Quantum Dilogarithm

The quantum theory of the Liouville model with imaginary field is considered using the Quantum Inverse Scattering Method. An integrable structure with nontrivial spectral parameter dependence is developed for lattice Liouville theory by scaling the L-matrix of lattice sine-Gordon theory. This L-matrix yields Bethe Ansatz equations for Liouville theory, by the methods of the algebraic Bethe Ansatz. Using the string picture of exited Bethe states, the lattice Liouville Bethe equations are mapped to the corresponding spin 1 2 XXZ chain equations. The well developed theory of finite size corrections in spin chains is used to deduce the conformal properties of the lattice Liouville Bethe states. The unitary series of conformal field theories emerge for Liouville couplings of the form γ = π ν ν+1 , corresponding to root of unity XXZ anisotropies. The Bethe states give the full spectrum of the corresponding unitary conformal field theory, with the primary states in the Kač table parameterized by a string length K, and the remnant of the chain length mod (ν + 1). *

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“…For more discussion on solutions of (22) and other similar functional equations we refer to [9,10,31,32].…”

Section: Bethe Ansatz For Lattice Liouville Theorymentioning

confidence: 99%

Connections of the Liouville model and XXZ spin chain

Faddeev

Tirkkonen

1995

Nuclear Physics B

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“…The same data also appear in the context of SL(2) Chern-Simons theory or quantum Teichmüller theory [5][6][7][8][9]. The basic building block in this case is Faddeev's quantum dilogarithm [10] This special function plays an important role in mathematical physics. The quantum dilogarithm has many beautiful properties, in particular it satisfies a five term (pentagon) relation [10,11].…”

Section: Jhep10(2014)091mentioning

confidence: 77%

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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“…While the naive "quantization" of the dynamics over the times (2.2) leads (in absence of the "smooth variables" (2.18)) just to the string solution of the full hierarchy of the Toda or KP type with the tau-functions presented by the matrix elements in the two-dimensional theory of free fermions, the dispersive analogs of the full Krichever tau-function are matrix elements in non-trivial two-dimensional conformal theory, generally with extended symmetry. They also seem to be directly related with the quantization of Teichmüller spaces [40] (higher Teichmüller spaces for the case of W -gravity), Liouville theory [30,41], and quantum-mechanical integrable dynamics in the systems of Toda type (see e.g. [42,43,44]).…”

Section: Resultsmentioning

confidence: 99%

Gauge theories as matrix models

Marshakov

2011

Theor Math Phys

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The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. The matrix models of Eguchi-Yang type are derived quasiclassically as describing the instantonic contribution to the deformed partition functions of supersymmetric gauge theories. The constructed explicitly exact solution for the case of conformal four-dimensional theory is studied in detail, and some aspects of its relation with the recently proposed logarithmic beta-ensembles are considered. We discuss also the "quantization" of this picture in terms of twodimensional conformal theory with extended symmetry, and stress its difference from common picture of perturbative expansion a la matrix models. Instead, the representation for Nekrasov functions in terms of conformal blocks or Whittaker vector suggests some nontrivial relation with Teichmüller spaces and quantum integrable systems.

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Quantum Dilogarithm

Cited by 336 publications

References 5 publications

Connections of the Liouville model and XXZ spin chain

Connections of the Liouville model and XXZ spin chain

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 $$ {\mathcal{U}}_q\left(osp\left(1\Big|2\right)\right) $$

Gauge theories as matrix models

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