Geometry & Topology

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

Andreas Malmendier and Ken Ono

Full-text: Open access

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]).

Article information

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

Dates
Received: 28 April 2010
Revised: 1 May 2012
Accepted: 21 June 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
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

Subjects
Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Keywords
Donaldson invariant mock theta function

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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