Floer homology is a powerful tool in symplectic geometry. It was developed by Andreas Floer at the end of the 1980's in order to prove Arnold's conjecture on the number of fixed points of so-called Hamiltonian diffeomorphisms.

In a joint work with Dietmar Salamon and Gregor Noetzel, we had generalized Floer homology to hyperkähler geometry. More precisely, we had defined and computed it on flat, compact hyperkähler manifolds. In contrast to the classical Floer setting where the critical points of the symplectic action functional are periodic Hamiltonian orbits, the critical points of the hypersymplectic action functional are 3-dimensional tori or spheres solving a certain triholomorphic equation. Since the whole setting still seems somewhat mysterious (at least to me), the idea is to come up with another way to describe the solutions.

In this talk, we will give an intuitive introduction to hyperkähler Floer homology. Then we will show, using an explicit construction, that the triholomorphic tori can be seen as periodic solutions of a Hamiltonian system on the iterated loop space.