Towards a fully stringy computation of Yukawa couplings on non factorized tori and non abelian twist correlators (I): the classical solution and action

TL;DR

This paper develops classical solutions for string configurations with non-abelian monodromies on non-factorized tori, revealing suppressed Yukawa couplings in certain supersymmetric brane setups.

Contribution

It provides explicit classical solutions and actions for non-abelian twist fields on non-factorized tori, extending previous methods to more complex geometries.

Findings

Classical string solutions are non-holomorphic despite supersymmetry.

Classical action is not proportional to the triangle area of brane intersections.

Yukawa couplings are suppressed compared to factorized tori configurations.

Abstract

We consider the simplest possible setting of non abelian twist fields which corresponds to $SU(2)$ monodromies. We first review the theory of hypergeometric function and of the solutions of the most general Fuchsian second order equation with three singularities. Then we solve the problem of writing the general solution with prescribed $U(2)$ monodromies. We use this result to compute the classical string solution corresponding to three $D2$ branes in $R^4$. Despite the fact the configuration is supersymmetric the classical string solution is not holomorphic. Using the equation of motion and not the KLT approach we give a very simple expression for the classical action of the string. We find that the classical action is not proportional to the area of the triangle determined by the branes intersection points since the solution is not holomorphic. Phenomenologically this means that the Yukawa couplings for these supersymmetric configurations on non factorized tori are suppressed with respect to the factorized case.