Reflection positivity and invertible topological phases

Freed, Daniel S; Hopkins, Michael J

doi:10.2140/gt.2021.25.1165

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Reflection positivity and invertible topological phases

Daniel S Freed and Michael J Hopkins

Geometry & Topology 25 (2021) 1165–1330

DOI: 10.2140/gt.2021.25.1165

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 type. 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.

Keywords

topological phases, invertible field theory, reflection positivity, symmetry protected topological phases, free fermions, topological field theory

Mathematical Subject Classification 2010

Primary: 55N22, 57R90, 81T45, 81T50, 82B99

References

Publication

Received: 12 July 2016

Revised: 3 October 2019

Accepted: 29 October 2019

Published: 20 May 2021

Proposed: Peter Teichner

Seconded: Ralph Cohen, Ulrike Tillmann

Authors

Daniel S Freed
Department of Mathematics University of Texas Austin, TX United States

Michael J Hopkins
Department of Mathematics Harvard University Cambridge, MA United States

Volume 25, issue 3 (2021)