A derivation of Witten's conjecture relating Donaldson and Seiberg-Witten invariants

TL;DR

This paper provides a formal derivation of Witten's conjecture linking Donaldson and Seiberg-Witten invariants, extending it to non-simple type manifolds using equivariant localization techniques without relying on non-abelian monopoles.

Contribution

It introduces a new derivation method for Witten's conjecture applicable to non-simple type manifolds, bypassing the need for non-abelian monopole theory.

Findings

Derivation of Witten's conjecture using equivariant localization

Extension of the conjecture to non-simple type manifolds

Avoidance of non-abelian monopole theory in the derivation

Abstract

We generalize Witten's conjectured formula relating Donaldson and Seiberg-Witten invariants to manifolds of non-simple type, via equivariant localization techniques. This approach does not use the theory of non-abelian monopoles, but works directly on the Donaldson-Witten and Seiberg-Witten moduli spaces. We give a formal derivation of Witten's conjecture and its generalization, making use of an infinite dimensional version of the abelian localization theorem.