Conformal Correlation Functions, Frobenius Algebras and Triangulations
arXiv:hep-th/0110133 · doi:10.1016/S0550-3213(01)00638-1
Abstract
We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the Moore-Seiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of T-duality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIM-reps of the fusion rules, respectively.
17 pages, LaTeX2e; v2: more references and Note added in proof
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- Kramers-Wannier duality from conformal defects
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- Correspondences of ribbon categories
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- Boson Condensation in Topologically Ordered Quantum Liquids
- Lectures on Branes in Curved Backgrounds
- Full field algebras
- Non-compact String Backgrounds and Non-rational CFT
- Open-string vertex algebras, tensor categories and operads
- Invertible defects and isomorphisms of rational CFTs
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- Anomalies and entanglement renormalization
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