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Seiberg-Witten equations in all dimensions

par Ghosh, Partha
Président du jury Haydys, Andriy
Promoteur Fine, Joel
Publication Non publié, 2024-06-07

Thèse de doctorat

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Résumé :

Starting with an oriented Riemannian manifold of any dimension greater than two, with a Spin-c structure, we describe an elliptic system of equations which recover the Seiberg-Witten equations when the dimension is three or four. The equations are for spinors, connections and some odd-degree forms on the base manifold. The equations have two parts: the Dirac equation and the curvature equation. The left hand side of the curvature equation is principal part of the Weitzenböck remainder for the corresponding Dirac operator. We then prove a collection of a priori estimates for solutions to these equations. Unfortunately, they are not sufficient to prove compactness modulo gauge, instead leaving the possibility that bubbling might occur. We also construct several examples of solutions of these equations in dimensions five, six and eight. And finally we describe a modified version of the Seiberg-Witten equations on manifolds with a Spin(7)-structure and construct a solution when the Spin(7)-structure is torsion free.