Introduction to Conformal Field Theory and Infinite Dimensional Algebras

1990 ◽  
pp. 241-261
Author(s):  
David Olive
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Infinite Dimensional ◽  
Conformal Field

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.

2D SIGMA MODEL APPROACH TO 4D INSTANTONS

1992 ◽  
Vol 07 (07) ◽  
pp. 1415-1447 ◽  
Author(s):  
Q-HAN PARK
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Sigma Models ◽  
Sigma Model ◽  
Gauge Groups ◽  
Yang Mills ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Toda Field Theory ◽  
Model Approach

4D self-dual theories are proposed to generalize 2D conformal field theory. We identify 4D self-dual gravity as well as self-dual Yang-Mills theory with 2D sigma models valued in infinite-dimensional gauge groups. It is shown that these models possess infinite-dimensional symmetries with associated algebras—“CP1 extensions” of respective gauge algebras of 2D sigma models—which generalize the Kac-Moody algebra as well as W∞. We address various issues concerning 2D sigma models, twistors and sheaf cohomology. An attempt to connect 4D self-dual theories with 2D conformal field theory is made through sl (∞) Toda field theory.

EXACT SOLUTIONS OF CONFORMAL FIELD THEORY IN TWO DIMENSIONS AND CRITICAL PHENOMENA

1989 ◽  
Vol 01 (02n03) ◽  
pp. 197-234 ◽  
Author(s):  
A. B. ZAMOLODCHIKOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Critical Phenomena ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Minimal Models ◽  
Infinite Dimensional ◽  
Superconformal Field Theory ◽  
Conformal Field ◽  
Modern Development

Modern development of conformal field theory in two dimensions and its applications to critical phenomena are briefly reviewed. The specific properties of the renormalization group in two dimensions and the fundamentals of 2-dimensional conformal field theory are presented. The properties of degenerate representations of the Virasoro algebra and other infinite dimensional algebras, "minimal" models of conformal and superconformal field theory, "parafermionic" and other symmetries are discussed. We also investigate a perturbation theory around conformal solutions of field theory.

INFINITE DIMENSIONAL LIE ALGEBRAS IN 4D CONFORMAL FIELD THEORY

2008 ◽  
Vol 05 (08) ◽  
pp. 1361-1371
Author(s):  
IVAN TODOROV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Recent Work ◽  
Lie Algebras ◽  
Scalar Fields ◽  
Operator Product ◽  
Positive Energy ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Compact Gauge Group

It is known that there are no scalar Lie fields in more than two space-time dimensions [4]. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. Recent work, [2, 3], is reviewed, in which we classify such algebras and their unitary positive energy representations in a theory of a system of scalar fields of dimension two. The results are linked to the Doplicher–Haag–Roberts theory of superselection sectors governed by a (global) compact gauge group.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

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