# Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting
for the theory with a fundamental matter

Mochizuki's formula express Donaldson invariants in terms of
Seiberg-Witten invariants and
certain integrals over Hilbert schemes of points.
We write the latter by the instanton counting partition
function of the theory with a fundamental matter.
We then compute the partition function in terms of
elliptic integrals associated with
Seiberg-Witten curves for this theory.
For the case of Mochizuki's formula,
the Seiberg-Witten curve becomes singular,
and everything become explicit.
We then prove Witten's conjecture and Marino-Moore-Peradze's
superconformal simple type condition
(for projective surfaces).
This is a joint work with Kota Yoshioka

nakajima@math.kyoto-u.ac.jp