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

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

2008-02-29

arxiv.org/abs/0803.0025v1

Mathematics

Differential Geometry

23 pages

Scientific paper

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.

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