6j Symbols for the Modular Double, Quantum Hyperbolic Geometry, and Supersymmetric Gauge Theories

Teschner, J.; Vartanov, G. S.

doi:10.1007/s11005-014-0684-3

Cited by 72 publications

(147 citation statements)

References 50 publications

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“…In section 5 we will compute R kskt by taking a large c limit of the CFT R-matrix that expresses the monodromy of 2D conformal blocks under analytic continuation over the lightcone. This 2D crossing kernel is explicitly known, thanks to the work of Ponsot and Teschner [33], see also [34,35]. As shown in [33], the 2D kernel can be expressed as a quantum 6j-symbol of the non-compact quantum group U q (sl(2, R)).…”

Section: Jhep08(2017)136mentioning

confidence: 99%

Solving the Schwarzian via the conformal bootstrap

Mertens

Turiaci

Verlinde

2017

J. High Energ. Phys.

317

632

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We obtain exact expressions for a general class of correlation functions in the 1D quantum mechanical model described by the Schwarzian action, that arises as the low energy limit of the SYK model. The answer takes the form of an integral of a momentum space amplitude obtained via a simple set of diagrammatic rules. The derivation relies on the precise equivalence between the 1D Schwarzian theory and a suitable large c limit of 2D Virasoro CFT. The mapping from the 1D to the 2D theory is similar to the construction of kinematic space. We also compute the out-of-time ordered four point function. The momentum space amplitude in this case contains an extra factor in the form of a crossing kernel, or R-matrix, given by a 6j-symbol of SU(1,1). We argue that the R-matrix describes the gravitational scattering amplitude near the horizon of an AdS 2 black hole. Finally, we discuss the generalization of some of our results to N = 1 and N = 2 supersymmetric Schwarzian QM.

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Section: Jhep08(2017)136mentioning

confidence: 99%

Solving the Schwarzian via the conformal bootstrap

Mertens

Turiaci

Verlinde

2017

J. High Energ. Phys.

317

632

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“…This is a manifestation of the celebrated but mysterious modular invariance in representation theory of quantum groups: certain aspects of the representation theory of U q g depend naturally not on q but only on the corresponding elliptic curve C/q Z . This modularity is expressed by Faddeev's modular double [44] and many subsequent works (see, for example, [50,93] and the references therein) and the Langlands duality for quantized cluster varieties of [46]. 1.4.5.…”

Section: Betti Geometricmentioning

confidence: 99%

Integrating quantum groups over surfaces

Ben‐Zvi

Brochier

Jordan

2018

Journal of Topology

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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“…More recently, Teschner and Vartanov found an interesting alternative expression for the Racah-Wigner coefficients [21]. We will discuss this representation along with its extension to the supersymmetric case in an accompanying paper.…”

Section: Jhep10(2014)091mentioning

confidence: 99%

Self-dual continuous series of representations for U q s l 2 $$ {\mathcal{U}}_q\left(sl(2)\right) $$ and U q o s p 1 | 2 $$ {\mathcal{U}}_q\left(osp\left(1\Big|2\right)\right) $$

Hadasz

Pawelkiewicz²,

Schomerus³

2014

J. High Energ. Phys.

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Abstract:We determine the Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of the quantum deformed algebras U q (sl(2)) and U q (osp(1|2)). While our results for the former algebra reproduce formulas by Ponsot and Teschner, the expressions for the orthosymplectic algebra are new. Up to some normalization factors, the associated Racah-Wigner coefficients are shown to agree with the fusing matrix in the Neveu-Schwarz sector of N = 1 supersymmetric Liouville field theory.

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6j Symbols for the Modular Double, Quantum Hyperbolic Geometry, and Supersymmetric Gauge Theories

Cited by 72 publications

References 50 publications

Solving the Schwarzian via the conformal bootstrap

Solving the Schwarzian via the conformal bootstrap

Integrating quantum groups over surfaces

Self-dual continuous series of representations for U q s l 2 $$ {\mathcal{U}}_q\left(sl(2)\right) $$ and U q o s p 1 | 2 $$ {\mathcal{U}}_q\left(osp\left(1\Big|2\right)\right) $$

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