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Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases

Anton Kapustin, Alex Turzillo

TL;DR

The paper develops an algebraic framework to classify bosonic SRE/SPT phases with symmetry G by mapping them to invertible G‑equivariant TQFTs, including time-reversal via unoriented theories. It shows that oriented low‑dimensional invertible TQFTs are classified by ordinary group cohomology $H^{D+1}(BG,U(1))$, while unoriented cases require ρ‑twisted cohomology $H^{D+1}(BG,U(1)_{\rho})$ with $\rho:G\to\mathbb{Z}_2$. Concrete results are established for $D=0$ and $D=1$ in both oriented and unoriented settings, with $D=2$ discussed as a natural extension through $G$‑modular categories. A key achievement is the explicit correspondence in $D=1$ between invertible unoriented TQFTs and $\rho$‑twisted 2‑cocycles, thereby aligning SRE/SPT classifications with twisted group cohomology and suggesting a path toward higher‑dimensional, categorified formulations. Overall, the work provides a coherent algebraic route to understand how symmetry and orientation influence the topological classification of gapped, short‑range entangled phases.

Abstract

Short-range entangled topological phases of matter are closely connected to Topological Quantum Field Theory. We use this connection to classify bosonic Symmetry Protected Topological Phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev's description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or less spatial dimensions are classified by twisted group cohomology, in agreement with the group cohomology proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or less are classified by ordinary group cohomology.

Equivariant Topological Quantum Field Theory and Symmetry Protected Topological Phases

TL;DR

The paper develops an algebraic framework to classify bosonic SRE/SPT phases with symmetry G by mapping them to invertible G‑equivariant TQFTs, including time-reversal via unoriented theories. It shows that oriented low‑dimensional invertible TQFTs are classified by ordinary group cohomology , while unoriented cases require ρ‑twisted cohomology with . Concrete results are established for and in both oriented and unoriented settings, with discussed as a natural extension through ‑modular categories. A key achievement is the explicit correspondence in between invertible unoriented TQFTs and ‑twisted 2‑cocycles, thereby aligning SRE/SPT classifications with twisted group cohomology and suggesting a path toward higher‑dimensional, categorified formulations. Overall, the work provides a coherent algebraic route to understand how symmetry and orientation influence the topological classification of gapped, short‑range entangled phases.

Abstract

Short-range entangled topological phases of matter are closely connected to Topological Quantum Field Theory. We use this connection to classify bosonic Symmetry Protected Topological Phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev's description of equivariant TQFT to the unoriented case. We show that invertible unoriented equivariant TQFTs in one or less spatial dimensions are classified by twisted group cohomology, in agreement with the group cohomology proposal of Chen, Gu, Liu and Wen. We also show that invertible oriented equivariant TQFTs in spatial dimension two or less are classified by ordinary group cohomology.

Paper Structure

This paper contains 11 sections, 2 theorems, 50 equations, 3 figures.

Table of Contents

  1. Introduction and overview
  2. Oriented equivariant TQFT and SRE phases with an internal symmetry
  3. $D=0$
  4. $D=1$
  5. $D=2$
  6. On the definition of SRE phases with symmetry $G$
  7. Unoriented equivariant $D=0$ TQFT
  8. Unoriented equivariant $D=1$ TQFT
  9. Definition of unoriented equivariant TQFT
  10. Algebraic description for $D=1$
  11. Invertible unoriented equivariant $D=1$ TQFT

Key Result

Proposition 1

Unoriented equivariant $D=1$ TQFTs with symmetry $(G,\rho: G\rightarrow{\mathbb Z}_2)$ are in bijective correspondence with algebraic TQFT data $({\mathcal{A}},\eta,\alpha,\theta_g, g\notin G_0)$.

Figures (3)

  • Figure 1: Axiom \ref{['eqax5']}. To obtain the right figure from the left, the puncture with holonomy $k$ is pulled through the crosscap along the path with holonomy $g$.
  • Figure 2: Axiom \ref{['eqax6']}. Two projective planes are punctured and sewed along their boundaries, the diagonal lines, to obtain their connected sum, the Klein bottle.
  • Figure :

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2