Algèbre de Virasoro

Résumé

L′algèbre de Virasoro est une algèbre de Lie complexe de dimension infinie qui joue un rôle essentiel dans certaines théories physiques, notamment en théorie des cordes, et d'une manière générale dans les théories conformes des champs, ainsi qu'en mathématiques dans l'étude du groupe Monstre (au travers du module moonshine) et des algèbres vertex. Elle tient son nom du physicien argentin qui les a introduit en théorie des cordes en 1970. Définition L'algèbre de Virasoro \mathrm{Vir} s'obtient comme extension centrale de l' complexe : :\mathcal D = \bigoplus_{s \in\Z} d_s munie des opérateurs de dérivation :d_s = - z^{n+1} \frac{\partial}{\partial z}. On a la suite exacte courte : :0 \to \mathbb Cc \to \mathrm{Vir} \to \mathcal D\to 0 avec c la , le crochet de Lie associé étant : :\left[ d_i, d_j \right] = (i - j) d_{i+j} + \frac1{12}(i^3 - i)\delta_{i,-j} c. Le facteur 1/12 est conventionnel,

Source officielle

https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Virasoro

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Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

Tudor Ratiu

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 -245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3 ary bracket. Further, we derive the so-called rho compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.

Taylor & Francis Ltd2016

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Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333-380, 1984). Both exhibit exactly solvable structures in two dimensions. A long-standing question (Itoyama and Thacker in Phys Rev Lett 58:1395-1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771-833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191-225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087-1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.

SPRINGER2022

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Lattice Representations of the Virasoro Algebra I : Discrete Gaussian Free Field

Clément Hongler

Most two-dimensional massless field theories carry represe ntations of the Virasoro algebra as consequences of their conformal symmetry. Recently, conformal symmetry has been rigorously established for scaling limit s of lattice models by means of discrete complex analysis, which efficiently expresses th e integrability of these models. In this paper we study the discrete Gaussian free field on the square grid. We show that the lattice integrability of this model gives explicit representations of the Virasoro algebra acting on the Gibbs measures of the model. Thus, some what surprisingly, the algebraic structure of Conformal Field Theory describing the scaling limit of the model is already present on lattice level.

2015

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