D3-instantons, Mock Theta Series and Twistors

TL;DR

This paper explores the mathematical structure of D-instanton corrections in string theory, demonstrating modular invariance and the role of mock theta series in the twistorial description of hypermultiplet moduli spaces.

Contribution

It proves the SL(2,Z) symmetry in the one-instanton approximation and links modular anomalies to mock theta series within the twistorial framework.

Findings

SL(2,Z) symmetry is established in the one-instanton approximation.

Modular anomalies are canceled by contact transformations involving mock theta series.

The work connects Donaldson-Thomas invariants, twistors, and mock theta functions in string theory.

Abstract

The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2,Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2,Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.