#### Volume 25, issue 3 (2021)

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Reflection positivity and invertible topological phases

### Daniel S Freed and Michael J Hopkins

Geometry & Topology 25 (2021) 1165–1330
##### 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.

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##### 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