#### Volume 16, issue 3 (2012)

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

### Andreas Malmendier and Ken Ono

Geometry & Topology 16 (2012) 1767–1833
##### 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 $2{N}_{f}$ 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]).

##### Keywords
Donaldson invariant, mock theta function
Primary: 57R57
##### Publication
Received: 28 April 2010
Revised: 1 May 2012
Accepted: 21 June 2012
Published: 3 August 2012
Proposed: Walter Neumann
Seconded: Jim Bryan, Simon Donaldson
##### Authors
 Andreas Malmendier Department of Mathematics and Statistics Colby College Waterville MN 04901 USA Ken Ono Mathematics and Computer Science Emory University Atlanta GA 30322 USA http://www.mathcs.emory.edu/~ono/