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Zuber: Boundary conditions in rational conformal field theories, Nucl. Phys. B570
, 2000
"... We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy’s equation, expressing consistency of a RCFT on a cy ..."
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Cited by 126 (11 self)
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We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy’s equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sℓ(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the ADE classification. We also review the current status for WZW sℓ(3) theories. Finally, a systematic generalization of the formalism of CardyLewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulkboundary coefficients and reproduce the relevant algebraic structures from the sewing constraints. 1
The electronic properties of graphene
 Rev. Mod. Phys. 2009
"... This article reviews the basic theoretical aspects of graphene, a oneatomthick allotrope of carbon, with unusual twodimensional Diraclike electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or t ..."
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Cited by 212 (1 self)
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This article reviews the basic theoretical aspects of graphene, a oneatomthick allotrope of carbon, with unusual twodimensional Diraclike electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge �surface � states in graphene depend on the edge termination �zigzag or armchair � and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electronelectron and electronphonon interactions in single layer and multilayer graphene are also
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
"... ..."
AUTHOR Campbell, Paul B.; And Others
"... An analysis assessed the effects of a high school vocational curriculum over time as labor market experience accumulates. Since two additional yerrs of labor market experience had become available for respondents to the National Longitudinal Survey of Labor Market ExperienceYouth Cohort (NLSYouth) ..."
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An analysis assessed the effects of a high school vocational curriculum over time as labor market experience accumulates. Since two additional yerrs of labor market experience had become available for respondents to the National Longitudinal Survey of Labor Market ExperienceYouth Cohort (NLSYouth) and longer trends of effects could be observed, the study replicated the exact specifications of an earlier analysis and added the dimension of expected lifetime earnings. Data were from the NLSYouth and High School and Beyond databases. Findings indicated that vocational education provided, in the short term, a direct wage advantage for vocational students. The advantage became indirect as time in the labor market accrued and appeared to operate through increased hours of work and fuller employment rather than differential wage rates. An optimum mix between vocational and academic courses in terms of
JeanPaul Bacher, JeanPierre Pansart, JeanPierre Peigneux,
, 2011
"... et autres détecteurs optiques de particules ..."
Logarithmic minimal models
"... Working in the dense loop representation, we use the planar TemperleyLieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p ′). Specifically, we construct YangBaxter integrable TemperleyLieb models on the strip acting on link states and consider their associated ..."
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Cited by 88 (25 self)
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Working in the dense loop representation, we use the planar TemperleyLieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p ′). Specifically, we construct YangBaxter integrable TemperleyLieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the TemperleyLieb algebra are inherently nonlocal and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 − 6(p−p ′ ) 2 pp ′ where p,p ′ = 1,2,... are coprime. The first few members of the principal series LM(m,m + 1) are critical dense polymers (m = 1, c=−2), critical percolation (m = 2, c=0) and logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights, r,s = 1,2,.... The associated conformal partition functions are given in terms of Virasoro characters of highestweight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion. ∆r,s = ((m+1)r−ms)2 −1
IN MEMORY OF PAUL EVAN PETERS (19471996), FOUNDING EXECUTIVE DIRECTOR OF THE COALITION FOR NETWORKED INFORMATION, WHOSE VISIONARY LEADERSHIP AT THE DAWN OF THE
, 2010
"... (before alteration) by NASA. This work is licensed under the Creative Commons AttributionNonCommercial 3.0 United States License. To view a copy of this license, visit ..."
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(before alteration) by NASA. This work is licensed under the Creative Commons AttributionNonCommercial 3.0 United States License. To view a copy of this license, visit
On the classification of bulk and boundary conformal field theories.” Phys
 Lett
, 1998
"... The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy’s equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only ..."
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Cited by 63 (8 self)
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The classification of rational conformal field theories is reconsidered from the standpoint of boundary conditions. Solving Cardy’s equation expressing the consistency condition on a cylinder is equivalent to finding integer valued representations of the fusion algebra. A complete solution not only yields the admissible boundary conditions but also gives valuable information on the bulk properties. The classification of conformal field theories (CFTs) remains an important issue, both in the study of bulk and boundary critical phenomena [1] and in string theory [2]. The guiding principle is that of consistency of the theory on an arbitrary 2D surface, with or without boundaries. To define a rational conformal field theory, one first specifies a chiral algebra A, e.g. the Virasoro algebra or one of its extensions, at a certain level. Rationality means that at this level, A has only a finite set I of admissible irreducible representations Vi, i ∈ I. We denote by Vi ∗ the representation conjugate to Vi and i = 1 refers to the vacuum representation. 1 We suppose that the characters χi(q) of these representations, the symmetric matrix Si j of
The many faces of ocneanu cells
 Nuclear Phys. B
"... We define generalised chiral vertex operators covariant under the Ocneanu “double triangle algebra” A, a novel quantum symmetry intrinsic to a given rational 2d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures e ..."
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Cited by 63 (6 self)
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We define generalised chiral vertex operators covariant under the Ocneanu “double triangle algebra” A, a novel quantum symmetry intrinsic to a given rational 2d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3jsymbols of the weak Hopf algebra A, reappear in several guises. With A and its dual algebra Â one associates a pair of graphs, G and ˜G. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs ˜G classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining Â. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.
Results 1  10
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