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.