Casson-type invariants from the Seiberg-Witten equations

TL;DR

This paper surveys recent work on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends, introducing a new invariant related to classical invariants and providing calculations for 4-dimensional mapping tori.

Contribution

It introduces a new invariant derived from Seiberg-Witten theory, connecting it to classical and Yang-Mills invariants, with explicit calculations for specific 4-manifolds.

Findings

New invariant related to Rohlin and Furuta-Ohta invariants

Calculations for 4-dimensional mapping tori

Connections between gauge theory and classical invariants

Abstract

This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gauge theory. We give some new calculations of our invariant for 4-dimensional mapping tori.