Mathematics – Quantum Algebra
Scientific paper
2006-03-03
Adv. Math.213:271-340, 2007
Mathematics
Quantum Algebra
80 pages, 69 figures, a mistake and some misprints are corrected
Scientific paper
10.1016/j.aim.2006.12.007
We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted $\R\times \R$-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 $V^L\otimes V^R$, 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 $V^L\otimes V^R$ 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 $V^L\otimes V^R$ equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of $V^L\otimes V^R$-modules. The so-called diagonal constructions of conformal full field algebras are given in tensor-categorical language.
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