{
  "id": "hep-th/9908036",
  "version": "v3",
  "published": "1999-08-04T18:20:20.000Z",
  "updated": "2000-04-13T02:51:04.000Z",
  "title": "Boundary Conditions in Rational Conformal Field Theories",
  "authors": [
    "Roger E. Behrend",
    "Paul A. Pearce",
    "Valentina B. Petkova",
    "Jean-Bernard Zuber"
  ],
  "comment": "71 pages. Minor changes with respect to 2nd version. Recently published in Nucl.Phys.B but mistakenly as 1st version. Will be republished in Nucl.Phys.B as this (3rd) version",
  "journal": "Nucl.Phys.B570:525-589,2000; Nucl.Phys.B579:707-773,2000",
  "doi": "10.1016/S0550-3213(99)00592-1",
  "categories": [
    "hep-th"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2000-04-13T02:51:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational conformal field theories",
      "conformal boundary conditions",
      "verlinde algebra",
      "relevant algebraic structures",
      "cylinder partition functions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier",
      "journal": "Nucl. Phys. B"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 505088
    }
  }
}