# Publications of the Research Institute for Mathematical Sciences

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**Volume 47, Issue 1, 2011, pp. 307–359**

**DOI: 10.2977/PRIMS/37**

Donaldson = Seiberg–Witten from Mochizuki's Formula and Instanton Counting

Lothar Göttsche (1), Hiraku Nakajima (2) and Kota Yoshioka (3)

(1) International Centre for Theoretical Physics, Strada Costiera 11, 34151, TRIESTE, ITALY(2) Research Institute for Mathematical Sciences, Kyoto University, 606-8502, KYOTO, JAPAN

(3) Department of Mathematics, Kobe University, Rokko, 657-8501, KOBE, JAPAN

We propose an explicit formula connecting Donaldson invariants and Seiberg–Witten invariants of a 4-manifold of simple type via Nekrasov's deformed partition function for the *N* = 2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizuki's formula, which makes sense and was proved when the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Witten's conjecture and sum rules for Seiberg–Witten invariants (superconformal simple type condition), conjectured by Mariño, Moore and Peradze.

*Keywords: *Donaldson invariants, Seiberg{Witten invariants, instanton counting