L’algèbre des symétries quantiques d’Ocneanu 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)

Introduction

2d CFT Quantum 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.

1998 … Zuber, Petkova : interpreted in CFT language as part. funct. of systems with defect lines

Plan

• 2d CFT and partition functions

• From graphs to partition functions

• Weak hopf algebra aspects

• Open problems

2d CFT

and

partition functions

Set of coefficients

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

• 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

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

Boundary conditions and defect lines

Boundary conditions labelled by a,b

Defect lines labelled by x,y

matrices Fi

Fi representation of fusion algebra

Matrices Wij or Wxy

Wij representation of square fusion algebra

x = y = 0

• 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)

From graphs

to

partition functions

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)

b) SU(3)

Irreps (i) 1 identity, 3 e 3 generators

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

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

• 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

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

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

Realization of the algebra of quantum symmetries

Exemple: E6 case of ``su(2)´´

G = E6 A(G) = A11

Order of verticesAdjacency matrix

E6 is a module under action of A11

Restriction

• Matrices Fi

• Essential matrices Ea

Sub-algebra of E6 defined by modular properties

Realization of Oc(E6)

0 : identity

1, 1´ : generators

1 =

1´ =

Multiplication by generator 1 : full lines

Multiplication by generator 1´: dashed lines

..

.

.

.

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

.

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

``su(3) example´´: the case

24*24 = 576 partition functions

1 of them modular invariant

Gannon classification

Weak Hopf algebras aspects

Paths on diagrams

``su(2)´´ cases G = ADE diagram example of A3 graph

Elementary paths = succession of adjacent vertices on the graph

0 1 2A3 ( = 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

(Fn)ij : number of essential paths of length n from vertex i to vertex j

Coefficients of fusion algebra

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 0 1 2

Number of Ess. paths 3 4 3

dim(B(A3)) = 3² + 4² + 3² = 34

Algebraic structures on B

Product on B : composition of endomorphism

B as a weak Hopf algebra

B vector space <B,B*> C B* dual

<< , >> scalar product

product

coproduct

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)

• 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