Volume 16, issue 3 (2012)

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
$\mathrm{SO}(3)$–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/