SO(3)-Donaldson invariants of CP^2 and Mock Theta Functions

Abstract

We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta functions and harmonic Maass forms. We also derive further such generating functions for the SO(3)-Donaldson invariants with 2N_f massless monopoles using the geometry of certain rational elliptic surfaces (N_f=0,2,3,4). We show that the partition function for N_f=4 is nearly modular. When combined with one of Ramanujan's mock theta functions, we obtain a weight 1/2 modular form. This fact is central to the proof of the conjecture.

56 pages, accepted for publication In Geometry and Topology

References in corpus (4)

• Issues in Topological Gauge Theory
• Softly Broken N=2 QCD with Massive Quark Hypermultiplets, I
• Integrating over the Coulomb branch in N=2 gauge theory
• Duality in twisted N=4 supersymmetric gauge theories in four dimensions

Cited by in corpus (10)

• Holomorphic projections and Ramanujan's mock theta functions
• Vafa-Witten theory and iterated integrals of modular forms
• Donaldson-Witten theory and indefinite theta functions
• Duality and Mock Modularity
• Elliptic Loci of SU(3) Vacua
• Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD
• Four flavours, triality and bimodular forms
• The $u$-plane integral, mock modularity and enumerative geometry
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers