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Reflection positivity and invertible topological phases

Daniel S Freed and Michael J Hopkins

Geometry & Topology 25 (2021) 1165–1330

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.

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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
Received: 12 July 2016
Revised: 3 October 2019
Accepted: 29 October 2019
Published: 20 May 2021
Proposed: Peter Teichner
Seconded: Ralph Cohen, Ulrike Tillmann
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