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

TL;DR

The authors develop a fermionic generalization of Dijkgraaf-Witten theories by gauging G-equivariant invertible spin-TQFTs, showing that fermionic SPTs with finite symmetry G are classified by the torsion part of spin bordism groups $\\Omega^{Spin}_n(BG)$ and can be computed explicitly for several abelian groups via Adams spectral sequence techniques. They derive explicit partition functions on closed manifolds, construct symmetry-preserving boundary TQFTs, and extend the framework to time-reversal and non-orientable settings using pin$^\pm$ bordism. The work further connects fSPTs to crystalline SPTs through dimensional extension and layer-stacking, providing detailed cobordism classifications and invariants for 3D and 4D systems, including ABK, Arf, and Arf–Brown–Kervaire refinements. Overall, the results illuminate how interacting fermionic orders with finite and crystalline symmetries arise, classify, and bound potential boundary phenomena and SETs in higher dimensions, with clear implications for quantum matter and topological phases.

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.

Figures (28)

• Figure 1: The $\mathcal{A}(1)$-module structure of $\operatorname{H}^*(B(\mathbb{Z}_2\times\mathbb{Z}_4))$
• Figure 2: The $E_2$ page of the Adams spectral sequence for $\mathbb{Z}_2\times\mathbb{Z}_4$
• Figure 3: The $\mathcal{A}(1)$-module structure of $\operatorname{H}^*(B\mathbb{Z}_4^2)$
• Figure 4: The $E_2$ page of the Adams spectral sequence for $\mathbb{Z}_4^2$
• Figure 5: The $\mathcal{A}(1)$-module structure of $\operatorname{H}^*(B(\mathbb{Z}_2^2\times\mathbb{Z}_4))$

Theorems & Definitions (22)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10