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Full field 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 VL ⊗ VR, where VL and VR are two vertex operator algebras satisfying certain natural finite and reductive conditions. We also study the geometry interpretation of conformal full field algebras over VL ⊗ VR equipped with a nondegenerate invariant bi- linear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL ⊗ VR equipped with a non- degenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of VL ⊗ VR-modules. The so-called diagonal constructions [Y.-Z. Huang and L. Kong, Full field algebras, arXiv:math.QA/0511328.] of conformal full field algebras are given in tensor-categorical language.