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Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D

Meng Guo, Pavel Putrov, Juven Wang

TL;DR

The paper develops a cobordism-based, symmetry-centric framework to classify and realize interacting 3+1D SPTs for 10 Cartan symmetry classes and their SU(N) generalizations. It connects UV lattice onsite symmetries to IR field theory cobordism data, providing explicit SPT invariants and eta-invariants on various 4-manifolds. By gauging SU(N) subgroups, the authors derive a rich zoo of SU(N) Yang-Mills phases, including potential deconfined gapless TR-symmetric CFTs and symmetry-enriched TQFTs, with distinct behavior on orientable vs non-orientable spacetimes. The work also builds a web of symmetry reductions and embeddings showing how higher symmetry classes flow to lower ones and clarifies how bosonic and fermionic SPTs relate under GTot extensions. Overall, the framework offers a precise, topology-driven route to understand the nonperturbative phases of TR-invariant non-Abelian gauge theories and their boundary anomalies.

Abstract

We introduce a web of strongly correlated interacting 3+1D topological superconductors/insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their new SU(N) generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of field theories of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism calculations, matching their full classification. We also present explicit 4-manifolds that detect these SPTs. On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of SU(N) Yang-Mills (YM) gauge theories, analogous to quantum spin liquids with emergent gauge fields. We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy. For example, we point out a possibility of having two deconfined gapless time-reversal symmetric SU(2) YM theories at $θ=π$ as two distinct conformal field theories, which although are secretly indistinguishable by correlators of local operators on orientable spacetimes nor by gapped SPT states, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.

Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D

TL;DR

The paper develops a cobordism-based, symmetry-centric framework to classify and realize interacting 3+1D SPTs for 10 Cartan symmetry classes and their SU(N) generalizations. It connects UV lattice onsite symmetries to IR field theory cobordism data, providing explicit SPT invariants and eta-invariants on various 4-manifolds. By gauging SU(N) subgroups, the authors derive a rich zoo of SU(N) Yang-Mills phases, including potential deconfined gapless TR-symmetric CFTs and symmetry-enriched TQFTs, with distinct behavior on orientable vs non-orientable spacetimes. The work also builds a web of symmetry reductions and embeddings showing how higher symmetry classes flow to lower ones and clarifies how bosonic and fermionic SPTs relate under GTot extensions. Overall, the framework offers a precise, topology-driven route to understand the nonperturbative phases of TR-invariant non-Abelian gauge theories and their boundary anomalies.

Abstract

We introduce a web of strongly correlated interacting 3+1D topological superconductors/insulators of 10 particular global symmetry groups of Cartan classes, realizable in electronic condensed matter systems, and their new SU(N) generalizations. The symmetries include SU(N), SU(2), U(1), fermion parity, time reversal and relate to each other through symmetry embeddings. We overview the lattice Hamiltonian formalism. We complete the list of field theories of bulk symmetry-protected topological invariants (SPT invariants/partition functions that exhibit boundary 't Hooft anomalies) via cobordism calculations, matching their full classification. We also present explicit 4-manifolds that detect these SPTs. On the other hand, once we dynamically gauge part of their global symmetries, we arrive in various new phases of SU(N) Yang-Mills (YM) gauge theories, analogous to quantum spin liquids with emergent gauge fields. We discuss how coupling YM theories to time reversal-SPTs affects the strongly coupled theories at low energy. For example, we point out a possibility of having two deconfined gapless time-reversal symmetric SU(2) YM theories at as two distinct conformal field theories, which although are secretly indistinguishable by correlators of local operators on orientable spacetimes nor by gapped SPT states, can be distinguished on non-orientable spacetimes or potentially by correlators of extended operators.

Paper Structure

This paper contains 83 sections, 5 theorems, 132 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction and Summary
  2. Overview of Global Symmetries of the UV Lattice and IR Field Theories
  3. Global Symmetries of the UV Lattice, Hamiltonian and Hilbert space
  4. Symmetries of charge/spin, fermion-pairing, and spin-orbital coupling orders
  5. Global Symmetries of the IR Field Theories, Path Integral and Cobordisms
  6. $C, T$, and $CT$ for Dirac fermion coupled to $U(1)$ gauge field
  7. $C, T$, and $CT$ for Dirac fermion coupled to $SU(2)$ gauge field
  8. $SU(N)$ and more general groups
  9. Complete List of Symmetry-Protected Topological Invariants for 10 global symmetries of Cartan classes
  10. CI class: $\frac{SU(2) \times {\mathbb{Z}}_4^T}{{\mathbb{Z}}_2}$ and ${\mathrm{Pin}^{+}\times_{{\mathbb{Z}}_2^F} SU(2)}$
  11. SPT vacua and topological terms
  12. CII class: ${SU(2) \times {\mathbb{Z}}_2^{T}}$ and ${\mathrm{Pin}^{-}\times_{{\mathbb{Z}}_2^F} SU(2)}$
  13. SPT vacua and topological terms
  14. C class: $SU(2)$ and ${\mathrm{Spin} \times_{{\mathbb{Z}}_2^F} SU(2)}$
  15. AI class: $U(1) \rtimes {\mathbb{Z}}^{{T}}_{2}$ and ${\mathrm{Pin}^{-}\ltimes_{{\mathbb{Z}}_2^F} U(1)}$
  16. ...and 68 more sections

Key Result

Theorem 1

The bordism groups of $MTH$ are

Figures (5)

  • Figure 1: $H=\textrm{Pin}^+\times_{\{\pm1\}} SU(2)$
  • Figure 2: $H=\textrm{Pin}^-\times_{\{\pm1\}} SU(2)$
  • Figure 3: $H=\textrm{Pin}^+\times_{\{\pm1\}} SO(4)$
  • Figure 4: $H=\textrm{Pin}^+\times SU(3)$
  • Figure 5: $H=\textrm{Pin}^+\times_{\{\pm1\}} SU(4)$

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5