Hyperbolic String Field Theory

Abstract

This thesis develops string field theory whose elementary interactions are parameterized using hyperbolic geometry. We introduce a systematic procedure to characterize its off-shell data: the local coordinates around punctures on Riemann surfaces as a function of complex structure and the vertex regions in the relevant moduli spaces over which the moduli integration is performed. This procedure exploits the relation between hyperbolic geometry and the semi-classical Liouville theory. We demonstrate that the (generalized) hyperbolic three-string vertex is exactly solvable, while the higher-order vertices can be obtained via the conformal bootstrap of Liouville theory in terms of classical conformal blocks and the DOZZ formula. The four-string and tadpole vertices are constructed explicitly using the known expressions of the associated blocks. Our method suggests the existence of a hidden cubic structure within hyperbolic string field theory.

We also take the WKB-like limit of our construction and demonstrate that it can be used to characterize Strebel quadratic differentials on Riemann surfaces. These differentials encode the geometry of polyhedral vertices of classical closed string field theory. The implication is that they can be embedded into the hyperbolic paradigm. The validity of our results in this regime is further confirmed by developing a topology-independent machine learning algorithm characterizing Strebel differentials. Such algorithm provides an alternative, numerically scalable approach for computing closed string field theory interactions. Finally, our work investigates the open-closed string field theory in the presence of large number of D-branes. We establish its consistency by solving the relevant geometric version of the Batalin-Vilkovisky master equation using hyperbolic geometry and investigate its limits.