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

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

2008-08-11

arxiv.org/abs/0808.1442v3

Mathematics

Differential Geometry

55 pages

Scientific paper

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 $2 N_f$ massless monopoles using the geometry of certain rational elliptic surfaces ($N_f \in \{0,2,3,4\}$). 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.

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