We study the duality between string theory on AdS 3 × S 3 × S 3 and twodimensional conformal theories with large N = 4 superconformal algebra A γ . We discuss configurations of intersecting branes which give rise to such near-horizon geometries. We compute the Kaluza-Klein spectrum and propose that the boundary superconformal theory can be described by a sigma model on a suitable symmetric product space with a particular choice of anti-symmetric two-form.

We explore the thermodynamics of a general class of two-dimensional dilatonic black holes. A simple prescription is given that allows us to compute the mass, entropy and thermodynamic potentials, with results in agreement with those obtained by other methods, when available.

Building on a recent work ofČ. Crnković, M. Douglas and G. Moore, a study of multicritical multi-cut one-matrix models and their associated sl(2, C) integrable hierarchies, is further pursued. The double scaling limits of hermitian matrix models with different scaling ansätze, lead, to the KdV hierarchy, to the modified KdV hierarchy and part of the non-linear Schrödinger hierarchy. Instead, the anti-hermitian matrix model, in the two-arc sector, results in the Zakharov-Shabat hierarchy, which contains both KdV and mKdV as reductions. For all the hierarchies, it is found that the Virasoro constraints act on the associated tau-functions. Whereas it is known that the ZS and KdV models lead to the Virasoro constraints of an sl(2, C) vacuum, we find that the mKdV model leads to the Virasoro constraints of a highest weight state with arbitrary conformal dimension.

N=2 supersymmetry in two dimensions can be seen as a quantum realization of the geometry of singularity theory. We show this using non-perturbative methods. N=2 susy is related to the Picard-Lefschetz theory much in the same way as N=1 susy is related to Morse theory. All the concepts of singularity theory fit in the physics of the N=2 Landau-Ginsburg models. The critical behaviour of the theory is encoded in a certain natural “gauge-connection” in coupling-constant space. It is flat for a quashihomogeneous superpotential, but not in general. We find an explicit formula relating it to the Gauss-Manin connection of the singularity associated to the superpotential. Our results are valid for both the quasihomogeneous and the non-quasihomogeneous case, but in the former our equations simplify dramatically. We discuss some preliminary applications.

We present the technical tools needed to compute any one-loop amplitude involving external spacetime fermions in a four-dimensional heterotic string modelà la Kawai-Lewellen-Tye. As an example, we compute the one-loop three-point amplitude with one "photon" and two external massive fermions ("electrons"). As a check of our computation, we verify that the one-loop contribution to the Anomalous Magnetic Moment vanishes if the model has spacetime supersymmetry, as required by the supersymmetric sum rules.