SU(2)-Donaldson invariants of the complex projective plane

Abstract

Abstract There are two families of Donaldson invariants for the complex projective plane, corresponding to the SU(2)-gauge theory and the SO(3)-gauge theory with non-trivial Stiefel–Whitney class. In 1997 Moore and Witten conjectured that the regularized u-plane integral on ℂ P 2 $\mathrm {P}^2$ gives the generating functions for these invariants. In earlier work, the second two authors proved the conjecture for the SO(3)-gauge theory. Here we complete the proof of the conjecture by confirming the claim for the SU(2)-gauge theory. As a consequence, we find that the SU(2)-Donaldson invariants for ℂ P 2 $\mathrm {P}^2$ are explicit linear combinations of the Hurwitz class numbers which arise in the theory of imaginary quadratic fields and orders.