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Lalgèbre des symétries quantiques dOcneanu et la classification des systèms conformes à 2D Gil Schieber Directeurs : R. Coquereaux R. Amorim (J. A. Mignaco)

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

2 Introduction 2d CFTQuantum Symmetries Classification of partition functions 1987: Cappelli-Itzykson-Zuber modular invariant of affine su(2) 1994: Gannon modular invariant of affine su(3) Algebra of quantum symmetries of diagrams (Ocneanu) Ocneanu graphs From unity, we get classification of modular invariants partition functions Other points generalized part. funct … 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: Classification problem Find matrices M such that: Caracteres of affine su(n) algebra

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 Defect lines labelled by x,y matrices F i Fi representation of fusion algebra Matrices W ij or W xy Wij representation of square fusion algebra x = y = 0

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

10 From graphs to partition functions

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

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

13 II. Quantum case Lie groupsQuantum 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) Graph algebra Fusion algebra of CFT h = Coxeter number of SU(n) = gen. Coxeter number of

15 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) Search of graph G (vertices ) such that: (Generalized) Coxeter-Dynkin graphs G Fix graph vertices norm = max. eigenvalue of adjacency matrix Partition functions of models with boundary conditions a,b

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18 Ocneanu graph Oc(G) To each generalized Dynkin graph G Ocneanu graph Oc(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

19 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

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

21 E6 is a module under action of A11 Restriction Matrices Fi Essential matrices Ea

22 Sub-algebraof E6 defined by modular properties

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

24 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 Action of A(G) on Oc(G) Partition functions of models with defect lines and modular invariant Partition functions with defect lines x,y Modular invariant : x = y = 0.

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

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

27 Weak Hopf algebras aspects

28 Paths on diagrams ``su(2)´´ cases G = ADE diagram example of A3 graph Elementary paths = succession of adjacent vertices on the graph 0 12 A3 ( = 4 ) : 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 Essential paths : paths kernel of Jones projectors n Theorem [Ocneanu] No essential paths with length bigger than - 2 ( F n ) ij : number of essential paths of length n from vertex i to vertex j Coefficients of fusion algebra

29 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 Elements of B A3 length 012 Number of Ess. paths 343 dim(B(A3)) = 3² + 4² + 3² = 34

30 Algebraic structures on B Product on B : composition of endomorphism B as a weak Hopf algebra B vector space C B* dual > scalar product product coproduct

31 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)

32 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. open problems


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