D-instantons and twistors
Sergei Alexandrov, Boris Pioline, Frank Saueressig, Stefan Vandoren
TL;DR
The work develops a duality‑covariant twistorial framework for nonperturbative corrections to the hypermultiplet moduli space in Type II string theory on Calabi–Yau threefolds. By encoding the quaternionic–Kähler metric in the complex contact structure of the twistor space and systematically incorporating D‑instanton corrections through A‑type contributions, EM duality, and mirror symmetry, the authors express all D‑instantons as a sum of dilogarithms $\mathrm{Li}_2$. They derive explicit instanton corrected twistor lines and the corrected contact potential in the leading instanton approximation, and outline a rigorous exact twistor construction via BPS rays that implements Kontsevich–Soibelman wall‑crossing in this geometric setting. The discussion connects these nonperturbative corrections to 3D black hole partition functions, BPS degeneracies, and the arising DT invariants, while identifying NS5-brane contributions as a major open challenge with potential links to quantum invariants.
Abstract
Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-Kahler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.