Modular-invariant closed-string field theory

Attempts so far at constructing a covariant closed-string field theory have been frustrated by the fact that modular invariance always appears to be violated. At both the tree and loop levels, moduli space is either overcounted an infinite number of times, or undercounted because of a missing region. We solve this problem by demonstrating that a new Iclosed four-string interactionR is necessary to reproduce the closed-string amplitude which precisely fills the missing region. This closed four-string interaction, which has the topology of a tetrahedron, is predicted by geometric string field theory. The tetrahedron graph is generated by gauge fixing the geometric theory's local gauge group, the unified string group, and is the exact counterpart of the instantaneous four-fermion Coulomb term found in QED. We prove the existence of this tetrahedron graph both analytically and by direct computer calculation and show that it is the key to reproducing the Shapiro-Virasoro amplitude