# Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms

## TL;DR

This paper develops a framework for classifying and computing fermionic topological quantum field theories (TQFTs) with finite group symmetries using cobordism theory, providing explicit formulas and new insights into anomalies and crystalline SPT phases.

## Contribution

It introduces a fermionic generalization of Dijkgraaf-Witten TQFTs via cobordism classification, computes relevant groups, and constructs explicit formulas for partition functions and anomalies.

## Key findings

- Classified fermionic SPTs using spin and pin cobordism groups.
- Derived explicit formulas for TQFT partition functions on closed manifolds.
- Constructed new anomalous boundary spin-TQFTs and explored crystalline SPTs.

## Abstract

We formulate a family of spin Topological Quantum Filed Theories (spin-TQFTs) as fermionic generalization of bosonic Dijkgraaf-Witten TQFTs. They are obtained by gauging $G$-equivariant invertible spin-TQFTs, or, in physics language, gauging the interacting fermionic Symmetry Protected Topological states (SPTs) with a finite group $G$ symmetry. We use the fact that the latter are classified by Pontryagin duals to spin-bordism groups of the classifying space $BG$. We also consider unoriented analogues, that is $G$-equivariant invertible pin$^\pm$-TQFTs (fermionic time-reversal-SPTs) and their gauging. We compute these groups for various examples of abelian $G$ using Adams spectral sequence and describe all corresponding TQFTs via certain bordism invariants in dimensions 3, 4, and other. This gives explicit formulas for the partition functions of spin-TQFTs on closed manifolds with possible extended operators inserted. The results also provide explicit classification of 't Hooft anomalies of fermionic QFTs with finite abelian group symmetries in one dimension lower. We construct new anomalous boundary deconfined spin-TQFTs (surface fermionic topological orders). We explore SPT and SET (symmetry enriched topologically ordered) states, and crystalline SPTs protected by space-group (e.g. translation $\mathbb{Z}$) or point-group (e.g. reflection, inversion or rotation $C_m$) symmetries, via the layer-stacking construction.