Reflection positivity and invertible topological phases

TL;DR

This paper develops a framework combining reflection positivity and stable homotopy theory to classify invertible topological phases and SPT phases, providing explicit formulas and computations for fermionic systems.

Contribution

It introduces an extended reflection positivity approach for invertible topological quantum field theories and derives a general formula for SPT phases using Thom's bordism spectra.

Findings

Computed the abelian group of deformation classes using stable homotopy theory.

Provided explicit calculations for fermionic systems in relevant dimensions.

Established a topological spin-statistics theorem.

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.