Abelian current algebra and the Virasoro algebra on the lattice

Faddeev, L. D.; Volkov, А. Yu.

doi:10.1016/0370-2693(93)91618-w

Cited by 124 publications

(134 citation statements)

References 12 publications

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“…As such, this L-matrix is not viable for the Bethe Ansatz. However, let us comment that work with Lax operator (16) leads naturally to the lattice deformation of Virasoro algebra [4,9,10,11].…”

Section: Massless Limit Of the Sine-gordon L-matrixmentioning

confidence: 99%

“…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%

“…In the spirit of this method, Liouville theory was put on lattice in [4], but due to peculiarities of the integrable structure of Liouville theory, only partial quantum results were achieved. The concept of lattice Liouville theory, and the parallel concept of lattice Virasoro algebra were further developed in [9,10,11].…”

Section: Introductionmentioning

confidence: 99%

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Connections of the Liouville model and XXZ spin chain

Faddeev

Tirkkonen

1995

Nuclear Physics B

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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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Section: Massless Limit Of the Sine-gordon L-matrixmentioning

confidence: 99%

“…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%

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Connections of the Liouville model and XXZ spin chain

Faddeev

Tirkkonen

1995

Nuclear Physics B

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“…We can easily see that this formula realizes the abstract relation (16) for operators in 7-/@ 7-I | 7-/, if we put u = U1U2V1-1V2, v = U2U3~-1 89 (17) in (15).…”

Section: (Li)mentioning

confidence: 82%

“…In particular, it is noted that the relation (33) previously obtained in [16], is an analog of the famous functional equation for the dilogarithm. Taking into account that in terms of the Heisenberg variables t -luv = e i(p+q), (34) and using (31), we obtain the following relation as an analog of (33):…”

Section: N(~) = Exp K(tk --T-k) 'mentioning

confidence: 97%

Yang-baxterization of the quantum dilogarithm

Volkov¹,

Faddeev²

1998

J Math Sci

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In the present paper, we prove the Yang-Baxter relations for a specific R-matrix dependent on the spectral parameter. For the simpliest Lax operator connected with the trigonometric case of the main commutation relation, this R-matrix plays the role of the fundamental R-matrix in the sense of [1]. Now we describe this Lax operator.Let P and Q be the Heisenberg pair of self-adjoint operators with the commutation relationWe regard the real number 7 as a dimensionless coupling constant and put the Planck constant h equal to 1. We construct the Weyl pair of unitary operatorssatisfying the commutation relationThe quantum space 7/, where P and Q are represented irreducibly, can be realized as L2(IR) with r such that Or = ~r pc(x) _ ~ d i ~ r (4)(coordinate representation). In the sequel, a specific realization of P and Q is not significant. The Lax operator acts in the-tensor product of the quantum space 7/and the auxiliary space 12 = C 2 and can be given by a 2 • 2-matrix whose entries are operators on 7/, U xV)Here, z is a complex number called the spectral parameter. The indices f and a stand for the quantum and auxiliary spaces where Lf, a(x ) acts. In this notation, the main commutation relation (cf.[2]) can conveniently be written in the following form:which is regarded as a relation in 7"l @ 1)1 | ~22. Here, Ral,a2(x) is an operator in the tensor product 1) | 1) = C 4 which can be given by the 4

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Algebraic Lessons from the Theory of Quantum Integrable Models

Faddeev

Progress in Mathematics

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Abelian current algebra and the Virasoro algebra on the lattice

Cited by 124 publications

References 12 publications

Connections of the Liouville model and XXZ spin chain

Connections of the Liouville model and XXZ spin chain

Yang-baxterization of the quantum dilogarithm

Algebraic Lessons from the Theory of Quantum Integrable Models

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