2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70665We introduce two $K$-theories, one for vector bundles whose fibers are modules of vertex operator algebras, another for vector bundles whose fibers are modules of associative algebras. We verify the cohomological properties of these $K$-theories, and construct a natural homomorphism from the VOA K-theory to the associative algebra K-theory.Corrected some typosDifferential GeometryQuantum Algebra19L99$K$-theory associated to vertex operator algebrastext