  The complete quantum theory of closed superstrings is constructed using
string diagrams endowed with metric having constant curvature $-1$. The
elementary string diagrams are equipped with the analytic local coordinates
induced from the hyperbolic metric and a distribution of a set of picture
changing operators constructed using the identities satisfied by the simple
closed geodesics on a hyperbolic Riemann surface. However, a slight
modification near the boundary of the string vertices is needed to make them
satisfy the geometric condition imposed by the quantum Batalin-Vilkovisky
master equation.
