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