Direct Compactification of Heterotic Strings, Modular Invariance and Three Families of Chiral Fermions

Abstract

Superstring theories^1,2 are the most promising candidates for the unified theory including gravity. To make contact with low energy physics one must compactify the ten dimensional space-time into the four dimensional one. This is usually carried out within the context of a d=10 field theory,³ but this approach may have problems.⁴ Recent calculations⁵ of the Beta function of the nonlinear Sigma-model in four-loop level showed that Calabi-Yau manifold may not be the solution for the compactification, because of the conformal anomalies arising from the four-loop level. It is much more desirable to carry out the compactification directly with strings. Compactifying string theories to four dimensions via tori has been studied.⁶ But it is impossible to get chiral fermions in this way. Dixon, Harrey, Vafa and Witten⁷ considered compactifying the d=10 heterotic strings on a six dimensional orbifold. They found 36 families of chiral fermions. But this orbifold has 27 singularities.