Integrating quantum groups over surfaces

Abstract

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...

Fourier transform for quantum $D$-modules via the punctured torus mapping class group

“…One such automorphism of D q (G) we call the quantum Fourier transform; its classical limit upon an appropriate degeneration is the classical Fourier transform on the Weyl algebra D(g). We expect that our quantum Fourier transform for D q (G) will be compatible with that on the braided dual of U q (g) defined in [LM94] This paper is a companion to [BZBJ15], in which we compute the value of a certain category valued 2-dimensional topological field theory attached to H -mod, and show that its value on a punctured torus is the category of H-equivariant E H -modules.…”

Equivariant higher Hochschild homology and topological field theories

“…This article brings together equivariant topological field theories and the powerful homotopy-theoretic techniques that have by now become standard in the study of non-equivariant topological field theory such as complete Segal spaces, factorization homology and factorization algebras, see [L-HA, AF15, Gi15,CG16] for the general background and [Lur09,Sc14,BZBJ15] for the relation to field theories. On the one hand, our motivation is the study of higher analogues of the algebraic structures produced by nonhomotopical equivariant field theory; on the other hand we also have a concrete class of examples in mind which actually requires a treatment within the framework of homotopy theory, see below.…”

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

“…These quantized algebras of functions L g,n and their generalizations appear in various works of mathematics and mathematical physics, see e.g. [BFKB98], [BNR02], [MW15], [BZBJ15], [BJ17], [CMR17].…”