Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1999-08-04
Nucl.Phys.B570:525-589,2000; Nucl.Phys.B579:707-773,2000
Physics
High Energy Physics
High Energy Physics - Theory
71 pages. Minor changes with respect to 2nd version. Recently published in Nucl.Phys.B but mistakenly as 1st version. Will be
Scientific paper
10.1016/S0550-3213(99)00592-1
We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph $G$ to each RCFT such that the conformal boundary conditions are labelled by the nodes of $G$. This approach is carried to completion for $sl(2)$ theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the $A$-$D$-$E$ classification. We also review the current status for WZW $sl(3)$ theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.
Behrend Roger E.
Pearce Paul A.
Petkova Valentina B.
Zuber Jean-Bernard
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