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

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 ± -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 Z) or point-group (e.g. reflection, inversion or rotation C m ) symmetries, via the layer-stacking construction. *

Anomaly of the Electromagnetic Duality of Maxwell Theory

“…Let us now consider the infinitely massive fermions encoding the anomaly of a (3+1)-dimensional Weyl fermion of unit charge under Z 2k , which was studied in [47][48][49]. The corresponding eta invariants on S 5 /Z k are also tabulated in Table I.…”

Pin TQFT and Grassmann integral

“…The notion of fermionic topological phase of matter has attracted great interest, since fermionic systems admit novel phases that have no counterpart in bosonic systems [1][2][3][4][5][6][7][8].…”

On gapped boundaries for SPT phases beyond group cohomology

“…We can call them as unoriented bosonic invertible phases, and they are described by Hom(Ω unoriented d+1 (X), U (1)) where X = BG. Luckily, an explicit and complete description of this group was already given in the algebraic topology literature in the 1960s [32] 11 This allows us to construct gapped boundaries for all of them.…”

“…[35] or [36]. 11 The 2nd edition of the textbook [33] contains a very readable account in its Chapter I.18.…”

“…in [2][3][4][5]. On the other hand, there is also a large class of nontrivial SPT phases which are known to admit gapped boundaries, some of which are described in [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”