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

TL;DR

This paper completes the proof of Moore and Witten's conjecture relating the SU(2)-Donaldson invariants of the complex projective plane to the u-plane integral, revealing their connection to Hurwitz class numbers.

Contribution

It proves the conjecture for SU(2)-gauge theory invariants on CP^2, linking them explicitly to Hurwitz class numbers in number theory.

Findings

SU(2) Donaldson invariants are expressed as linear combinations of Hurwitz class numbers.

The proof confirms the conjecture for the SU(2) case, completing previous work on the SO(3) case.

The invariants are connected to the theory of imaginary quadratic fields.

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 the complex projective plane 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 CP^2 are explicit linear combinations of the Hurwitz class numbers which arise in the theory of imaginary quadratic fields and orders.