$PU(N)$ monopoles, higher rank instantons, and the monopole invariants

TL;DR

This paper explores the extension of Witten's conjecture to higher rank gauge theories by studying $PU(N)$ monopoles, providing evidence that the polynomial invariants can be expressed in terms of Seiberg-Witten invariants for complex four-manifolds.

Contribution

It generalizes the cobordism program to $PU(N)$ monopoles, offering new insights and evidence supporting a higher rank version of Witten's conjecture.

Findings

Evidence supporting the generalization of Witten's conjecture to $PU(N)$ monopoles.

Analysis of differences between $PU(2)$ and higher rank cases.

Extension of the cobordism program to higher rank gauge theories.

Abstract

A famous conjecture in gauge theory mathematics, attributed to Witten, 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. Mathematicians have sought a proof of the conjecture by means of a `cobordism program' involving $PU(2)$ monopoles. A higher rank version of the Donaldson invariants was recently introduced by Kronheimer. Before being 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 adopt a generalisation of the cobordism program to the higher rank situation by studying $PU(N)$ monopoles. We analyse the differences to the $PU(2)$ situation, yielding evidence that a generalisation of Witten's conjecture should hold.