Full field algebras, operads and tensor categories |
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Authors: | Liang Kong |
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Institution: | a Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, D-04103 Leipzig, Germany b Institut des Hautes Études Scientifiques, Le Bois-Marie, 35, Route De Chartres, F-91440 Bures-sur-Yvette, France |
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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 VL⊗VR, where VL and VR are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over VL⊗VR equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL⊗VR equipped with a nondegenerate 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, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language. |
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Keywords: | Vertex operator algebra |
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