Stochastic differential equations on two-dimensional theory space and Morse theory

Abstract

The renormalization group equations are shown to be saddle points of the action of a superparticle moving in the presence of a Kahler potential. This allows us to view the "theory space" in the language of Morse theory where the Kahler potential in the Morse function. In the case of two-dimensional field theory, Zamolodchikov's c function can be used as a Morse function. The Morse polynomial tr (t^F) can be computed entirely from the universal Virasoro data at the various fixed points and this provides a topological characterization of the space of potential functions. This idea is applied to the two-dimensional theory space which contains the c<1 minimal models.