Semi-Infinite Homology and 2d Quantum Gravity

The primary goal of this thesis is to study the constraint problem for two-dimensional quantum gravity in two gauges--the conformal and the light-cone gauge. In each, case, we examine a natural class of representations of the gravitational sector. We then ask for the physical spectrum when gravity is coupled to conformal matter via the BRST procedure. We classify the free field representations which result in non-trivial physical states, and compute the quantum numbers of these states. Our work also uncovers some infinitely many new physical states beyond those proposed by Knizhnik, Polyakov and Zamolodchikov. These new states are shown to be the continuum counterparts of most of the physical observables in the random matrix models. The principal tool for our investigations is the theory of semi-infinite homology, first introduced and developed by Feigin, and later by Frenkel, Garland and Zuckerman. In solving the constraint problems, we also discuss a number of new results in semi-infinite homology theory.