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Reflection positivity and invertible topological phases

Daniel S. Freed, Michael J. Hopkins

TL;DR

The paper constructs a rigorous, homotopy-theoretic framework for classifying invertible, reflection-positive quantum field theories using Thom and Madsen-Tillmann spectra, with deformation classes identified as maps into the Anderson dual of a sphere. It provides a computable formula for the group of SPT phases in terms of Thom bordism spectra, depending only on spacetime dimension and symmetry, and demonstrates this with explicit fermionic cases in low dimensions. It develops an extensive apparatus of extended reflection structures, real/Hermitian positivity, and equivariant stable homotopy theory to handle time-reversal, spin, and Pin structures, culminating in a relativistic 10-fold way for fermions. The work connects lattice-model intuition with stable-homotopy classifications, offering a principled pathway to compute invertible phase groups and to understand low-energy effective theories via continuous invertible topological theories.

Abstract

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.

Reflection positivity and invertible topological phases

TL;DR

The paper constructs a rigorous, homotopy-theoretic framework for classifying invertible, reflection-positive quantum field theories using Thom and Madsen-Tillmann spectra, with deformation classes identified as maps into the Anderson dual of a sphere. It provides a computable formula for the group of SPT phases in terms of Thom bordism spectra, depending only on spacetime dimension and symmetry, and demonstrates this with explicit fermionic cases in low dimensions. It develops an extensive apparatus of extended reflection structures, real/Hermitian positivity, and equivariant stable homotopy theory to handle time-reversal, spin, and Pin structures, culminating in a relativistic 10-fold way for fermions. The work connects lattice-model intuition with stable-homotopy classifications, offering a principled pathway to compute invertible phase groups and to understand low-energy effective theories via continuous invertible topological theories.

Abstract

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.

Paper Structure

This paper contains 83 sections, 66 theorems, 467 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Symmetry groups in relativistic quantum field theory
  3. Stabilization of Wick-rotated symmetry groups
  4. Curved manifolds and bordism categories with $H_n$-structure
  5. Unitarity and Wick rotation
  6. Wick rotation in quantum mechanics
  7. Reflection positivity in Euclidean quantum field theory
  8. The extended symmetry group $\widehat{H}_n$
  9. Reflection symmetry on manifolds
  10. An involution on $H_n$-manifolds
  11. Duals and opposites
  12. Reflection structures and positivity
  13. Doubles
  14. Invertible topological field theories and stable homotopy theory
  15. Extended field theories
  16. ...and 68 more sections

Key Result

Theorem 1.1

There is a 1:1 correspondence

Figures (7)

  • Figure 1: Correlation functions
  • Figure 2: Reflection positivity in Euclidean space
  • Figure 3: Evaluation, coevaluation, and the gluing to the identity
  • Figure 4: The double of $X$
  • Figure 5: The Adams spectral sequences
  • ...and 2 more figures

Theorems & Definitions (221)

  • Theorem 1.1
  • Definition 2.4
  • Theorem 2.7
  • Corollary 2.12
  • Remark 2.13
  • proof : Proof of Theorem \ref{['thm:5']}
  • Lemma 2.14
  • proof
  • Remark 2.15
  • Proposition 2.16
  • ...and 211 more