Topic
Conformal map
About: Conformal map is a research topic. Over the lifetime, 10514 publications have been published within this topic receiving 194215 citations. The topic is also known as: conformal & angle-preserving.
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TL;DR: In this article, a correspondence between conformal field theory observables and those of supergravity was proposed, where correlation functions in conformal fields are given by the dependence of the supergravity action on the asymptotic behavior at infinity.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.
8,751 citations
TL;DR: In this paper, the authors conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of SCFTs recently defined by one of the authors.
Abstract: We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of \({\mathcal{N}=2}\) SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.
1,881 citations
Book•
01 Jan 1997
TL;DR: In this paper, the authors consider the effects of an external field (or weight) on the minimum energy problem and provide a unified approach to seemingly different problems in constructive analysis, such as the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points, the existence and construction of fast decreasing polynomial, the numerical conformal mapping of simply and doubly connected domains, generalization of the Weierstrass approximation theorem to varying weights, and the determination of convergence rates for best approximating rational functions.
Abstract: This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions.
1,560 citations
TL;DR: Based on the conformal algebra approach, a general technique for the calculation of multipoint correlation functions in 2D statistical models at the critical point is given in this article, where particular conformal operator algebras are found for operators of the 2D q-component Potts model.
1,317 citations
TL;DR: This work attempts to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations, and constructs a family of metrics which realize these symmetries as isometries.
Abstract: We attempt to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations. Such systems govern ultracold atoms at unitarity, nucleon scattering in some channels, and, more generally, a family of universality classes of quantum critical behavior. We construct a family of metrics which realize these symmetries as isometries. They are solutions of gravity with a negative cosmological constant coupled to pressureless dust. We discuss realizations of the dust, which include a bulk superconductor. We develop the holographic dictionary and find two-point correlators of the correct form. A strange aspect of the correspondence is that the bulk geometry has two extra noncompact dimensions.
1,010 citations