SO(3)–Donaldson invariants of ℂP2 and mock theta functions

Malmendier, Andreas; Ono, Ken

doi:10.2140/gt.2012.16.1767

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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

DOI: 10.2140/gt.2012.16.1767

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 $SO (3)$ –Donaldson invariants of $ℂ ‘ P^{2}$ . We also derive generating functions for the $SO (3)$ –Donaldson invariants with $2 N_{f}$ massless monopoles using the geometry of certain rational elliptic surfaces ( $N_{f} \in {0, 2, 3, 4}$ ), 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

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/

Volume 16, issue 3 (2012)