Classification of (2+1)D invertible fermionic topological phases with symmetry

Abstract

We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G f and general values of the chiral central charge c−. Here G f is a central extension of a bosonic symmetry group G b by fermion parity, (−1) F , specified by a second cohomology class [ω2] ∈ H 2 (G b , Z2). Our approach proceeds by gauging fermion parity and classifying the resulting G b symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using G-crossed braided tensor categories and Spin(2c−)1 Chern-Simons theory coupled to a background G gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge c− = 0. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of c− and G f . Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with G f symmetry.

Computing the group of minimal non-degenerate extensions of a super-Tannakian category

“…Hence, Z sym (C) = Vect, i.e., C is non-degenerate. It is a minimal extension since FPdim(C) = FPdim(E) 2 .…”

“…For example, a symmetric fusion subcategory E ⊂ C satisfies E ⊆ E ′ and so can be thought of as a categorical analog of a coisitropic subspace. When C is non-degenerate, one has E = E ′ , i.e., E is a Lagrangian subcategory, if and only if FPdim(E) 2 = FPdim(C), where FPdim denotes the Frobenius-Perron dimension. An embedding E ֒→ C with this property will be called a minimal non-degenerate extension (or simply a minimal extension) of E. Lan, Kong, and Wen observed in [26] that there is a natural product of minimal extensions of E, so that the set of their equivalence classes is an abelian group Mext(E) (in fact, minimal extensions of E form a symmetric 2-categorical group Mext(E)).…”

“…Groups of minimal extensions were studied by many authors, including [1,2,3,11,18,21,26,27,30]. From the physics point of view, minimal extensions appear in the description of 2+1D topological orders and symmetry-protected trivial (SPT) orders [26], and symmetric invertible fermionic phases [1,2].…”

“…Groups of minimal extensions were studied by many authors, including [1,2,3,11,18,21,26,27,30]. From the physics point of view, minimal extensions appear in the description of 2+1D topological orders and symmetry-protected trivial (SPT) orders [26], and symmetric invertible fermionic phases [1,2]. It is known that for a Tannakian category E = Rep(G), where G is a finite group, the group Mext(E) is isomorphic to H 3 (G, × ), the third cohomology group of G. For E = sVect, the category of super-vector spaces, it is isomorphic to Z/16Z (this statement is known in physics as Kitaev's 16-fold way [25]).…”

“…In Section 4 we analyze the structure of the group Mext(E) for a pointed symmetric category E. We consider a filtration (2) Mext triv (E)…”