Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 4, pages 87-102.

Topological Field Theories and formulae of Casson and Meng-Taubes

S K Donaldson

Abstract. The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In particular, we relate the Seiberg-Witten equations on a 3-manifold with the Abelian vortex equations on a Riemann surface. These techniques also give a new proof of the surgery formula for the Casson invariant, interpreted as an invariant of a homology S^2 x S^1.

Keywords. Seiberg-Witten invariant, Casson invariant, Alexander polynomial, Milnor torsion, topological quantum field theory, moduli space, vortex equation

AMS subject classification. Primary: 57R57. Secondary: 57M25, 57N10, 58D29.

E-print: arXiv:math.GT/9911248

Submitted: 5 March 1999. (Revised: 24 June 1999.) Published: 17 November 1999.

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S K Donaldson
Department of Mathematics
Imperial College, London SW7 2BZ, UK
Email: s.donaldson@ic.ac.uk

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