Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting

TL;DR

This paper derives an explicit formula linking Donaldson and Seiberg-Witten invariants of 4-manifolds using Nekrasov's partition function and Mochizuki's formula, leading to proofs of Witten's conjecture under certain assumptions.

Contribution

It introduces a new explicit formula connecting Donaldson and Seiberg-Witten invariants via Nekrasov's function, extending Mochizuki's formula to non-complex projective 4-manifolds.

Findings

Proves Witten's conjecture assuming the formula holds for simple type 4-manifolds.

Establishes sum rules for Seiberg-Witten invariants under superconformal simple type conditions.

Provides a bridge between gauge theory invariants and algebraic geometry via Nekrasov's partition function.

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 Mari\~no, Moore and Peradze.