Four Dimensional Non-Critical Strings

Abstract

This is a set of lectures on the gauge/string duality and non-critical strings, with a particular emphasis on the discretized, or matrix model, approach. After a general discussion of various points of view, I describe the recent generalization to four dimensional non-critical (or five dimensional critical) string theories of the matrix model approach. This yields a fully non-perturbative and explicit definition of string theories with eight (or more) supercharges that are related to four dimensional CFTs and their relevant deformations. The space-time as well as world-sheet dimensions of the supersymmetry preserving world-sheet couplings are obtained. Exact formulas for the central charge of the space-time supersymmetry algebra as a function of these couplings are calculated. They include infinite series of string perturbative contributions as well as all the non-perturbative effects. An important insight on the gauge theory side is that instantons yield a non-trivial 1/N expansion at strong coupling, and generate open string contributions, in addition to the familiar closed strings from Feynman diagrams. We indicate various open problems and future directions of research.

This work was supported in part by a Robert H. Dicke fellowship and by the Swiss National Science Foundation. The author is on leave of absence from Centre National de la Recherche Scientifique, École Normale Supérieure, Paris, France.