# The Classical Solution for the Bosonic String in the Presence of Three D-branes Rotated by Arbitrary SO(4) Elements

## TL;DR

This paper analyzes the classical bosonic string solution in a setup with three D-branes rotated by SO(4) elements, revealing non-holomorphic solutions that inform supersymmetry and affect Yukawa coupling suppression.

## Contribution

It provides the first explicit classical solution for three D-branes with arbitrary SO(4) rotations, extending previous Abelian twist field results and exploring supersymmetry indications.

## Key findings

- Classical solution expressed via hypergeometric functions.
- Solution is non-holomorphic, indicating supersymmetry detection.
- Yukawa couplings are more suppressed compared to factorized cases.

## Abstract

We consider the classical instantonic contribution to the open string configuration associated with three D-branes with relative rotation matrices in SO(4) which corresponds to the computation of the classical part of the correlator of three non Abelian twist fields. We write the classical solution as a sum of a product of two hypergeometric functions. Differently from all the previous cases with three D-branes, the solution is not holomorphic and suggests that the classical bosonic string knows when the configuration may be supersymmetric. We show how this configuration reduces to the standard Abelian twist field computation. From the phenomenological point of view, the Yukawa couplings between chiral matter at the intersection in this configuration are more suppressed with respect to the factorized case in the literature.