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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
$\mathrm{SO}(3)$–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 P2, and we confirm the conjecture that it is the generating function for the SO(3)–Donaldson invariants of P2. We also derive generating functions for the SO(3)–Donaldson invariants with 2Nf massless monopoles using the geometry of certain rational elliptic surfaces (Nf {0,2,3,4}), and we show that the partition function for Nf = 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
Mathematical Subject Classification 2010
Primary: 57R57
References
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/