algebras , operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R × R-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over V L ⊗ V R , where V L and V R are two vertex operator algebras satisfying certain natural finite and reductive conditions. We also study the geometry interpretation of conformal full field algebras over V L ⊗ V R equipped with a nondegenerate invariant bi-linear form. By assuming slightly stronger conditions on V L and V R , we show that a conformal full field algebra over V L ⊗ V R equipped with a non-degenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of V L ⊗ V R-modules. The so-called diagonal constructions [HK2] of conformal full field algebras are given in tensor-categorical language. 0 Introduction In [HK2], Huang and the author introduced the notion of conformal full field algebra and some variants of this notion. We also studied their basic properties and gave constructions. We explained briefly without giving details in the introduction of [HK2] that our goal is to construct conformal field theories [BPZ][MS]. It is one of the purpose of this work to explain the connection between conformal full field algebras and genus-zero conformal field theories. [HK2] and this work are actually a part of Huang's program ([H1]-[H13]) of constructing rigorously conformal field theories in the sense of Kontsevich and Segal [S1][S2]. Around 1986, I. Frenkel initiated a program of using vertex operator algebras to construct, in a suitable sense, geometric conformal field theories, the precise mathematical definition of which was actually given independently by Kontsevich 1 and Segal [S1][S2] in 1987. According to Kontsevich and Segal, a conformal field theory is a projective tensor functor from a category consisting of finite many ordered copies of S 1 as objects and the equivalent classes of Riemann surfaces with parametrized boundaries as morphisms to the category of Hilbert spaces. This beautiful and compact definition of conformal field theory encloses enormously rich structures of conformal field theory. In order to construct such theories, it is more fruitful to look at some substructures of conformal field theories at first. In the first category, the set of …