SO(3)-Donaldson invariants of CP^2 and Mock Theta Functions

TL;DR

This paper proves a conjecture relating the Moore-Witten u-plane integral on CP^2 to the generating function of SO(3)-Donaldson invariants using mock theta functions and harmonic Maass forms, and explores related invariants and modular properties.

Contribution

It confirms the Moore-Witten conjecture by employing mock theta functions and harmonic Maass forms to compute and interpret the generating functions of SO(3)-Donaldson invariants.

Findings

Confirmed the Moore-Witten conjecture for CP^2.

Derived generating functions for invariants with massless monopoles.

Showed the partition function for N_f=4 is nearly modular.

Abstract

We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta functions and harmonic Maass forms. We also derive further such generating functions for the SO(3)-Donaldson invariants with 2N_f massless monopoles using the geometry of certain rational elliptic surfaces (N_f=0,2,3,4). We show that the partition function for N_f=4 is nearly modular. When combined with one of Ramanujan's mock theta functions, we obtain a weight 1/2 modular form. This fact is central to the proof of the conjecture.