Supersymmetric Generalizations of Matrix Models

Abstract

In this thesis generalizations of matrix and eigenvalue models involving supersymmetry are discussed. Following a brief review of the Hermitian one matrix model, the c=-2 matrix model is considered. Built from a matrix valued superfield this model displays supersymmetry on the matrix level. We stress the emergence of a Nicolai-map of this model to a free Hermitian matrix model and study its diagrammatic expansion in detail. Correlation functions for quartic potentials on arbitrary genus are computed, reproducing the string susceptibility of c=-2 Liouville theory in the scaling limit. The results may be used to perform a counting of supersymmetric graphs. We then turn to the supereigenvalue model, today's only successful discrete approach to 2d quantum supergravity. The model is constructed in a superconformal field theory formulation by imposing the super-Virasoro constraints. The complete solution of the model is given in the moment description, allowing the calculation of the free energy and the multi-loop correlators on arbitrary genus and for general potentials. The solution is presented in the discrete case and in the double scaling limit. Explicit results up to genus two are stated. Finally the supersymmetric generalization of the external field problem is addressed. We state the discrete super-Miwa transformations of the supereigenvalue model on the eigenvalue and matrix level. Properties of external supereigenvalue models are discussed, although the model corresponding to the ordinary supereigenvalue model could not be identified so far.