We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten.
Abstract. Spin Hurwitz numbers count ramified covers of a spin surface, weighted by the size of their automorphism group (like ordinary Hurwitz numbers), but signed ±1 according to the parity of the covering surface. These numbers were first defined by Eskin-Okounkov-Pandharipande in order to study the moduli of holomorphic differentials on a Riemann surface. They have also been related to Gromov-Witten invariants of of complex 2-folds by work of Lee-Parker and Maulik-Pandharipande. In this paper, we construct a (spin) TQFT which computes these numbers, and deduce a formula for any genus in terms of the combinatorics of the Sergeev algebra, generalizing the formula of Eskin-Okounkov-Pandharipande. During the construction, we describe a procedure for averaging any TQFT over finite covering spaces based on the finite path integrals of Freed-Hopkins-Lurie-Teleman.
Given a reductive group G, we give a description of the abelian category of Gequivariant D-modules on g " LiepGq, which specializes to Lusztig's generalized Springer correspondence upon restriction to the nilpotent cone. More precisely, the category has an orthogonal decomposition in to blocks indexed by cuspidal data pL, Eq, consisting of a Levi subgroup L, and a cuspidal local system E on a nilpotent L-orbit. Each block is equivalent to the category of D-modules on the center zplq of l which are equivariant for the action of the relative Weyl group N G pLq{L. The proof involves developing a theory of parabolic induction and restriction functors, and studying the corresponding monads acting on categories of cuspidal objects. It is hoped that the same techniques will be fruitful in understanding similar questions in the group, elliptic, mirabolic, quantum, and modular settings.Main results. In his seminal paper [Lus84], Lusztig proved the Generalized Springer Correspondence, which gives a description of the category of G-equivariant perverse sheaves on the nilpotent cone N G Ď g " LiepGq, for a reductive group G:The sum is indexed by cuspidal data: pairs pL, Eq of a Levi subgroup L of G and simple cuspidal local system on a nilpotent orbit for L, up to simultaneous conjugacy. For each such Levi L, W G,L " N G pLq{L denotes the corresponding relative Weyl group.The main result of this paper is that Lusztig's result extends to a description of the abelian category Mpgq G of all G-equivariant D-modules on g:Theorem A. There is an equivalence of abelian categories:where the sum is indexed by cuspidal data pL, Eq.Here zplq denotes the center of the Lie algebra l of a Levi subgroup L which carries an action of the finite group W G,L , 1 and Mpzplqq WG,L denotes the category of W G,L -equivariant D-modules on zplq, or equivalently, modules for the semidirect product D zplq¸WG,L . If we restrict to the subcategory of modules with support on the nilpotent cone (which can be identified with the category of equivariant 1 In fact, in the cases when L carries a cuspidal local system, W G,L is a Coxeter group and zplq its reflection representation.
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]}.
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