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

Brochier, Adrien; Jordan, David A.

doi:10.4171/qt/92

Cited by 9 publications

(19 citation statements)

References 16 publications

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“…Using this universal property, it is proved in [24] that there is an action by algebra automorphisms of the mapping class group Γ 1 0,1 , which is the universal central extension SL 2 (Z) of the modular group, on D q (G). Recall that SL 2 (Z) is generated by X, Y with relations X 4 = (XY ) 3 (X 2 , Y ) = 1.…”

Section: Factorization Homology Of the Punctured Torusmentioning

confidence: 99%

“…Definition 6.12 [24]. For an integer k, the k-twisted braided dual Hopf algebra H k is H • as a vector space, with multiplication given by…”

Section: Factorization Homology Of Non-blackboard-framed Annulimentioning

confidence: 99%

“… (1)For the disk, we have

a_{D^{2}} ≅ {bold1}_{scriptA}

, which is just the tautological equivalence

A ≃ {bold1}_{scriptA} {prefix-mod}_{scriptA}

. (2)For the annulus or cylinder, we have

a_{A n n} = {scriptO}_{q} false(G false)

, the reflection equation algebra . (3)For the punctured torus, we have

a_{T^{2} ∖ D^{2}} ≅ {scriptD}_{q} false(G false)

, the algebra of quantum differential operators on

G

. This algebra has received a lot of attention in recent years . (4)More generally, the moduli algebras of Alekseev can be recovered as algebras

a_{P}

associated to certain gluing patterns for higher genus surfaces. …”

Section: Introductionmentioning

confidence: 99%

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Integrating quantum groups over surfaces

Ben‐Zvi

Brochier

Jordan

2018

Journal of Topology

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161

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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...

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Section: Factorization Homology Of the Punctured Torusmentioning

confidence: 99%

“…Definition 6.12 [24]. For an integer k, the k-twisted braided dual Hopf algebra H k is H • as a vector space, with multiplication given by…”

Section: Factorization Homology Of Non-blackboard-framed Annulimentioning

confidence: 99%

“… (1)For the disk, we have

a_{D^{2}} ≅ {bold1}_{scriptA}

, which is just the tautological equivalence

A ≃ {bold1}_{scriptA} {prefix-mod}_{scriptA}

. (2)For the annulus or cylinder, we have

a_{A n n} = {scriptO}_{q} false(G false)

, the reflection equation algebra . (3)For the punctured torus, we have

a_{T^{2} ∖ D^{2}} ≅ {scriptD}_{q} false(G false)

, the algebra of quantum differential operators on

G

. This algebra has received a lot of attention in recent years . (4)More generally, the moduli algebras of Alekseev can be recovered as algebras

a_{P}

associated to certain gluing patterns for higher genus surfaces. …”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Integrating quantum groups over surfaces

Ben‐Zvi

Brochier

Jordan

2018

Journal of Topology

Self Cite

161

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“…Relation to factorization homology. Let us briefly explain the connection of the present work to [BJ,BZBJ1] and to [BZBJ2], where Theorem 4.1 has already been announced. Since this relation is only motivational in the present paper, we will be informal, refering to [BZBJ1], [BZBJ2] for complete details.…”

Section: Introductionmentioning

confidence: 94%

The Harish-Chandra isomorphism for quantum $GL_2$

Balagović

Jordan

2018

J. Noncommut. Geom.

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“…One can connect these theorems more directly, following [13,Section 6] and [5,Section 6]. One can define a suitable degeneration D q (SL N ) ❀ D(sl N ), and a degeneration H q,t ❀ RCA n (c), compatible in such a way that we obtain a degeneration of the functors F SL n ❀ F n .…”

Section: Introductionmentioning

confidence: 99%