Hyperbolic Geometry and Closed Bosonic String Field Theory I: The String Vertices Via Hyperbolic Riemann Surfaces

The main geometric ingredient of the closed string field theory are the string vertices, the collections of string diagrams describing the elementary closed string interactions, satisfying the quantum Batalian-Vilkovisky master equation. They can be characterized using the Riemann surfaces endowed with the metric solving the generalized minimal area problem. However, an adequately developed theory of such Riemann surfaces is not available yet, and consequently description of the string vertices via Riemann surfaces with the minimal area metric fails to provide practical tools for performing calculations. We describe an alternate construction of the string vertices satisfying the Batalian-Vilkovisky master equation using Riemann surfaces endowed with the metric having constant curvature $-1$ all over the surface. We argue that this construction provides an approximately gauge invariant closed string field theory.