# 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

Göttsche Lothar, Nakajima Hiraku, Yoshioka Kota: Donaldson = Seiberg–Witten from Mochizuki's Formula and Instanton Counting. *Publ. Res. Inst. Math. Sci.* 47 (2011), 307-359. doi: 10.2977/PRIMS/37