  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.
