UC Santa Cruz

The quantum dimensions of modules for vertex operator algebras are defined and their properties are discussed systematically. The quantum dimensions of the Heisenberg vertex operator algebra modules, the Virasoro vertex operator algebra modules and the lattice vertex operator algebra modules are computed. A criterion for simple current modules of a rational vertex operator algebra is given. The possible values of the quantum dimensions are obtained for rational vertex operator algebras. A full Galois theory for rational vertex operator algebras is established using the quantum dimensions.