Unitary matrix models: A study of the string equation

In this thesis I review the Symmetric Unitary One Matrix Models (UMM). In the beginning, I discuss matrix models in general, with particular emphasis on their relation to string theory and two dimensional quantum gravity. The crux of matrix models lies in a single ordinary non-linear differential equation which, in a certain limit known as the double scaling limit, embodies the entire dynamical content of the continuum theory. This differential equation, called the string equation, may be solved and analyzed, yielding much insight into string theory and related physical models. Integrable hierarchies arise naturally from the local operators of the theory and describe the flows between multicritical points. The relevant hierarchy for UMM is the modified-KdV hierarchy. The Sato Grassmannian description of the flows is most appropriate for the computation of the space of solutions to the string equation and I discuss its connection to the $\tau$-function formalism of the Japanese school and more conventional representations. The main results of this thesis are the discovery of the operator formalism for UMM, the computation of the space of solutions to the string equation and the derivation of the mKdV flows from the continuum limit of the local scaling operators.