The Arf-Brown TQFT of Pin$^-$ Surfaces

TL;DR

This paper explores a topological quantum field theory (TQFT) based on the Arf-Brown invariant for Pin$^-$ surfaces, connecting it to topological phases, Majorana chains, and fermion anomalies.

Contribution

It introduces a fully extended, invertible TQFT with the Arf-Brown invariant as its partition function, linking mathematical invariants to physical topological phases.

Findings

Constructs a TQFT with the Arf-Brown invariant as partition function

Connects the TQFT to Majorana chains and fermion anomalies

Provides a mathematical framework for topological phases of matter

Abstract

The Arf-Brown invariant $\mathit{AB}(\Sigma)$ is an 8th root of unity associated to a surface $\Sigma$ equipped with a pin$^-$ structure. In this note we investigate a certain fully extended, invertible, topological quantum field theory (TQFT) whose partition function is the Arf-Brown invariant. Our motivation comes from the recent work of Freed-Hopkins on the classification of topological phases, of which the Arf-Brown TQFT provides a nice example of the general theory; physically, it can be thought of as the low energy effective theory of the Majorana chain, or as the anomaly theory of a free fermion in 1 dimension.