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In this talk, we will explain how to construct Liouville manifolds which have vanishing symplectic cohomology but non-vanishing symplectic K-theory. In particular, we construct an exotic symplectic structure on Euclidean space which is not distinguished by traditional Floer homology invariants. Instead, it is detected by a module spectrum for complex K-theory, built as a variant of Cohen-Jones-Segal’s Floer homotopy type. The proof involves passage through (wrapped) Fukaya categories with coefficients in a ring spectrum, rather than an ordinary ring; we will outline the construction of such "spectral Fukaya categories" in the setting of exact symplectic manifolds.

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