$K$-theory associated to vertex operator algebras

Abstract

We 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. Since its introduction by Grothendieck, Atiyah and Hirzebruch, K-theory has found many applications in algebraic geometry, topology and differential geometry. K-theories in different settings lead to the Grothendieck-Hirzebruch- Riemann-Roch theorem and the Atiyah-Singer indextheory. Originally such theory was developed starting from vector bundles. Note that one can regard vector spaces, which are fibers of vector bundles, simply as C-modules. It is natural to consider bundles of modules over other algebras. The theory of vertexoperator algebras has been very much developed in the last eighteen years. Playing important roles in the study of elliptic genus and Witten genus, the highest weight representations of Heisenberg and affine Kac-Moody algebras provide important examples of vertex operator algebras. In this note we introduce a K-theory for vector bundles of modules of vertex operator algebras, and a K-theory for vector bundles of modules of associative algebras. We verify the cohomology theory properties of these K-theories - the exact sequences. We also give a natural homomorphism from the vertex operator algebra K-group to the associative algebra K-group when the associative algebra is the Zhu's algebra of the vertexoperator algebra. We follow closely the construction of topological K-theory by Atiyah in (1) to verify the cohomological properties of the two new K-groups. We need certain deep property of vertexoperator algebras, such as the existence of the nondegen- erate symmetric bilinear form, to get exact sequences. Our original motivation