Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions

Friedan, Daniel; Qiu, Zhihai; Shenker, Stephen H.

doi:10.1103/physrevlett.52.1575

Cited by 1,118 publications

(751 citation statements)

References 9 publications

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“…The scaling properties of the antiferromagnetic XXZ vacuum with κ = 1 yield the central charges of the minimal models [22] belonging to the unitary series of Friedan Qiu and Shenker [23].…”

Section: Introductionmentioning

confidence: 99%

Connections of the Liouville model and XXZ spin chain

Faddeev

Tirkkonen

1995

Nuclear Physics B

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The quantum theory of the Liouville model with imaginary field is considered using the Quantum Inverse Scattering Method. An integrable structure with nontrivial spectral parameter dependence is developed for lattice Liouville theory by scaling the L-matrix of lattice sine-Gordon theory. This L-matrix yields Bethe Ansatz equations for Liouville theory, by the methods of the algebraic Bethe Ansatz. Using the string picture of exited Bethe states, the lattice Liouville Bethe equations are mapped to the corresponding spin 1 2 XXZ chain equations. The well developed theory of finite size corrections in spin chains is used to deduce the conformal properties of the lattice Liouville Bethe states. The unitary series of conformal field theories emerge for Liouville couplings of the form γ = π ν ν+1 , corresponding to root of unity XXZ anisotropies. The Bethe states give the full spectrum of the corresponding unitary conformal field theory, with the primary states in the Kač table parameterized by a string length K, and the remnant of the chain length mod (ν + 1). *

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Section: Introductionmentioning

confidence: 99%

Connections of the Liouville model and XXZ spin chain

Faddeev

Tirkkonen

1995

Nuclear Physics B

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“…En 1984, les physiciens D. Friedan, Z. Qiu et S. Shenker ont énoncé une condition nécessaire d'unitarisabilité des modules de Verma sur l'algèbre de Virasoro [3]. Ils ont fait la même recherche pour les super-algèbres de Virasoro N = 1 [4] R. Langlands [8] a démontré le théorème énoncé par [3] en 1988.…”

Section: Introductionunclassified

“…Ils ont fait la même recherche pour les super-algèbres de Virasoro N = 1 [4] R. Langlands [8] a démontré le théorème énoncé par [3] en 1988. Suivant les idées de [8], F. Sauvageot [11] a démontré celui de [4] en 1989.…”

Section: Introductionunclassified

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Modules de plus haut poids unitarisables sur la super-algèbre de Virasoro N=2 tordue

Iohara

2008

Ann. inst. Fourier

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“…• this was answered in part in the work of Friedan, Qiu and Shenker [3], who showed that in theories having reflection positivity (eg. Q = 2, 3, 4 Potts models, or n = 1, 2 in the O(n) model), we should look for unitary representations, and at least when c < 1…”

Section: Conformal Field Theorymentioning

confidence: 99%

Conformal Invariance in Percolation, Self-Avoiding Walks, and Related Problems

Cardy¹

2003

Ann. Henri Poincaré

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Over the years, problems like percolation and self-avoiding walks have provided important testing grounds for our understanding of the nature of the critical state. I describe some very recent ideas, as well as some older ones, which cast light both on these problems themselves and on the quantum field theories to which they correspond. These ideas come from conformal field theory, Coulomb gas mappings, and stochastic Loewner evolution.This talk is about 'geometric' critical phenomena. These are random spatial processes, where either (1) the probability distribution is determined by equilibrium statistical mechanics, and we ask questions about geometrical properties, or (2) the probability distribution is itself geometrical in nature. The simplest example of (1) is clustering in percolation (see Fig. 1), in which the probability distribution * Plenary talk given at

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Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions

Abstract: Conformal invariance and unitarity severely limit the possible values of critical exponents in two-dimensional systems.

Cited by 1,118 publications

References 9 publications

Connections of the Liouville model and XXZ spin chain

Connections of the Liouville model and XXZ spin chain

Modules de plus haut poids unitarisables sur la super-algèbre de Virasoro N=2 tordue

Conformal Invariance in Percolation, Self-Avoiding Walks, and Related Problems

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