scholarly journals The Schrödinger-Virasoro Lie algebra: a mathematical structure between conformal field theory and non-equilibrium dynamics

2006 ◽  
Vol 40 ◽  
pp. 156-162 ◽  
Author(s):  
Jérémie Unterberger
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Mathematical Structure ◽  
Conformal Field ◽  
Equilibrium Dynamics ◽  
Non Equilibrium

scholarly journals Inhomogeneous Conformal Field Theory Out of Equilibrium

2021 ◽  
Author(s):  
Per Moosavi
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Charge Transport ◽  
Chemical Potential ◽  
Momentum Tensor ◽  
Translation Invariance ◽  
Full Time ◽  
Conformal Field ◽  
Equilibrium Dynamics ◽  
Non Equilibrium

AbstractWe study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity v(x) explicitly breaking translation invariance. Such inhomogeneous CFT is argued to effectively describe 1+1-dimensional quantum many-body systems with certain inhomogeneities varying on mesoscopic scales. Both heat and charge transport are studied, where, for concreteness, we suppose that our CFT has a conserved U(1) current. Based on projective unitary representations of diffeomorphisms and smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states defined by smooth inverse-temperature and chemical-potential profiles $$\beta (x)$$ β ( x ) and $$\mu (x)$$ μ ( x ) . Using this recipe, the following exact analytical results are obtained: (i) the full time evolution of densities and currents for heat and charge transport, (ii) correlation functions for components of the energy–momentum tensor and the U(1) current as well as for any primary field, and (iii) the thermal and electrical conductivities. The latter are computed by direct dynamical considerations and alternatively using a Green–Kubo formula. Both give the same explicit expressions for the conductivities, which reveal how inhomogeneous dynamics opens up the possibility for diffusion as well as implies a generalization of the Wiedemann–Franz law to finite times within CFT.

scholarly journals LINEARIZED FORM OF THE GENERIC AFFINE-VIRASORO ACTION

1994 ◽  
Vol 09 (14) ◽  
pp. 2451-2466 ◽  
Author(s):  
J. DE BOER ◽  
K. CLUBOK ◽  
M.B. HALPERN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
World Sheet ◽  
Conformal Field ◽  
Highly Nonlinear

Halpern and Yamron have given a Lorentz, conformal, and Diff S2-invariant world-sheet action for the generic irrational conformal field theory, but the action is highly nonlinear. In this paper, we introduce auxiliary fields to find an equivalent linearized form of the action, which shows in a very clear way that the generic affine-Virasoro action is a Diff S2-gauged WZW model. In particular, the auxiliary fields transform under Diff S2 as local Lie g× Lie g connections, so that the linearized affine-Virasoro action bears an intriguing resemblance to the usual (Lie algebra) gauged WZW model.

scholarly journals RENORMALIZED HIGHER POWERS OF WHITE NOISE (RHPWN) AND CONFORMAL FIELD THEORY

2006 ◽  
Vol 09 (03) ◽  
pp. 353-360 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANDREAS BOUKAS
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
White Noise ◽  
Lie Algebra ◽  
Noise Analysis ◽  
Quantum Probability ◽  
Conformal Field ◽  
Quantization Of Gravity ◽  
Renormalization Technique ◽  
Satisfactory Theory

The Virasoro–Zamolodchikov *-Lie algebra w∞ has been widely studied in string theory and in conformal field theory, motivated by the attempts of developing a satisfactory theory of quantization of gravity. The renormalized higher powers of quantum white noise (RHPWN) *-Lie algebra has been recently investigated in quantum probability, motivated by the attempts to develop a nonlinear generalization of stochastic and white noise analysis. We prove that, after introducing a new renormalization technique, the RHPWN Lie algebra includes a second quantization of the w∞ algebra. Arguments discussed at the end of this note suggest the conjecture that this inclusion is in fact an identification.

Lie algebra extensions of current algebras on S3

2015 ◽  
Vol 12 (09) ◽  
pp. 1550087 ◽  
Author(s):  
Tosiaki Kori ◽  
Yuto Imai
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Current Algebra ◽  
Central Extension ◽  
Mathematical Tool ◽  
Two Dimensional ◽  
Root Space ◽  
Moody Algebra ◽  
Conformal Field

An affine Kac–Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac–Moody algebras give it for two-dimensional conformal field theory.

scholarly journals GEOMETRIC CONSTRUCTION OF MODULAR FUNCTORS FROM CONFORMAL FIELD THEORY

2007 ◽  
Vol 16 (02) ◽  
pp. 127-202 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN ◽  
KENJI UENO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Fractional Power ◽  
Geometric Construction ◽  
Simple Lie Algebra ◽  
Conformal Field ◽  
Abelian Theory ◽  
Modular Functor

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [16, 19] by a certain fractional power of the abelian theory first considered in [13] and further studied in [2].

scholarly journals INTRODUCTION TO VERTEX ALGEBRAS, BORCHERDS ALGEBRAS AND THE MONSTER LIE ALGEBRA

1993 ◽  
Vol 08 (31) ◽  
pp. 5441-5503 ◽  
Author(s):  
REINHOLD W. GEBERT
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Point Of View ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Conformal Field ◽  
Formal Calculus ◽  
Developing Area ◽  
Definition Of

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory In this context Borcherds algebras arise as certain “physical” subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction to this rapidly developing area of mathematics. Based on the machinery of formal calculus, we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analyzed from the point of view of symmetry in quantum theory and the construction of the monster Lie algebra is sketched.

scholarly journals Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$

2009 ◽  
Vol 16 (04) ◽  
pp. 549-566 ◽  
Author(s):  
Shoulan Gao ◽  
Cuipo Jiang ◽  
Yufeng Pei
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Statistical Physics ◽  
Two Dimensional ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Infinite Dimensional ◽  
Conformal Field

We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

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