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 hyperk\"ahler Floer theory of Hohloch, Noetzel, and Salamon. Apart from establishing $C^0$-bounds in order to be able to deal with noncompact hyperk\"ahler manifolds, the core ingredient is a regularization scheme for the polysymplectic formalism due to Bridges, which allows us to link Euclidean field theory with hyperk\"ahler Floer theory. As a concrete result, we prove a cuplength estimate.