From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometry

Abstract

Hamiltonian Floer theory plays an important role for finding periodic solutions of Hamilton's equation, which can be seen as a generalization of Newton's equation. Generalizing Newton's equation to Laplace's equation with non-linearity, we show, building on the work of Ginzburg and Hein, that this role is taken over by the hyperkähler Floer theory of Hohloch, Noetzel, and Salamon. Apart from establishing $C^0$-bounds in order to be able to deal with noncompact hyperkähler manifolds, the core ingredient is a regularization scheme for the polysymplectic formalism due to Bridges, which allows us to link Euclidean field theory with hyperkähler Floer theory. As a concrete result, we prove a cuplength estimate.

Minor revision after noticing a comment by Ginzburg and Hein that had to be referenced