On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants

TL;DR

This paper explores the connection between Rozansky-Witten invariants and 3D Seiberg-Witten invariants, establishing their relationship and proposing higher rank gauge theories to understand more complex invariants.

Contribution

It introduces a general relationship between Rozansky-Witten and 3D Seiberg-Witten invariants, including the equality of the SU(2) Casson and 3D SW invariants, and suggests higher rank gauge theories for advanced invariants.

Findings

Proved the equality of SU(2) Casson and 3D SW invariants.

Connected RW invariants with 3D monopole invariants.

Proposed higher rank Abelian gauge theories for complex invariants.

Abstract

The Seiberg-Witten analysis of the low-energy effective action of d=4 N=2 SYM theories reveals the relation between the Donaldson and Seiberg-Witten (SW) monopole invariants. Here we apply analogous reasoning to d=3 N=4 theories and propose a general relationship between Rozansky-Witten (RW) and 3-dimensional Abelian monopole invariants. In particular, we deduce the equality of the SU(2) Casson invariant and the 3-dimensional SW invariant (this includes a special case of the Meng-Taubes theorem relating the SW invariant to Milnor torsion). Since there are only a finite number of basic RW invariants of a given degree, many different topological field theories can be used to represent essentially the same topological invariant. This leads us to advocate using higher rank Abelian gauge theories to shed light on the higher (non-Abelian) RW invariants and we write down candidate higher rank SW equations.