Reflection positivity and invertible topological phases

Abstract

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