Yang-Mills-Higgs theory for symplectic fibrations

TL;DR

This paper develops a unified framework extending Yang-Mills-Higgs and Gromov theories, introduces a new functional, studies the moduli space of solutions, and defines invariants for symplectic manifolds with Hamiltonian circle actions.

Contribution

It generalizes key concepts from Yang-Mills-Higgs and Gromov theories, establishing a Hitchin-Kobayashi correspondence and constructing a moduli space with compactification for these equations.

Findings

Established a Hitchin-Kobayashi correspondence in the Kähler case.

Constructed a smooth moduli space of solutions.

Defined invariants generalizing Gromov-Witten invariants.

Abstract

Our aim in this work is to study a system of equations which generalises at the same time the vortex equations of Yang-Mills-Higgs theory and the holomorphicity equation in Gromov theory of pseudoholomorphic curves. We extend some results and definitions from both theories to a common setting. We introduce a functional generalising Yang-Mills-Higgs functional, whose minima coincide with the solutions to our equations. We prove a Hitchin-Kobayashi correspondence allowing to study the solutions of the equations in the Kaehler case. We give a structure of smooth manifold to the set of (gauge equivalence classes of) solutions to (a perturbation of) the equations (the so-called moduli space). We give a compactification of the moduli space, generalising Gromov's compactification of the moduli of holomorphic curves. Finally, we use the moduli space to define (under certain conditions) invariants of compact symplectic manifolds with a Hamiltonian almost free action of S^1. These invariants generalise Gromov-Witten invariants. This is the author's Ph.D. Thesis. A chapter of it is contained in the paper math.DG/9901076. After submitting his thesis in April 1999, the author knew that K. Cieliebak, A. R. Gaio and D. Salamon had also arrived (from a different point of view) at the same equations, and had developed a very similar programme (see math.SG/9909122).