Automorphic Instanton Partition Functions on Calabi-Yau Threefolds

TL;DR

This paper reviews how automorphic forms can be used to resum infinite D-brane and NS5-brane instanton series in string theory compactified on Calabi-Yau threefolds, revealing deep symmetry structures in the moduli space.

Contribution

It proposes that the total instanton partition function in certain string theories is given by automorphic forms associated with U-duality groups, connecting physical instanton sums to mathematical automorphic representations.

Findings

Instanton partition functions correspond to automorphic forms of U-duality groups.

In D=4, N=2 theories, the relevant automorphic forms are in the quaternionic discrete series.

The partition function can be viewed as a holomorphic section on the twistor space over the moduli space.

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.