We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane-Yetter-Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld-Jimbo quantum group Uq(g) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang-Mills theory, the setting introduced by Kapustin-Witten for the geometric Langlands program.For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to Uq(g), and from the punctured torus we recover the algebra of quantum differential operators associated to Uq(g). From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules. Contents INTEGRATING QUANTUM GROUPS OVER SURFACES 875[21]. These connections (which are discussed further in Sections 1.4.2 and 1.4.4) suggest many rich structures for quantum character varieties, some of which we discuss in this paper, and many which we plan to explore in future papers. Factorization homology of surfacesFactorization homology was originally introduced by Beilinson and Drinfeld [17] in the setting of chiral conformal field theory, as an abstraction (and geometric interpretation) of the functor of conformal blocks of a vertex algebra. Factorization homology in the topological, rather than conformal, setting is developed in [71] and further in [11,12]. In this paper we use the terminology and formalism of [11,12] (see [54] for a survey and [29] for more general applications to quantum field theory).The algebraic input to factorization homology of surfaces (in the terminology of [11] and subsequent papers) is a '2-disk algebra' in an appropriate symmetric monoidal higher category C . Informally speaking, a 2-disk algebra is an object A ∈ C equipped with operations A k → A parametrized in a locally constant fashion by embeddings of disjoint unions of k disks into a large disk, and satisfying a composition law governed by composition of disk embeddings. There are several variants of 2-disk algebras, named according to the kind of tangential structure carried by the disks and embeddings: framed 2-disk algebras are better known as E 2 algebras (algebras over the little 2-disk operad), while oriented 2-disk algebras are (confusingly) known as framed E 2 -algebras (algebras over the framed little 2-disk operad). We adopt the terminology of [11] as it reflects the type of surfaces over which the corresponding algebras may be int...

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants S A of a surface S, determined by the choice of a braided tensor category A, and computed via factorization homology.We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for A, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction.Characters of braided A-modules are objects of the torus category T 2 A. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of A = Rep q G with the category D q (G/G) -mod of equivariant quantum D-modules. When G = GL n , we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) SH q,t .

Abstract. The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface M × I but the corresponding category of diagrams lacks some finiteness properties which are essential to the above construction. We suggest to overcome this obstruction by studying families of Vassiliev invariants which, roughly, are associated to finite coverings of M . In the case M = C * , it leads for each positive integer N to a filtration on the space of tangles in C * × I (or "B-tangles"). We first prove an extension of the Shum-Reshetikhin-Turaev theorem in the framework of braided module category leading to B-tangles invariants. We introduce a category of "N -chord diagrams", and use a cyclotomic generalization of Drinfeld associators, introduced by Enriquez, to put a braided module category structure on it. We show that the corresponding functor from the category of B-tangles is a universal invariant with respect to the N filtration. We show that Vassiliev invariants in the usual sense are well approximated by N finite type invariants. We show that specializations of the universal invariant can be constructed from modules over a metrizable Lie algebra equipped with a finite order automorphism preserving the metric. In the case the latter is a "Cartan" automorphism, we use a previous work of the author to compute these invariants explicitly using quantum groups. Restricted to links, this construction provides polynomial invariants. IntroductionFinite type invariants are those numerical knot invariants whose "(n + 1)th derivative", in some specific sense, vanishes for some n. Since their discovery around 1989 by Vassiliev, it became clear that there is a deep relation between finite type invariants, Lie theory and quantum topology. Indeed, the collection of all finite type invariants dominates all quantum invariants, including the various knot polynomials. Kontsevich [Kon] then gave an essentially complete description of the space of finite type invariants, by constructing a "universal" invariant taking its values in some space of Feynman diagrams. The Kontsevich integral not only provide a powerful knot invariant, but shed some light on the topological background of deformation-quantization theory.Let K be a field of characteristic 0 and V be the K-linear span of isotopy classes of knots in R 3 , then K-valued knots invariants may be identified with linear map V → K. A singular knot can be identified with a formal linear combination of non-singular knots by repetitive use of the Vassiliev skein relation = − 1 Here as usual we draw only the part of the knots which is involved in the relation, and assume that they are identical outside the picture. This define a filtration on V by letting I n to be the linear span of knots having a least n singularities. A finite type invariant of type (or de...

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Abstract. We construct a certain cross product of two copies of the braided dualH of a quasitriangular Hopf algebra H, which we call the elliptic double E H , and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to H. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in [Jor09], and hence construct a homomorphism from E H to the Heisenberg double D H , which is an isomorphism if H is factorizable.The universal property of E H endows it with an action by algebra automorphisms of the mapping class group SL 2 (Z) of the punctured torus. One such automorphism we call the quantum Fourier transform; we show that when H = Uq(g), the quantum Fourier transform degenerates to the classical Fourier transform on D(g) as q → 1.