A vanishing result for a Casson-type instanton invariant

TL;DR

This paper proves that a specific Casson-type instanton invariant, defined via moduli spaces of flat instantons, must vanish due to a cobordism argument involving PU(2) Seiberg-Witten monopoles.

Contribution

It introduces a cobordism approach using PU(2) Seiberg-Witten monopoles to demonstrate the vanishing of a Casson-type invariant in certain 4-manifolds.

Findings

Casson-type invariant vanishes for the studied 4-manifolds

Cobordism constructed between instanton moduli space and empty set

Uses PU(2) Seiberg-Witten monopoles to establish the result

Abstract

Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by a count of a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of PU(2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.