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On gapped boundaries for SPT phases beyond group cohomology

Ryohei Kobayashi, Kantaro Ohmori, Yuji Tachikawa

TL;DR

The paper develops a unified symmetry-extension framework to realize gapped boundaries for SPT phases beyond group cohomology, including time-reversal-invariant bosonic SPTs and fermionic Gu-Wen SPTs for arbitrary finite internal symmetry groups. It generalizes the extension construction to higher-form symmetries and leverages invertible $G$-equivariant TQFTs to trivialize bulk anomalies on the boundary. For unoriented bosonic SPTs, it uses Brown–Peterson results and recursive $\mathbb{Z}_2$ higher-form extensions to kill Stiefel–Whitney anomalies, enabling gapped boundaries in broad dimensional regimes. For Gu-Wen spin SPTs, it extends the Gu-Wen Grassmann integral to boundary/bulk couplings and provides explicit boundary actions that cancel the bulk anomaly, yielding gapped, symmetry-preserving boundaries for all finite $G$. Collectively, the work extends the landscape of SPTs admitting gapped boundaries beyond group cohomology and offers concrete constructions for practical implementation.

Abstract

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

On gapped boundaries for SPT phases beyond group cohomology

TL;DR

The paper develops a unified symmetry-extension framework to realize gapped boundaries for SPT phases beyond group cohomology, including time-reversal-invariant bosonic SPTs and fermionic Gu-Wen SPTs for arbitrary finite internal symmetry groups. It generalizes the extension construction to higher-form symmetries and leverages invertible -equivariant TQFTs to trivialize bulk anomalies on the boundary. For unoriented bosonic SPTs, it uses Brown–Peterson results and recursive higher-form extensions to kill Stiefel–Whitney anomalies, enabling gapped boundaries in broad dimensional regimes. For Gu-Wen spin SPTs, it extends the Gu-Wen Grassmann integral to boundary/bulk couplings and provides explicit boundary actions that cancel the bulk anomaly, yielding gapped, symmetry-preserving boundaries for all finite . Collectively, the work extends the landscape of SPTs admitting gapped boundaries beyond group cohomology and offers concrete constructions for practical implementation.

Abstract

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups.

Paper Structure

This paper contains 20 sections, 66 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. General construction
  3. The symmetry extension method
  4. Extension by higher-form symmetries
  5. Symmetry breaking as symmetry extension
  6. Cases beyond group cohomology
  7. Preliminaries:
  8. An abstract construction:
  9. A more concrete version:
  10. The case of cohomological SPT phases:
  11. The action of the $K$-gauge theory:
  12. Further generalizations:
  13. Summary:
  14. For time-reversal invariant bosonic SPT phases
  15. For Gu-Wen spin SPT phases
  16. ...and 5 more sections

Figures (4)

  • Figure 1: The geometric configuration defining $P(h)$ in \ref{['eq:defP']}.
  • Figure 2: The geometric configuration defining $\omega_K$ in \ref{['eq:defomegaK']}.
  • Figure 3: Assignment of Grassmann variables on 1-simplices in the case of $d=2$. $\theta$ (resp. $\overline{\theta}$) is represented as a black (resp. white) dot.
  • Figure 4: $(a)$: An example of $K$ such that $\partial K= (N\times\{0\})\sqcup (M\times[0,1])\sqcup (N\times\{1\})$. $(b)$: Triangulation of $M\times[0,1]$ and $N\times\{1\}$ near the attaching region $M$, in the case of $d=2$. Note that $\theta$ (red dot) and $\overline{\theta}$ (white dot) are flipped from the original assignment rule on $M$ in the side of $N\times\{1\}$, which makes $\overline{\sigma}(N\times\{1\})=1$ when $\alpha(e)$ is nonzero on a single $(d-1)$-simplex on $M$. In contrast, we have $\sigma(M\times[0,1])=-1$ in such a situation.