The Arf-Brown TQFT of pin⁻ surfaces

Abstract

The Arf-Brown invariant AB(Σ) is an 8th root of unity associated to a surface Σ equipped with a Pin − structure. In this note we investigate a certain fully extended, invertible, topological quantum field theory (TQFT) whose partition function is the Arf-Brown invariant. Our motivation comes from the recent work of Freed-Hopkins on the classification of topological phases, of which the Arf-Brown TQFT provides a nice example of the general theory; physically, it can be thought of as the low energy effective theory of the Majorana chain, or as the anomaly theory of a free fermion in 1 dimension.Date: March 30, 2018. Preliminaries2.1. Clifford algebras, pin groups, and pin structures. Pin structures are generalizations of spin structures to unoriented vector bundles and manifolds. In this section, we define the pin groups and state a few useful results about them. For proofs and a more detailed exposition, see [ABS].Definition 2.1. Let k be a field of characteristic not equal to 2, S be a finite set, and o : S → {±1} be a function. The Clifford algebra Cℓ(k, S, o) is defined to be the k-algebrawhere T (k[S]) denotes the tensor algebra of the space of functions S → k, and we identify s with the function equal to 1 at s and 0 elsewhere. For S := {1, . . . , m} ∪ {−1, . . . , −n} and o(x) := sign(x), we'll write Cℓ m,n (k) := Cℓ(k, S, o), as well as Cℓ n (k) := Cℓ n,0 (k) and Cℓ −n (k) := Cℓ 0,n (k). If k = C, we'll suppress C from the notation, e.g. writing Cℓ m,n , Cℓ n , and Cℓ −n .The ideal in the quotient in (2.2) contains only even-degree elements of the tensor algebra, so the Clifford algebras are Z/2-graded algebras, or superalgebras. If a is a homogeneous element in a Z/2-graded algebra or module, we will let |a| ∈ Z/2 denote its degree.Lemma 2.3 ([ABS, Proposition 1.6]). Let S 1 and S 2 be finite sets and o i :For this to be true, we must use the graded tensor product, whose multiplication contains a sign: if a, b, a ′ , b ′ are homogeneous elements, thenLet α ∈ End (Cℓ(k, S, o)) be the grading operator, whose action on a homogeneous element a is multiplication by (−1) |a| . Definition 2.5. The Clifford group is Γ(k, S, o) := {x ∈ Cℓ(k, S, o) × | α(x)yx −1 ∈ k[S] ⊂ Cℓ(k, S, o) for all y ∈ k[S]}.

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

“…That is, we want to find an enhancement of a given TQFT functor Bord Spin n (BG) → Vect Z 2 C to a functor Z : Bord Spin,∂ n (BG) → Vect Z 2 C , where Bord Spin,∂ n is the bordism category whose objects can have boundaries and morphisms can have corners. 32 Further, since the fSPT is trivial when its background is ignored (i.e. maps to BG are trivial), physically we expect that we can obtain the boundary TQFT Z ∂ : Bord Spin n−1 → Vect Z 2 C from the bulk-boundary TQFT Z.…”

“…where C[n] denotes a one-dimensional complex vector space with Z 2 grading n, and Z Arf is the nontrivial (fully extendable) invertible 2d spin-TQFT such that its value on a closed spin-surface Σ is Z Arf (Σ) = (−1) Arf(Σ) . Such TQFT was considered in detail in [32] (see also [33]). The value of SL 4 on morphisms (using the conventions in (5.23)) is Then the functor SL 4 categorifies link invariant SL 3 in the sense that…”

“…C is the functor representing the pin − fSPT defined by the ABK invariant. Such (fully extended) 2d TQFT was considered in detail in [32]. The functor Z evaluated on a morphism (M 3 , g) ∈ Hom Bord Spin,∂ 3 ((M 2 1 , g), (M 2 2 , g)) is similarly constructed as:…”

Pin TQFT and Grassmann integral

“…In this section, we construct the 2d pin − invertible TQFT [23] for the Arf-Brown-Kervaire (ABK) invariant via the Grassmann integral on lattice, whose state sum definition was initially given in [15]. In condensed matter literature, this invertible theory describes (1+1)d topological superconductors in class BDI [16].…”

“…We further show that pin ± Gu-Wen SPT phases always admit a gapped boundary, by explicitly constructing the Grassmann integral for the coupled bulk and boundary system on an unoriented manifold. In addition, we propose a lattice definition of 2d pin − TQFT whose partition function is the Arf-Brown-Kervaire (ABK) invariant [15,22,23], which generates the Z 8 classification of (1+1)d topological superconductors. Finally, we discuss a way to compute the Z 16 -valued (2+1)d pin + anomaly from the data of (2+1)d anomalous theory.…”