Full field algebras, operads and tensor categories

Abstract

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 finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over V L ⊗ V R equipped with a nondegenerate invariant bilinear 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 nondegenerate 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 [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.