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

Malmendier, Andreas¹ ; Ono, Ken²

¹ Department of Mathematics and Statistics, Colby College, Waterville MN 04901, USA
² Mathematics and Computer Science, Emory University, Atlanta GA 30322, USA

Geometry & topology, Tome 16 (2012) no. 3, pp. 1767-1833.

Résumé

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 $2N_f$ 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]).

Détail

DOI : 10.2140/gt.2012.16.1767

Classification : 57R57
Keywords: Donaldson invariant, mock theta function

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