$\mathrm{SO}(3)$–Donaldson invariants of $\mathbb{C}\mathrm{P}^2$ and mock theta functions

Abstract

We compute the Moore–Witten regularized $u$–plane integral on $ℂ‘P^2$, and we confirm the conjecture that it is the generating function for the $SO\left(3\right)$–Donaldson invariants of $ℂ‘P^2$. We also derive generating functions for the $SO\left(3\right)$–Donaldson invariants with $2N_f$ massless monopoles using the geometry of certain rational elliptic surfaces ($N_f\in \left\{0,2,3,4\right\}$), 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 [Current developments in mathematics, 2008, Int. Press, Somerville, MA (2009) 347–454]).