RG flow on random surfaces with handles and closed string field theory

The renormalization group flow in two-dimensional field theories that are coupled to gravity has unusual features: First, the flow equations are second order in derivatives. Second, in the presence of handles the flow has quantum mechanical properties. Third, the beta functions contain the elementary higher-genus vertices of closed string field theory. This is demonstrated at simple examples and is applied to derive various results about gravitationally dressed beta functions. The possibility of interpreting closed string field theory as the theory of the renormalization group on random surfaces with random topology is considered.