Automorphic instanton partition functions on Calabi-Yau threefolds

We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X 7 S 1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that when the theory in three dimensions exhibits an arithmetic "U-duality" symmetry G(ℤ) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. In the case of four-dimensional N = 2 theories on ℝ 7 S 1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function is a holomorphic section on the twistor space Z over M