# Pin TQFT and Grassmann integral

## TL;DR

This paper develops a lattice construction for fermionic topological phases on unoriented manifolds using Grassmann integrals, extending spin TQFT methods to pin structures, and applies it to classify and analyze topological superconductors and TQFTs.

## Contribution

It extends Grassmann integral-based spin TQFT constructions to unoriented pin manifolds, enabling new classifications and boundary constructions for fermionic topological phases.

## Key findings

- Constructed gapped boundaries for time-reversal-invariant fermionic SPT phases.
- Provided lattice definition of 1+1d pin_- invertible theory with Arf-Brown-Kervaire invariant.
- Computed time-reversal anomaly indicator formula for 2+1d pin_+ TQFT.

## 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$_\pm$ 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 $\mathbb{Z}_8$ classification of (1+1)d topological superconductors. We also compute the indicator formula of $\mathbb{Z}_{16}$ valued time-reversal anomaly for (2+1)d pin$_+$ TQFT based on our construction.