Topological and Integrability Properties of Multi-Matrix Models

This thesis is a study of topological and integrability properties of the multi-matrix models. The main method used in deri\·ing these results is the Q-matrices approach. There are ten chapters. After a general introduction in the first chapter. in chapter two we deal ·with the Liouville theory. We make special emphasis on the dynamics of the boundaries, due to the recent interest in the p-branes theory. In chapter three we introduce the discrete states and argue that they could be interpreted as excitations of the boundary. In chapter four we pass to the matrix theory and introduce the general Q-matrices approach. After, ·we take some simple examples where \Ve apply this approach and the classical method of \Y-constraints. In chapter five we show how the discrete states appear in the 2q-matrix model and how to calculate the correlation functions. \Ve argue that the multi-matrix models are topological models and that they might accomodate the states of the Wn+1 minimal models coupled to topological gravity. In the next three chapters we apply the general method to some concrete models. In chapter six we calculate nonzero momentum correlation functions in the c = 1-matrix model and show the connection between our approach and the free fermion approach. Chapter seven deals with the star-matrix models which describe the Potts-model on a random surface. Chapter eight discusses the quantum chaos in multi-matrix models. Chapter nine shows that it is possible to classify the reductions of the Toda lattice hierarchy according to the Drinfeld-Sokolov generalized KdV hierarchies. In the last chapter of conclusions we discuss some of the unsolved problems.