On the Instanton Contributions to the Masses and Couplings of $E_6$ Singlets

Abstract

We consider the gauge neutral matter in the low--energy effective action for string theory compactification on a \cym\ with $(2,2)$ world--sheet supersymmetry. At the classical level these states (the \sing's of $E_6$) correspond to the cohomology group $H^1(\M,{\rm End}\>T)$. We examine the first order contribution of instantons to the mass matrix of these particles. In principle, these corrections depend on the \K\ parameters $t_i$ through factors of the form $e^{2\p i t_i}$ and also depend on the complex structure parameters. For simplicity we consider in greatest detail the quintic threefold $\cp4[5]$. It follows on general grounds that the total mass is often, and perhaps always, zero. The contribution of individual instantons is however nonzero and the contribution of a given instanton may develop poles associated with instantons coalescing for certain values of the complex structure. This can happen when the underlying \cym\ is smooth. Hence these poles must cancel between the coalescing instantons in order that the superpotential be finite. We examine also the \Y\ couplings involving neutral matter \ysing\ and neutral and charged fields \ymix, which have been little investigated even though they are of phenomenological interest. We study the general conditions under which these couplings vanish classically. We also calculate the first--order world--sheet instanton correction to these couplings and argue that these also vanish.