César Gómez

The quantum symmetry of rational conformal field theories

The quantum group symmetry of the c

Integrability in random matrix models

1 Rational Conformal Field Theory, in its Coulomb gas version, is formulated in terms of a new type of screened vertex operators, which define the representation spaces of a quantum group Q. The conformal properties of these operators show a deep interplay between the quantum group Q and the Virasoro algebra. The R-matrix, the comultiplication rules and the quantum Clebsch-Gordan coefficients of Q are obtained using contour deformation techniques. Finally, the relation between the chiral vertex operators and the quantum Clebsch-Gordan coefficients is shown.

New R-matrices associated with finite dimensional representations of Uq(sl(2)) at roots of unity

Abstract We prove the equivalence between the recent matrix model formulation of 2D gravity and lattice integrable models. For even potentials this system is the Volterra hierarchy, and many properties of the continuum matrix model like the Virasoro conditions on the partition function stem directly from the integrability properties of the lattice model and its hamiltonian properties.

A new solution to the star-triangle equation based on Uq(sl(2)) at roots of unity

Abstract We construct the R -matrix which intertwines between “semi-cyclic” representations of U q (sl(2)) with q 3 =1. When q p =1 (for odd p ), we call semi-cyclic representations those of dimension p which are characterized by the following value of the center of U q (sl(2)): E p =0, F p = x , K p = γ p . As in the cyclic case, the values of x and γ p of two different representations for which the intertwiner exists are constrained to lie on an algebraic curve, x = κ (1− λ p ). The main interest of our approach is that we do not need to resort to the affine extension of U q (sl(2)) in order to construct a spectral-dependent R -matrix.

Integrability and uniformization in Liouville theory: the geometrical origin of quantized symmetries

We find new solutions to the Yang-Baxter equation in terms of the interwiner matrix for semi-cyclic representations of the quantum group Uq(sl(2)) with q = e2πi/N. These interwiners serve to define the Boltzmann weights of a lattice model, which shares some similarities with the chiral Potts model. An alternative interpretation of these Boltzmann weights is as scattering matrices of solitonic structures whise kinematics is entirely governed by the quantum group. Finally, we consider the limit N → ∞ where we find an infinite-dimensional representation of the braid group, which may give rise to an invariant of knots and links.

New integrable deformations of higher spin Heisenberg-Ising chains

We prove, in the classical limit of the Liouville theory, that the BPZ null vector decoupling equations admit a geometrical interpretation as uniformization equations. Using the Feigin-Fuchs representation of the Liouville theory we obtain a uniformization which gives rise to a propagator-vertex picture of the underlying Riemann surface. The space of solutions of these uniformization equations turns out to be the classical limit of the screened vertex operators and defines the spin-12 representation of the classical limit of SU(2)q+. This is the perturbative part of the full quantum group Q of the Liouville theory. In the strong coupling regime 1<c<25 the non perturbative quantum group is shown to be isomorphic to a quantum deformation of the four-dimensional Lorentz group.

A NOTE ON THE STRING ANALOG OF N =2 SUPER-SYMMETRIC YANG-MILLS

Abstract We show that the anisotropic Heisenberg-Ising chains with higher spin allow, for special values of the anisotropy, integrable deformations intimately related to the theory of quantum groups at roots of unity. For the spin-one case we construct and study the symmetries of the hamiltonian which depends on a spectral variable belonging to an elliptic curve. One of the points of this curve yields the Fateev-Zamolodchikov hamiltonian of spin-one and anisotropy Δ= 1 2 (q 2 +q −2 ) with q a cubic root of unity. In some other special points the spin degrees of freedom as well as the hamiltonian splits into pieces governed by a larger symmetry.

Fusion rules in two-dimensional gravity

Abstract A connection between the conifold locus of the type II string on the W P112264 Calabi-Yau manifold and the geometry of the quantum moduli of N = 2 SU(2) super Yang-Mills is presented. This relation is obtained from the anomalous behaviour of the SU(2) super Yang-Mills special coordinates under S-duality transformation in Sl(2; Z) Γ 2 .

On the integrability of N = 2 Landau-Ginzburg models. A graph generalization of the Yang-Baxter equation

Abstract We analyze the fusion rules for a minimal conformal theory (not necessarily unitary) coupled to gravity in the continuum limit. We show that when the correlators are thought of in terms of string theory, and we include the ghost contributions, the standard fusion rules for the ordinary primary fields are recovered and most of the correlation functions involving null states of the conformal field theory vanish in any Riemann surface.

Duality and Quantum Groups

The study of the integrability properties of the N=2 Landau- Ginzburg models leads naturally to a graph generalization of the Yang-Baxter equation which synthetizes the well known vertex and RSOS Yang-Baxter equations. A non trivial solution of this equation is found for the