Hamiltonian elements in algebraic K-theory (2407.21003v2)

Abstract: Recall that topological complex $K$-theory associates to an isomorphism class of a complex vector bundle $E$ over a space $X$ an element of the complex $K$-theory group of $X$. Or from algebraic $K$-theory perspective, one assigns a homotopy class $[X \to K (\mathcal{K})]$, where $\mathcal{K}$ is the ring of compact operators on the Hilbert space. We show that there is an analogous story for algebraic $K$-theory of a general commutative ring $k$, replacing, and in a sense generalizing complex vector bundles by certain monotone/Calabi-Yau Hamiltonian fiber bundles. (In Calabi-Yau setting $k$ must be restricted.) In suitable cases, we may first assign elements in a certain categorified algebraic $K$-theory, analogous to To\"en's secondary $K$-theory of $k$. And there is a natural ``Hochschild'' map from this categorified algebraic $K$-theory to the classical variant. In particular, if $k$ is regular and $G$ is a compact Lie group we obtain a natural group homomorphism $\pi _{m} (BG) \to K _{m}(k) \oplus K _{m-1} (k) $. This story leads us to formulate a generalization of the homological mirror symmetry phenomenon to the algebraic $K$-theory context, based on ideas of gauged mirror symmetry of Teleman, and the formalism of Langlands dual groups.