Representations of vertex operator algebras over an arbitrary field

An associative algebra A(V)A(V) for a vertex operator algebra over an arbitrary algebraically closed field FF is constructed such that there is a one to one correspondence between irreducible A(V)A(V)-modules and irreducible admissible V  -modules. Moreover, A(V)A(V) is semisimple if V   is rational. An A(V)A(V)-bimodule A(M)A(M) is also constructed for any admissible V-module M   and its connection with the fusion rules is investigated. These results are then used to compute the fusion rules for the rational vertex operator algebra L(12,0)F associated to the irreducible highest weight module for the Virasoro algebra with central charge 12.