Matter Field Kahler Metric in Heterotic String Theory from Localisation

TL;DR

This paper introduces an analytic localisation method to compute matter field K"ahler metrics in heterotic string compactifications, simplifying calculations by focusing on specific points on Calabi-Yau manifolds.

Contribution

It presents a new localisation technique for calculating matter field K"ahler metrics, reducing complex integrals to point-localised computations in heterotic string theory.

Findings

Derived a general formula for the matter field K"ahler metric.

Showed localisation around specific points due to large gauge flux.

Provided explicit calculations for Calabi-Yau hypersurfaces in projective space.

Abstract

We propose an analytic method to calculate the matter field K\"ahler metric in heterotic compactifications on smooth Calabi-Yau three-folds with Abelian internal gauge fields. The matter field K\"ahler metric determines the normalisations of the ${\cal N}=1$ chiral superfields, which enter the computation of the physical Yukawa couplings. We first derive the general formula for this K\"ahler metric by a dimensional reduction of the relevant supergravity theory and find that its T-moduli dependence can be determined in general. It turns out that, due to large internal gauge flux, the remaining integrals localise around certain points on the compactification manifold and can, hence, be calculated approximately without precise knowledge of the Ricci-flat Calabi-Yau metric. In a final step, we show how this local result can be expressed in terms of the global moduli of the Calabi-Yau manifold. The method is illustrated for the family of Calabi-Yau hypersurfaces embedded in $\mathbb{P}^1\times\mathbb{P}^3$ and we obtain an explicit result for the matter field K\"ahler metric in this case.