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Hamiltonian Floer theory plays an important role for finding periodicsolutions of Hamilton's equation, which can be seen as a generalization ofNewton's equation. Generalizing Newton's equation to Laplace's equation withnon-linearity, we show, building on the work of Ginzburg and Hein, that thisrole is taken over by the hyperk\"ahler Floer theory of Hohloch, Noetzel, andSalamon. Apart from establishing $C^0$-bounds in order to be able to deal withnoncompact hyperk\"ahler manifolds, the core ingredient is a regularizationscheme for the polysymplectic formalism due to Bridges, which allows us to linkEuclidean field theory with hyperk\"ahler Floer theory. As a concrete result,we prove a cuplength estimate.

From Euclidean field theory to hyperk\"ahler Floer theory via regularized polysymplectic geometry