Full field algebras

We solve the problem of constructing a genus-zero full conformal field theory (a conformal field theory on genus-zero Riemann surfaces containing both chiral and antichiral parts) from representations of a simple vertex operator algebra satisfying certain natural finiteness and reductive conditions. We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For two vertex operator algebras, their tensor product is naturally a full field algebra and we introduce a notion of full field algebra over such a tensor product. We study the structure of full field algebras over such a tensor product using modules and intertwining operators for the two vertex operator algebras. For a simple vertex operator algebra V satisfying certain natural finiteness and reductive conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over the tensor product of two copies of V and an invariant bilinear form on this algebra.