The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to $\infty$ in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an old version of uuencode that produces blank characters some times suppressed by the mailer. Same content

Solving the Strongly Coupled 2D Gravity: 2. Fractional-Spin Operators, and Topological Three-Point Functions

Progress along the line of a previous article are reported. One main point is to include chiral operators with fractional quantum group spins (fourth or sixth of integers) which are needed to achieve modular invariance. We extend the study of the chiral bootstrap (recently completed by E. Cremmer, and the present authors) to the case of semi-infinite quantum-group representations which correspond to positive integral screening numbers. In particular, we prove the Bidenharn-Elliot and Racah identities for q-deformed 6-j symbols generalized to continuous spins. The decoupling of the family of physical chiral operators (with real conformal weights) at the special values C_{Liouville}= =7, 13, and 19, is shown to provide a full solution of Moore and Seiberg's equations, only involving operators with real conformal weights. Moreover, our study confirms the existence of the strongly coupled topological models. The three-point functions are shown to be given by a product of leg factors similar to the ones of the weakly coupled models. However, contrary to this latter case, the equality between the quantum group spins of the holomorphic and antiholomorphic components is not preserved by the local vertex operator. Thus the ``c=1'' barrier appears as connected with a deconfinement of chirality.Comment: 45 pages Latex file, 14 figures (uuencoded

The Braiding of Chiral Vertex Operators with Continuous Spins in 2D Gravity

Chiral vertex-operators are defined for continuous quantum-group spins $J$ from free-field realizations of the Coulomb-gas type. It is shown that these generalized chiral vertex operators satisfy closed braiding relations on the unit circle, which are given by an extension in terms of orthogonal polynomials of the braiding matrix recently derived by Cremmer, Gervais and Roussel. This leads to a natural extension of the Liouville exponentials to continuous powers that remain local.Comment: (14 pages, Latex file) preprint LPTENS-93/1

Operator Coproduct-Realization of Quantum Group Transformations in Two Dimensional Gravity, I.

A simple connection between the universal $R$ matrix of $U_q(sl(2))$ (for spins \demi and $J$) and the required form of the co-product action of the Hilbert space generators of the quantum group symmetry is put forward. This gives an explicit operator realization of the co-product action on the covariant operators. It allows us to derive the quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of $U_q(sl(2))$ realized by (what we call) fixed point commutation relations. This is explained by showing that the link between the algebra of field transformations and that of the co-product generators is much weaker than previously thought. The central charges of our extended $U_q(sl(2))$ algebra, which includes the Liouville zero-mode momentum in a nontrivial way are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry U_q(sl(2))\odot U_{\qhat}(sl(2)) related to the presence of both of the screening charges of 2D gravity.Comment: 33 pages, latex, no figure

Systematic classical continuum limits of integrable spin chains and emerging novel dualities

We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gl_n magnets. Certain classical and quantum integrable models emerging from special "dualities" of quantum spin chains, parametrized by c-number matrices, are also presented.Comment: 29 pages, Latex. Two references added, a few typos corrected, version to appear in NP

Gravity-Matter Couplings from Liouville Theory

The three-point functions for minimal models coupled to gravity are derived in the operator approach to Liouville theory which is based on its $U_q(sl(2))$ quantum group structure. The result is shown to agree with matrix-model calculations on the sphere. The precise definition of the corresponding cosmological constant is given in the operator solution of the quantum Liouville theory. It is shown that the symmetry between quantum-group spins $J$ and $-J-1$ previously put forward by the author is the explanation of the continuation in the number of screening operators discovered by Goulian and Li. Contrary to the previous discussions of this problem, the present approach clearly separates the emission operators for each leg. This clarifies the structure of the dressing by gravity. It is shown, in particular that the end points are not treated on the same footing as the mid point. Since the outcome is completely symmetric this suggests the existence of a picture-changing mechanism in two dimensional gravity.Comment: (40 pages, Latex file

Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure U_q(sl(2)) \odot U_{\qhat}(sl(2)) is a non trivial extension of $U_q(sl(2))$ and U_{\qhat}(sl(2)). We construct the corresponding generalized exponentials and the generalized Liouville field.Comment: 36 pages, LaTex, LPTENS 93/4

Integrable boundary conditions and modified Lax equations

We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding transfer matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix.Comment: 27 pages Latex. References added and typos correcte

Boundary Lax pairs for the An(1)A_{n}^{(1)} Toda field theories

Based on the recent formulation of a general scheme to construct boundary Lax pairs,we develop this systematic construction for the $A_n^{(1)}$ affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sine-Gordon ($A_1^{(1)}$) and the $A_2^{(1)}$ models. The $A_2^{(1)}$ Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct types of boundary conditions. We provide here novel expressions of boundary Lax pairs associated to both types of boundary conditions.Comment: 30 pages, Latex. Typos corrected, clarifications added. Version to appear in NB