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

We formulate a family of spin Topological Quantum Field 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 torsion part of the classification is given by Pontryagin duals to spin-bordism groups of the classifying space BG . We also consider unoriented analogues, that is G -equivariant invertible pin ± -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 3, 4 and other dimensions. 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 spin-TQFTs (surface fermionic topological orders). We explore SPT and symmetry enriched topologically (SET) ordered states, and crystalline SPTs protected by space-group (e.g. translation Z ) or point-group (e.g. reflection, inversion or rotation C m ) symmetries, via the layer-stacking construction.