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We consider the simplest possible setting of non abelian twist fields whichcorresponds to $SU(2)$ monodromies. We first review the theory ofhypergeometric function and of the solutions of the most general Fuchsiansecond order equation with three singularities. Then we solve the problem ofwriting the general solution with prescribed $U(2)$ monodromies. We use thisresult to compute the classical string solution corresponding to three $D2$branes in $R^4$. Despite the fact the configuration is supersymmetric theclassical string solution is not holomorphic. Using the equation of motion andnot the KLT approach we give a very simple expression for the classical actionof the string. We find that the classical action is not proportional to thearea of the triangle determined by the branes intersection points since thesolution is not holomorphic. Phenomenologically this means that the Yukawacouplings for these supersymmetric configurations on non factorized tori aresuppressed with respect to the factorized case.

nuclear physics b

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