Download Partition functions

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
Classification of partition functions
• 1987: Cappelli-Itzykson-Zuber
modular invariant of affine su(2)
• 1994: Gannon
modular invariant of affine su(3)
Quantum Symmetries
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
finite number of representations
affine su(n)
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:
Caracteres of affine su(n) algebra
Classification problem
Find matrices M such that:
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
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
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
From graphs
to
partition functions
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)
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)
h = Coxeter number of SU(n)
 = gen. Coxeter number of
Graph algebra  Fusion algebra of CFT
(Generalized) Coxeter-Dynkin graphs G
Fix graph
vertices 
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
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)´´
A(G) = A11
G = E6
Order of vertices
Adjacency matrix
E6 is a module under action of A11
• Matrices Fi
• Essential matrices Ea
Restriction
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
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
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
24*24 = 576 partition functions
1 of them modular invariant
Gannon classification
case
Weak Hopf
algebras aspects
Paths on diagrams
``su(2)´´ cases
A3 ( = 4)
G = ADE diagram
0
1
example of A3 graph
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
Endomorphism of essential paths
H = vector space of essential paths
H
graded by length
finite dimensional
Essential path of length i from vertex a to vertex b
B = vector space of graded endomorphism of essential paths
A3
length
Elements of B
Number of Ess. paths
0
1
2
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
product

coproduct
<< , >> scalar product
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)
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.