# Heterotic Stringy Corrections to Metrics of Toroidal Orbifolds and Their Resolutions

## TL;DR

This paper computes first-order stringy corrections to metrics on toroidal orbifolds and their resolutions, revealing effects of non-standard embeddings and potential applications in string phenomenology.

## Contribution

It provides explicit formulas for $O(eta')$ metric corrections on specific orbifolds and their resolutions, including non-standard embeddings and five-brane considerations.

## Key findings

- Closed-form $O(eta')$ corrections for $T^4/\mathbb{Z}_n$ orbifolds.
- Non-standard embedding requires five-branes on orbifolds.
- Non-trivial conformal factors possible on resolved singularities without five-branes.

## Abstract

We explicitly analyse $O(\alpha')$ corrections to heterotic supergravity on toroidal orbifolds and their resolutions, which play important roles in string phenomenology as well as moduli stabilisation. Using a conformal factor ansatz that is valid only for four dimensional geometries, we obtain a closed expression for the $O(\alpha')$ metric corrections in the case of several orbifold limits of K3, namely $T^4/\mathbb{Z}_n$ where $n=2,3,4,6$. However, we find that non-standard embedding requires the inclusion of five-branes on such orbifolds. We also numerically investigate the behaviour around orbifold fixed points by considering the metric correction on the resolution of a $\mathbb{C}^2/\mathbb{Z}_2$ singularity. In this case, a non-trivial conformal factor can be obtained in non-standard embedding even without five-branes. In the same manner, we generalise our analysis to study metric corrections on $T^6/\mathbb{Z}_3$ and its resolution described by a complex line bundle over $\mathbb{CP}^2$. Further prospects of utilising these $O(\alpha')$ corrected metrics as a novel approach in obtaining realistic or semi-realistic Yukawa couplings are discussed.