Donaldson invariants of CP^1 x CP^1 and Mock Theta Functions

TL;DR

This paper calculates Donaldson invariants for CP^1 x CP^1 using the u-plane integral and expresses the results through Mock modular forms, providing explicit formulas where traditional methods fail.

Contribution

It introduces a direct computation of the u-plane integral on CP^1 x CP^1 and derives explicit formulas for Donaldson invariants involving Mock modular forms.

Findings

Explicit formulas for SU(2) and SO(3)-Donaldson invariants

Novel computation method overcoming the failure of elliptic unfolding

Connection between Donaldson invariants and Mock modular forms

Abstract

We compute the Moore-Witten regularized u-plane integral on CP^1 x CP^1 directly in a chamber where the elliptic unfolding technique fails to work. This allows us to determine explicit formulas for its SU(2) and SO(3)-Donaldson invariants in terms of Mock modular forms.