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. 53D40, 32Q15

Floer homology groups in hyperkähler geometry

“…The gradient lines of H are solutions of this equation and in [16] we prove that, for ε > 0 sufficiently small, there are no other contractible solutions. This implies that our Floer homology groups HF * (M, X) are isomorphic to the singular homology H * (X; Z 2 ).…”

“…The details of the theory outlined here are worked out in [16] for compact flat target manifolds X. The extension to general hyperkähler manifolds X requires a careful understanding of the codimension 2 bubbing phenomenon for the solutions of equation (6).…”

“…If the target manifold X is flat the analysis in symplectic Floer theory can be adapted to the hyperkähler setting as explained below. The full details are given in [16]. To extend the theory to the non-flat case one must deal with new bubbling and compactness phenomena.…”

“…Since the exponent 3/2 is equal to the critical exponent (n + 2)/n for n = 4 (with regard to compactness questions and mean value estimates), the estimate (9) then gives rise to a crucial mean value inequality (see [23]), and hence to the relevant compactness result for the solutions of (6) (see [16]). …”

Hyperkähler Floer theory as infinite dimensional Hamiltonian system

“…Theorem 5 (Hyperkähler Arnold Conjecture, [HNS1,HNS2] Theorem 5 inspired Ginzburg & Hein [GH] to reprove the hyperkähler Arnold Conjecture on compact flat hyperkähler manifolds using Conley & Zehnder's method of finite dimensional approximation. They also established the degenerate version.…”