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

### Lothar Göttsche

International Centre for Theoretical Physics, Trieste, Italy### Hiraku Nakajima

Kyoto University, Japan### Kota Yoshioka

Kobe University, Japan

## Abstract

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.

## Cite this article

Lothar Göttsche, Hiraku Nakajima, Kota Yoshioka, Donaldson = Seiberg–Witten from Mochizuki's Formula and Instanton Counting. Publ. Res. Inst. Math. Sci. 47 (2011), no. 1, pp. 307–359

DOI 10.2977/PRIMS/37