2024/07/30 by Savelyev, Yasha · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2407.21003
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 → K (K)], where 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 Toën'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 πm (BG) → K m(k) ⊕ 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.