Abstract: There is a natural analogue of the symplectic action functional on the space of maps from a 3-manifold M, equipped with a volume preserving frame, into a hyperkaehler manifold X. The critical points and gradient flow lines of this functional are solutions of the Fueter equation in dimensions 3 and 4, possibly with Hamiltonian type perturbations.
When X is flat, one can construct Floer homology groups and use them to prove existence theorems for solutions of the Fueter equation, in analogy to the Arnold conjecture for the torus. Conjecturally, one would expect some version of hyperkaehler Floer theory to extend to non-flat manifolds, such as the moduli space of ASD instantons over a K3 surface S, and interact with the G2-type Donaldson-Thomas-Floer theory of M x S.
(The talk is based on joint work with Sonja Hohloch and Gregor Noetzel.)