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TFT CONSTRUCTION OF RCFT CORRELATORS II: Unoriented World Sheets
, 2003
"... A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that th ..."
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Cited by 20 (6 self)
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A full rational CFT, consistent on all orientable world sheets, can be constructed from the underlying chiral CFT, i.e. a vertex algebra, its representation category C, and the system of chiral blocks, once we select a symmetric special Frobenius algebra A in the category C [I]. Here we show that the construction of [I] can be extended to unoriented world sheets by specifying one additional datum: a reversion σ on A – an isomorphism from the opposed algebra of A to A that squares to the twist. A given full CFT on oriented surfaces can admit inequivalent reversions, which give rise to different amplitudes on unoriented surfaces, in particular to different Klein bottle amplitudes. We study the classification of reversions, work out the construction of the annulus, Möbius strip and Klein bottle partition functions, and discuss properties of defect lines on nonorientable world sheets. As an illustration, the Ising model is treated in detail.
Integrable lattice realizations of conformal twisted boundary conditions
"... We construct integrable lattice realizations of conformal twisted boundary conditions for ̂ sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical ADE lattice models with positive spectral parameter. The integrable seam bound ..."
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Cited by 14 (3 self)
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We construct integrable lattice realizations of conformal twisted boundary conditions for ̂ sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical ADE lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r,s,ζ) ∈ (Ag−2,Ag−1,Γ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A,D,E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a,b,γ) ∈ (Ag−2 ⊗G,Ag−2 ⊗G,Z2) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A2,A3) and 3state Potts (A4,D4) models.
Integrable and conformal twisted boundary conditions for sl(2) ADE lattice models
, 2002
"... We study integrable realizations of conformal twisted boundary conditions for sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u> 0 and Coxeter number g. Integr ..."
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Cited by 7 (3 self)
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We study integrable realizations of conformal twisted boundary conditions for sℓ(2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical G = A,D,E lattice models with positive spectral parameter u> 0 and Coxeter number g. Integrable seams are constructed by fusing blocks of elementary local face weights. The usual Atype fusions are labelled by the Kac labels (r,s) and are associated with the Verlinde fusion algebra. We introduce a new type of fusion in the two braid limits u → ±i ∞ associated with the graph fusion algebra, and labelled by nodes a,b ∈ G respectively. When combined with automorphisms, they lead to general integrable seams labelled by x = (r,a,b,κ) ∈ (Ag−2,H,H,Z2) where H is the graph G itself for Type I theories and its parent for Type II theories. Identifying our construction labels with the conformal labels of Petkova and Zuber, we find that the integrable seams are in onetoone correspondence with the conformal seams. The distinct seams are thus associated with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted partition functions are checked numerically for G  ≤ 6. We also show, in the case of D2ℓ, that the noncommutativity of the Ocneanu algebra of seams arises because the automorphisms do not commute with the fusions.
2003 Duality and conformal twisted boundaries in the Ising model
 GROUP 24: Physical and Mathematical Aspects of Symmetries, edited by JP Gazeau, R Kerner, JP Antoine, S Métens and JY Thibon (Bristol: IOP Publishing); hepth/0209048
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oro.open.ac.uk The Bubble Algebra: Structure of a TwoColour TemperleyLieb Algebra
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and other research outputs Duality and conformal twisted boundaries in the Ising model