Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting
Lothar Göttsche, Hiraku Nakajima, Kota Yoshioka
TL;DR
This work establishes an explicit bridge between Donaldson invariants and Seiberg–Witten invariants by expressing the Donaldson generating function ${\mathscr D}^{\xi}(\alpha)$ in terms of Nekrasov’s partition function for an $N=2$ gauge theory with one fundamental matter, grounded in Mochizuki’s algebro-geometric formula. The authors show that the leading Nekrasov data ${F^{\mathrm{inst}}_0}$, ${H^{\mathrm{inst}}}$, ${A^{\mathrm{inst}}}$, and ${B^{\mathrm{inst}}}$ reproduce Mochizuki’s coefficients, enabling a reformulation of Donaldson–Seiberg–Witten relations via fixed-point residues and blow-up formulas. Under the SW-simple type hypothesis and a conjecture extending Mochizuki’s result to smooth 4-manifolds, they derive Witten’s conjecture and the superconformal sum rules posited by Mariño, Moore, and Peradze, illustrating how Seiberg–Witten curves govern the differential structure underpinning the Donaldson invariants. The paper develops Seiberg–Witten curves and a genus-one differential, analyzes residues at singular and superconformal points, and introduces a rational differential on $\mathbb{P}^{1}$ in the auxiliary variable $\phi$, providing a computational framework that links gauge-theoretic and algebro-geometric approaches to 4-manifold topology. This fusion of Nekrasov’s partition function with Mochizuki’s formula advances the practical computation of Donaldson invariants from SW data and clarifies the role of blow-ups and singular fibers in the global invariant structure.
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.