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 $\mathbb{R}\times \mathbb{R}$-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.

80 pages, 69 figures, a mistake and some misprints are corrected

References in corpus (4)

• Vertex operator algebras, the Verlinde conjecture and modular tensor categories
• TFT construction of RCFT correlators IV: Structure constants and correlation functions
• Categorification and correlation functions in conformal field theory
• Rigidity and modularity of vertex tensor categories

Cited by in corpus (16)

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• A mathematical theory of gapless edges of 2d topological orders. Part I
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• Gapless edges of 2d topological orders and enriched monoidal categories
• Invertible defects and isomorphisms of rational CFTs
• A mathematical theory of gapless edges of 2d topological orders. Part II
• Gluing vertex algebras
• Cardy condition for open-closed field algebras
• Cardy algebras and sewing constraints, II
• Open-closed field algebras
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• The boundary phase transitions of the 2+1D ${\mathbb{Z}}_N$ topological order via topological Wick rotation
• Separable algebras in multitensor C${}^{\ast }$-categories are unitarizable