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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
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