A two parameter family of lightcone-like hyperbolic string vertices

TL;DR

This paper introduces a two-parameter family of hyperbolic string vertices, called hyperbolic Kaku vertices, which interpolate between different string field theory vertices and connect covariant and lightcone formulations.

Contribution

It defines hyperbolic Kaku vertices based on hyperbolic metrics, unifies open and closed string vertices, and explores their flat limits to connect various known vertices.

Findings

Open string lightcone vertex is the flat limit of hyperbolic Kaku vertices.

Open string Kaku vertices interpolate between Witten and lightcone vertices.

Closed string Kaku vertices interpolate between polyhedral and lightcone vertices.

Abstract

We introduce a two parameter family of string field theory vertices, which we refer to as hyperbolic Kaku vertices. It is defined in terms of hyperbolic metrics on the Riemann surface, but the geometry is allowed to depend on inputs of the states. The vertices are defined for both open and closed strings. In either case, the family contains the hyperbolic vertices. Then we show that the open string lightcone vertex is obtained as the flat limit of the hyperbolic Kaku vertices. The open string Kaku vertices, which interpolate between the Witten vertex and the open string lightcone vertex, is also obtained as a flat limit. We use the same limit on the case of closed strings to define the closed string Kaku vertices: a one parameter family of vertices that interpolates between the polyhedral vertices - which are covariant, but not cubic - and the closed string lightcone vertex - which is cubic, but not Lorentz covariant.