Understand
Heterotic string compactifications on a $K3$ surface $\mathfrak{S}$ depend on a choice of hyperk\"ahler metric, anti-self-dual gauge connection and Kalb-Ramond flux, parametrized by hypermultiplet scalars.
- The metric on hypermultiplet moduli space is in principle computable within the $(0,2)$ superconformal field theory on the heterotic string worldsheet, although little is known about it in practice.
- Using duality with type II strings compactified on a Calabi-Yau threefold, we predict the form of the quaternion-K\"ahler metric on hypermultiplet moduli space when $\mathfrak{S}$ is elliptically fibered, in the limit of a large fiber and even larger base.
- The result is in general agreement with expectations from Kaluza-Klein reduction, in particular the metric has a two-stage fibration structure, where the $B$-field moduli are fibered over bundle and metric moduli, while bundle moduli are themselves fibered over metric moduli.