Pin TQFT and Grassmann integral

Abstract

We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin_ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the ℤ8 classification of (1+1)d topological superconductors. We also compute the indicator formula of ℤ16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.