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L’algèbre des symétries quantiques d’Ocneanu et la classification des systèms conformes à 2D

L’algèbre des symétries quantiques d’Ocneanu et la classification des systèms conformes à 2D. Gil Schieber. IF-UFRJ (Rio de Janeiro) CPT-UP (Marseille). Directeurs : R. Coquereaux R. Amorim (J. A. Mignaco). Introduction. 2d CFT.

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L’algèbre des symétries quantiques d’Ocneanu et la classification des systèms conformes à 2D

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  1. L’algèbre des symétries quantiques d’Ocneanu et la classification des systèms conformes à 2D Gil Schieber IF-UFRJ (Rio de Janeiro) CPT-UP (Marseille) Directeurs : R. Coquereaux R. Amorim (J. A. Mignaco)

  2. Introduction 2d CFT Quantum Symmetries Classification of partition functions Algebra of quantum symmetries of diagrams (Ocneanu) • 1987: Cappelli-Itzykson-Zuber • modular invariant of affine su(2) • 1994: Gannon • modular invariant of affine su(3) Ocneanu graphs • From unity, we get classification of • modular invariants partition functions • Other points generalized part. funct. 1998  … Zuber, Petkova : interpreted in CFT language as part. funct. of systems with defect lines

  3. Plan • 2d CFT and partition functions • From graphs to partition functions • Weak hopf algebra aspects • Open problems

  4. 2d CFT and partition functions Set of coefficients

  5. 2d CFT • Conformal invariance  lots of constraints in 2d • algebra of symmetries : Virasoro ( dimensionnal) • Models with affine Lie algebra g : Vir  g affine su(n) • finite number of representations at a fixed level : RCFT • Hilbert space : • Information on CFT encoded in OPE coefficients of fields fusion algebra

  6. Geometry in 2d  torus ( modular parameter  ) • invariance under modular group SL(2,Z) Modular group generated by S, T The (modular invariant) partition function reads: Caracteres of affine su(n) algebra Classification problem Find matrices M such that:

  7. Classifications of modular invariant part. functions Affine su(2) : ADE classification by Cappelli-Itzykson-Zuber (1987) Affine su(3) : classification by Gannon (1994) 6  series , 6 exceptional cases graphs

  8. Boundary conditions and defect lines Boundary conditions labelled by a,b matrices Fi Fi representation of fusion algebra Defect lines labelled by x,y Matrices Wij or Wxy Wij representation of square fusion algebra x = y = 0

  9. Classification of partition functions Set of coefficients (non-negative integers) • They form nimreps of certain algebras • They define maps structures of a weak Hopf algebra • They are encoded in a set of graphs

  10. From graphs to partition functions

  11. Irreducible representations and graphs A I. Classical analogy a) SU(2) (n)  Irr SU(2) n = dimension = 2j+1 j = spin Graph algebra of SU(2)

  12. b) SU(3) Irreps (i) 1 identity, 3 e 3 generators

  13. II. Quantum case Lie groups Quantum groups Finite dimensional Hopf quotients Finite number of irreps graph of tensorisation Graph of tensorisation by the fundamental irrep identity Level k = 3

  14. Truncation at level k of classical graph of tensorisation of irreps of SU(n) h = Coxeter number of SU(n)  = gen. Coxeter number of Graph algebra  Fusion algebra of CFT

  15. (Generalized) Coxeter-Dynkin graphs G Fix graphvertices  norm = max. eigenvalue of adjacency matrix Search of graph G (vertices ) such that: • same norm of • vector space of vertex   G is a module under the action of the algebra • with non-negative integer coeficients • 0 .a = a 1 . a = 1 . a • Local cohomological properties (Ocneanu) Partition functions of models with boundary conditions a,b

  16. Ocneanu graph Oc(G) Ocneanu graph Oc(G) To each generalized Dynkin graph G Definition: algebraic structures on the graph G two products  and  diagonalization of the law  encoded by algebra of quantum symmetries graph Oc(G) = graph algebra Ocneanu: published list of su(2) Ocneanu graphs never obtained by explicit diagonalization of law used known clasification of modular inv. partition functions of affine su(2) models

  17. Works of Zuber et. al. , Pearce et. al., …  Ocneanu graph as an input  Method of extracting coefficients that enters definition of partition functions (modular invariant and with defect lines)  Limited to su(2) cases Our approach  Realization of the algebra of quantum symmetries Oc(G) = G J G  Coefficients calculated by the action (left-right) of the A(G) algebra on the Oc(G) algebra  Caracterization of J by modular properties of the G graph  Possible extension to su(n) cases

  18. Realization of the algebra of quantum symmetries Exemple: E6 case of ``su(2)´´ A(G) = A11 G = E6 Adjacency matrix Order of vertices

  19. E6 is a module under action of A11 • Matrices Fi • Essential matrices Ea Restriction

  20. Sub-algebra of E6 defined by modular properties

  21. Realization of Oc(E6) . . . 0 : identity 1, 1´ : generators . 1 = 1´ = . Multiplication by generator 1 : full lines Multiplication by generator 1´: dashed lines

  22. Partition functions G = E6 module under action of A(G) = A11  E6  A11 . Elements x  Oc(E6) Action of A11 (left-right ) on Oc(E6) We obtain the coefficients Partition functions with defect lines x,y Modular invariant : x = y = 0 Action of A(G) on Oc(G) Partition functions of models with defect lines and modular invariant

  23. Generalization All su(2) cases studied Cases where Oc(G) is not commutative: method not fully satisfactory Some su(3) cases studied G A(G) Oc(G) x = y = 0

  24. ``su(3) example´´: the case 24*24 = 576 partition functions 1 of them modular invariant Gannon classification

  25. Weak Hopf algebras aspects

  26. Paths on diagrams ``su(2)´´ cases G = ADE diagram example of A3 graph A3 ( = 4) 0 1 2 Elementary paths = succession of adjacent vertices on the graph : number of elementary paths of length 1 from vertex i to vertex j : number of elementary paths of length n from vertex i to vertex j n   Essential paths : paths  kernel of Jones projectors Theorem [Ocneanu] No essential paths with length bigger than  - 2 (Fn)ij : number of essential paths of length n from vertex i to vertex j Coefficients of fusion algebra

  27. Endomorphism of essential paths H = vector space of essential paths graded by length finite dimensional  H Essential path of length i from vertex a to vertex b B = vector space of graded endomorphism of essential paths A3 length 0 1 2 Elements of B Number of Ess. paths 3 4 3 dim(B(A3)) = 3² + 4² + 3² = 34

  28. Algebraic structures on B Product  on B : composition of endomorphism B as a weak Hopf algebra B vector space <B,B*>  C B* dual product << , >> scalar product  coproduct

  29. Graphs A(G) and Oc(G) (example of A3) • B(G) : vector space of graded endomorphism of essential paths • Two products  and  defined on B(G) • B(G) is semi-simple for this two algebraic structures • B(G) can be diagonalized in two  ways : sum of matrix blocks • First product  : blocks indexed by length i projectors i • Second product  : blocks indexed by label x projectors x A(G) Oc(G)

  30. open problems • Give a clear definition product  product and verify that all axioms defining a weak Hopf algebra are satisfied. • Obtain explicitly the Ocneanu graphs from the algebraic structures of B. • Study of the others su(3) cases + su(4) cases. • Conformal systems defined on higher genus surfaces.

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