Operator Methods in String Theory

Abstract

The motivation is two-fold. On the one hand, we would like to develop an operator formalism describing high orders of string perturbation theory, as well as conformal field theory on Riemann Surfaces of genus bigger than one. On the other hand, developments in the theory of soliton solutions of the K-P hierarchy (Kadomtsev-Petviashvili equations; for a detailed geometrical account of this theory and references to the literature see for example [4,5] ) made it clear that many of the geometrical features of Riemann Surfaces and their moduli spaces can be formulated in terms of the properties of certain two-dimensional quantum field theories [6], so that the geometrical complexity of a Riemann surface with a field on it can be coded into a state of a standard Fock space of the field theory [1,7,8,9,10,11,12]. This approach gives an important understanding of the action of the Virasoro algebra on the moduli space of surfaces [13,14]. The space of the solutions of the K. P. equation can be described in terms of an infinite dimensional Grassmannian Gr. To any algebraic curve X with a point P selected on it, a local coordinate around P, and a line bundle L over X, we can associate a point in Gr ( Krichever construction [4,5] ). Moreover, the collection of all those points in Gr is a dense set. It is thus plausible to expect that some subspace of Gr provides an explicit model for the Universal Moduli Space of Friedan and Shenker [15,16] which plays a central role in their non-perturbative approach to string theory.