Hypercontact structures and Floer homology

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. 1