## Geometry & Topology

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

#### Abstract

We compute the Moore–Witten regularized $u$–plane integral on $ℂ‘P2$, and we confirm the conjecture that it is the generating function for the $SO(3)$–Donaldson invariants of $ℂ‘P2$. 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]).

#### Article information

Source
Geom. Topol., Volume 16, Number 3 (2012), 1767-1833.

Dates
Revised: 1 May 2012
Accepted: 21 June 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732448

Digital Object Identifier
doi:10.2140/gt.2012.16.1767

Mathematical Reviews number (MathSciNet)
MR2967063

Zentralblatt MATH identifier
1255.57028

#### Citation

Malmendier, Andreas; Ono, Ken. $\mathrm{SO}(3)$–Donaldson invariants of $\mathbb{C}\mathrm{P}^2$ and mock theta functions. Geom. Topol. 16 (2012), no. 3, 1767--1833. doi:10.2140/gt.2012.16.1767. https://projecteuclid.org/euclid.gt/1513732448

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