Yukawa Couplings in Heterotic Compactification

TL;DR

The paper addresses the challenge of computing Yukawa couplings in heterotic compactifications with non-standard embeddings on Calabi-Yau three-folds. It develops a practical, algebraic method that translates Yukawa maps into polynomial computations via the cohomology of positive monad bundles over favourable CICYs, enabling computer implementation. The authors demonstrate the approach across $E_6$, $SO(10)$, and $SU(5)$ GUT scenarios, deriving explicit polynomial representations for relevant cohomology groups and showing how to extract Yukawa coefficients from polynomial products and projections. A detailed example with one Higgs multiplet and one heavy family illustrates the method’s capacity to engineer phenomenologically interesting spectra and to obtain rank-one Yukawa matrices. This framework opens the door to systematic, large-scale analyses of Yukawa structures in non-standard heterotic models and can be extended to broader Calabi-Yau and bundle constructions.

Abstract

We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automated investigation of Yukawa couplings for large classes of smooth heterotic compactifications a viable possibility.