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Homotopy Theoretic Classification of Symmetry Protected Phases

Jonathan A. Campbell

TL;DR

This work applies Freed–Hopkins’ homotopy-theoretic framework to classify symmetry-protected topological phases by translating invertible TQFT data into cobordism-theoretic problems handled with stable homotopy tools. The authors develop and leverage A(1)-level Adams spectral sequence computations, explicitly resolving Ext groups and tracing the resulting low-dimensional homotopy groups of cobordism spectra MTH(d) that classify bosonic and fermionic SPTs with various symmetry groups. They translate these calculations into concrete classifications for a broad set of cases, including bosonic and fermionic systems with time-reversal, U(1), and finite group symmetries, and provide several new results (e.g., for (C₂)^{×k} in (2+1)D and for Z/4_f and Z/2^n cases) that align with and extend prior physics literature. The approach demonstrates how cobordism data captured by spectra maps to physical SPT invariants, offering a robust mathematical backbone for the classification of topological phases and their lattice-model realizations. The paper also emphasizes the role of the Anderson dual Iℤ and the KO/ko framework as practical computational tools in low dimensions.

Abstract

We classify a number of symmetry protected phases using Freed-Hopkins' homotopy theoretic classification. Along the way we compute the low-dimensional homotopy groups of a number of novel cobordism spectra.

Homotopy Theoretic Classification of Symmetry Protected Phases

TL;DR

This work applies Freed–Hopkins’ homotopy-theoretic framework to classify symmetry-protected topological phases by translating invertible TQFT data into cobordism-theoretic problems handled with stable homotopy tools. The authors develop and leverage A(1)-level Adams spectral sequence computations, explicitly resolving Ext groups and tracing the resulting low-dimensional homotopy groups of cobordism spectra MTH(d) that classify bosonic and fermionic SPTs with various symmetry groups. They translate these calculations into concrete classifications for a broad set of cases, including bosonic and fermionic systems with time-reversal, U(1), and finite group symmetries, and provide several new results (e.g., for (C₂)^{×k} in (2+1)D and for Z/4_f and Z/2^n cases) that align with and extend prior physics literature. The approach demonstrates how cobordism data captured by spectra maps to physical SPT invariants, offering a robust mathematical backbone for the classification of topological phases and their lattice-model realizations. The paper also emphasizes the role of the Anderson dual Iℤ and the KO/ko framework as practical computational tools in low dimensions.

Abstract

We classify a number of symmetry protected phases using Freed-Hopkins' homotopy theoretic classification. Along the way we compute the low-dimensional homotopy groups of a number of novel cobordism spectra.

Paper Structure

This paper contains 23 sections, 25 theorems, 124 equations, 34 figures.

Table of Contents

  1. Introduction and Outline
  2. Introduction
  3. Outline
  4. Prerequisites, Conventions, Notation
  5. Acknowledgements
  6. The Steenrod Algebra
  7. $ko$ and $\mathcal{A}(1)$
  8. Some Graded Algebra
  9. The Adams Spectral Sequence
  10. $\mathcal{A}(1)$-resolutions
  11. Computations from Freed-Hopkins
  12. Action of Steenrod Algebra
  13. Computations
  14. New Computations
  15. Bosonic, with Time Reversal
  16. ...and 8 more sections

Key Result

Theorem 1.1

Deformation classes of invertible, reflection positive, topological quantum field theories of space-time dimension $n$ with structure group $G$ are in bijection with homotopy classes of maps

Figures (34)

  • Figure 3.1: A diagram of $\mathcal{A}(1)$
  • Figure 3.2: The $\mathcal{A}(1)$-module structure of $H^\ast (\mathbf{R} P^\infty)$
  • Figure 3.3: The $\mathcal{A}(1)$-module $Q$
  • Figure 3.4: The Joker
  • Figure 3.5: And $\mathcal{A}(1)$-module with a particularly nice resolution
  • ...and 29 more figures

Theorems & Definitions (83)

  • Theorem 1.1: Hopkins-Freed
  • Remark 1.2
  • Theorem 1.3: Hopkins-Freed
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 2.1
  • ...and 73 more