher central charges xi_n and a new invariant eta_n, culminating in a generalized anomaly formula that involves a distinguished anyon subset M^n. It systematically analyzes the crosscap data and symmetry fractionalization through M_a, eta(a), and the crosscap state, and establishes the algebraic structure and constraints of M^n (via Im^n(1+T) torsors and related sets E^n). The findings connect dimension counts on non-orientable surfaces to the higher central charges and provide a refined picture of when gapped boundaries can exist via a generalized Lagrangian-subgroup perspective. The results suggest a deeper role for M^n in anomaly cancellation and boundary phenomena, with future work aimed at physical interpretation and generalization to non-abelian and spin-TQFT settings.

Abstract

We study time-reversal symmetry in (2+1)-dimensional abelian bosonic topological phases. Z_2 x Z_2 classifies the time-reversal anomalies in such systems symmetry-protected topological (SPT) phases in (3+1) dimensions and can be diagnosed via partition functions on manifolds such as RP^4 and CP^2. These partition functions are related by an anomaly formula of the form Z(RP^4) * Z(CP^2) = theta_M, where theta_M is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges, denoted xi_n, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP^2) = xi_1, we propose a generalization of the anomaly formula involving both the higher central charges xi_n and a new time-reversal invariant eta_n. By introducing a distinguished subset M^n of the set A of anyons, we establish the relation eta_n * xi_n = (sum of theta(a)^n over a in M^n) divided by its absolute value. This generalizes the known anomaly formula. We analyze the algebraic structure of the subset M^n, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.

Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs

TL;DR

The work extends the established time-reversal anomaly framework for (2+1)D abelian bosonic TQFTs by incorporating higher central charges xi_n and a new invariant eta_n, culminating in a generalized anomaly formula that involves a distinguished anyon subset M^n. It systematically analyzes the crosscap data and symmetry fractionalization through M_a, eta(a), and the crosscap state, and establishes the algebraic structure and constraints of M^n (via Im^n(1+T) torsors and related sets E^n). The findings connect dimension counts on non-orientable surfaces to the higher central charges and provide a refined picture of when gapped boundaries can exist via a generalized Lagrangian-subgroup perspective. The results suggest a deeper role for M^n in anomaly cancellation and boundary phenomena, with future work aimed at physical interpretation and generalization to non-abelian and spin-TQFT settings.

Abstract

We study time-reversal symmetry in (2+1)-dimensional abelian bosonic topological phases. Z_2 x Z_2 classifies the time-reversal anomalies in such systems symmetry-protected topological (SPT) phases in (3+1) dimensions and can be diagnosed via partition functions on manifolds such as RP^4 and CP^2. These partition functions are related by an anomaly formula of the form Z(RP^4) * Z(CP^2) = theta_M, where theta_M is the Dehn twist phase associated with the crosscap state. Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges, denoted xi_n, which serve as computable invariants encoding obstruction data. Motivated by the known relation Z(CP^2) = xi_1, we propose a generalization of the anomaly formula involving both the higher central charges xi_n and a new time-reversal invariant eta_n. By introducing a distinguished subset M^n of the set A of anyons, we establish the relation eta_n * xi_n = (sum of theta(a)^n over a in M^n) divided by its absolute value. This generalizes the known anomaly formula. We analyze the algebraic structure of the subset M^n, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula.

Paper Structure

This paper contains 36 sections, 130 equations, 3 figures, 1 table.

Table of Contents

  1. Introdution and Summary
  2. Organization of the paper :
  3. Basic of time-reversal symmetry
  4. Defining data of abelian systems
  5. Time-reversal action
  6. Local Kramers degeneracy
  7. Definition of Local Kramers degeneracy :
  8. Symmetry fractionalization :
  9. Analysis of Crosscap
  10. Crosscap state
  11. $M_a$ :
  12. $\eta(a)$ :
  13. Constraints on $M_a$ and $\mathcal{M}$
  14. Gapped boundary
  15. Lagrangian subgroup
  16. ...and 21 more sections

Figures (3)

  • Figure 1: Symmetry fractionalization: each gray sheet represents a domain wall labeled by a group element, and an anyon is acted upon by that element when passing through the wall.
  • Figure 2: Local Kramers degeneracy: even when an anyon is invariant under time-reversal symmetry, a nontrivial phase may appear upon acting with $\mathsf{T}$ twice.
  • Figure 3: Trapped anyon: The phase associated with local Kramers degeneracy can be interpreted as the mutual braiding between an anyon and another anyon trapped at the intersection of two time-reversal domain walls.