Twistor approach to string compactifications: A review

We review a progress in obtaining the complete non-perturbative effective action of type II string theory compactified on a Calabi–Yau manifold. This problem is equivalent to understanding quantum corrections to the metric on the hypermultiplet moduli space. We show how all these corrections, which include D-brane and NS5-brane instantons, are incorporated in the framework of the twistor approach, which provides a powerful mathematical description of hyperkähler and quaternion-Kähler manifolds. We also present new insights on S-duality, quantum mirror symmetry, connections to integrable models and topological strings.