The Fokker-Planck formalism for closed bosonic strings

Every Riemann surface with genus g and n punctures admits a hyperbolic metric, if 2g − 2 + n > 0. Such a surface can be decomposed into pairs of pants whose boundaries are geodesics. We construct a string field theory for closed bosonic strings based on this pants decomposition. In order to do so, we derive a recursion relation satisfied by the off-shell amplitudes, using Mirzakhani's scheme for computing integrals over the moduli space of bordered Riemann surfaces. The recursion relation can be turned into a string field theory via the Fokker-Planck formalism. The Fokker-Planck Hamiltonian consists of kinetic terms and three-string vertices. Unfortunately, the worldsheet BRST symmetry is not manifest in the theory thus constructed. We will show that the invariance can be made manifest by introducing auxiliary fields.