Operator Methods in String Theory

TLDR: In this article, Gomez, Reina, Moore and Vafa developed an operator formalism describing high orders of string perturbation theory, as well as conformal field theories on Riemann Surfaces of genus bigger than one.

Abstract: These lectures are based on joint work with C. Gomez, C. Reina, G. Moore and C. Vafa [1], [2], [3]. 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 theories on Riemann Surfaces of genus bigger than one. On the other hand, developments in the theory of soliton solutions of the K-P hierarchy (KadomtsevPetviashvili 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 in the standard Fork 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 solutions of the K.P. equation can be described in terms of an infinite dimensional grassmannian GT. 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.