Persson posted a nice paper recently, exploring the relationships between Calabi-Yau threefolds, U-duality groups and automorphic instanton partition functions. The paper discusses recent attempts at describing the moduli space in type IIA/B string theory on X×S^1, or the hypermultiplet moduli space in type IIB/A on X. It is well known that in certain classes of N=2 supergravities (e.g. magic supergravities) one can use automorphic techniques to constrain quantum corrections. Given a D=3 U-duality group G_3(Z), BPS-degeneracies are recovered from the Fourier coefficients of its related automorphic forms. From this stems the conjecture that the instanton partition function in N=2 supergravity on R^3×S^1 should correspond to an automorphic representation in the quaternionic discrete series of G_3.
Abstract: 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 x 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 whenever the low-energy theory in D=3 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. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.