SL(2, $\mathbb C$) quartic vertex for closed string field theory

TL;DR

This paper constructs an explicit $ ext{SL}(2, ext{C})$ quartic vertex for bosonic closed string field theory, providing exact formulas for the vertex region and enabling future decomposition studies.

Contribution

It introduces a detailed construction of the $ ext{SL}(2, ext{C})$ quartic vertex with a generic stub parameter, including analytical formulas for the vertex region.

Findings

Derived exact formulas for the vertex region parametrization.

Provided volume calculations as a function of the stub parameter.

Facilitates future decomposition analysis using auxiliary fields.

Abstract

We construct the $\mathrm{SL}(2, \mathbb C)$ quartic vertex with a generic stub parameter for the bosonic closed string field theory by characterizing the vertex region in the moduli space of 4-punctured sphere, and providing the necessary and sufficient constraints for the local coordinate maps. While $\mathrm{SL}(2, \mathbb C)$ vertices are not known to have a nice geometric recursive construction like the minimal area or hyperbolic vertices, they can be studied analytically which makes them more convenient for simple computations. In particular, we obtain exact formulas for the parametrization and volume of the vertex region as a function of the stub parameter. The main objective of having an explicit quartic vertex is to later study its decomposition using auxiliary fields.