On higher rank instantons & the monopole cobordism program

TL;DR

This paper explores a higher rank generalization of Witten's conjecture relating Donaldson and Seiberg-Witten invariants, providing vanishing results that support the conjecture's extension to more complex four-manifolds.

Contribution

It extends the classical cobordism program to higher rank invariants and offers evidence supporting a generalized Witten's conjecture.

Findings

Obtained vanishing results for higher rank invariants.

Provided evidence for the validity of the generalized Witten's conjecture.

Abstract

Witten's conjecture suggests that the polynomial invariants of Donaldson are expressible in terms of the Seiberg-Witten invariants if the underlying four-manifold is of simple type. A higher rank version of the Donaldson invariants was introduced by Kronheimer. Before even having been defined, the physicists Mari\~no and Moore had already suggested that there should be a generalisation of Witten's conjecture to this type of invariants. We study a generalisation of the classical cobordism program to the higher rank situation and obtain vanishing results which gives evidence that the generalisation of Witten's conjecture should hold.