Connections of the Liouville model and XXZ spin chain

Abstract

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). *

Introduction to Quantum Integrability

“…Sine-Gordon and Liouville models: in what follows we shall briefly review how the lattice sine-Gordon [34] and Liouville models [35] are obtained in a natural way from the XXZ L matrix. Also the q harmonic oscillator realization will be obtained from the generalized XXZ form.…”

“…The interesting observation is that the entailed L operator (7.13) has a non trivial spectral (λ) dependence a fact that allows the application of Bethe ansatz techniques for the derivation of the spectrum (see also [35]). …”

Алгебра Янга - Бакстера И Генерация Квантовых Интегрируемых Моделей

“…В этом состоит действительно захватывающая возможность систематического построения различ-ных полезных операторов Лакса для одной и той же модели. Например, второй лиувиллевский оператор Лакса, который нам так легко построить, воспроизводит соответствующий оператор, полученный Фаддеевым с использованием ряда нова-торских приемов [10].…”

Boundary actions in Ponzano–Regge discretization, quantum groups and $AdS_3$

“…The XXZ spin chain is a possible starting point for such a calculation. This model is a chain of spins the solution to which involves the quantum group U q (sl (2)) and which is related to Liouville theory for c = 1 and c > 25 and also possibly for all c > 1 [38]. Within this framework we should be able to formulate the explicit calculation that is necessary to Figure 6: Two time slices of the boundary gas calculate the entropy of the boundary theory and thus black hole entropy.…”