# Multiple D3-instantons and mock modular forms II

## TL;DR

This paper proves that the modularity condition for D3-instanton corrections in string theory is sufficient at two-instanton order by constructing a holomorphic SL(2,Z) action using novel indefinite theta series.

## Contribution

It demonstrates the sufficiency of a mock modular form condition for D3-instanton corrections and introduces new theta series with modular properties.

## Key findings

- Proves the modularity condition is sufficient at two-instanton order.
- Constructs a holomorphic SL(2,Z) action on twistor space.
- Introduces new indefinite theta series with conjectured convergence.

## Abstract

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest.