SO(3)-Donaldson invariants of ℂP 2 and mock theta functions

Abstract

We compute the Moore-Witten regularized u-plane integral on ℂP 2, and we confirm the conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP 2. We also derive generating functions for the SO(3)-Donaldson invariants with 2N f massless monopoles using the geometry of certain rational elliptic surfaces (N f€{0,2,3,5}), and we show that the partition function for N f = 4 is nearly modular. Our results rely heavily on the theory of mock theta functions and harmonic Maass forms (for example, see Ono [40]).