Volume 13, issue 5 (2009)

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Hypercontact structures and Floer homology

Sonja Hohloch, Gregor Noetzel and Dietmar A Salamon

Geometry & Topology 13 (2009) 2543–2617
Abstract

We introduce a new Floer theory associated to a pair consisting of a Cartan hypercontact $3$–manifold $M$ and a hyperkähler manifold $X$. The theory is a based on the gradient flow of the hypersymplectic action functional on the space of maps from $M$ to $X$. The gradient flow lines satisfy a nonlinear analogue of the Dirac equation. We work out the details of the analysis and compute the Floer homology groups in the case where $X$ is flat. As a corollary we derive an existence theorem for the $3$–dimensional perturbed nonlinear Dirac equation.

Keywords
Floer homology, hyperkaehler, hypercontact
Mathematical Subject Classification 2000
Primary: 53D40, 32Q15
Publication
Revised: 15 April 2009
Accepted: 25 June 2009
Published: 21 July 2009
Proposed: Simon Donaldson
Seconded: Leonid Polterovich, Yasha Eliashberg
Authors
 Sonja Hohloch School of Mathematical Sciences Tel Aviv University Ramat Aviv Tel Aviv 69978 Israel Gregor Noetzel Mathematisches Institut Universität Leipzig Johannisgasse 26 04103 Leipzig Germany Dietmar A Salamon Departement Mathematik ETH Zentrum Rämistrasse 101 CH-8092 Zürich Switzerland