Finiteness of conformal blocks over compact Riemann surfaces

We study conformal blocks (the space of correlation functions) over compact Riemann surfaces associated to vertex operator algebras which are the sum of highest weight modules for the underlying Virasoro algebra. Under the fairly general condition, for instance, $C_2$-finiteness, we prove that conformal blocks are of finite dimensional. This, in particular, shows the finiteness of conformal blocks for many well-known conformal field theories including WZNW model and the minimal model.Comment: Latex2e 16page

Representations of vertex operator algebra V_L^+ for rank one lattice L

We classify the irreducible modules for the fixed point vertex operator subalgebra V_L^+ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the -1 isometry of L.Comment: Latex2e, 35 page

Classification of irreducible modules for the vertex operator algebra M(1)^+

We classify the irreducible modules for the fixed point vertex operator subebra of the rank 1 free bosonic VOA under the -1 automorphism.Comment: Latex 2e, 24 pages, several mistakes are correcte