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
–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
