From Weak to Strong Coupling in Two-Dimensional Gravity

The strong coupling physics of two dimensional gravity at C=7, 13, 19 is deciphered, by building up on previous works along the same line (for a recent review, of the background material, see hep-th/9408069). It is shown that chirality becomes deconfined. The string suceptibility is derived, and found to be real contrary to the continuation of the KPZ formula. Topological Liouville string theories (without transverse degree of freedom) are explicitely solved. Altough they involve strongly coupled gravity, they share many features with the standard matrix models.Comment: LPTENS-94/25, Latex file + figure

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

Lax equations in ten dimensional supersymmetric classical Yang-Mills theories

In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is pushed further by writing down a set of super partial linear differential equations whose consistency conditions may be derived from the SUSY Y-M equations in ten dimensions, and which are the analogues of the Lax pair of Belavin and Zakharov. On the simplest example of the two pole ansatz, it is shown that the same solution-generating techniques are at work, as for the derivation of the celebrated multi-instanton solutions carried out in the late seventies. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast with the Belavin Zakharov Lax pair.Comment: 8 pages Late

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

Backlund Transformations in 10D susy Yang-Mills Theories

This is a continuation of hep-th/9811108, hep-th/9903218, hep-th/9910235, on exact integration technics for modified dynamical equations in ten dimensional supersymmetric gauge theory. A B\"acklund transformation is derived for the Yang type (super) equations previously derived (hep-th/9811108) by M. Saveliev and the author, from the ten dimensional super Yang-Mills field equations in an on-shell light cone gauge. It is shown to be based upon a particular gauge transformation satisfying nonlinear conditions which ensure that the particular form of the equations is retained. These Yang type field equations are shown to be precisely such that they automatically provide a solution of these conditions. This B\"acklund transformation is similar to the one proposed by A. Lesnov for self-dual Yang-Mills in four dimensions. In the introduction a personal recollection on the birth of supersymmetry is given.Comment: 13 pages Latex, Contribution to Y. Golfand Memorial Volum

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