Vertex algebras and conformal field theory models in four dimensions

2006 ◽  
Vol 54 (5-6) ◽  
pp. 496-504 ◽  
Author(s):  
I. Todorov
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Vertex Algebras ◽  
Four Dimensions ◽  
Conformal Field

scholarly journals INTEGRABLE CONFORMAL FIELD THEORY IN FOUR DIMENSIONS AND FOURTH-RANK GEOMETRY

1993 ◽  
Vol 02 (04) ◽  
pp. 413-429 ◽  
Author(s):  
VICTOR TAPIA
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Critical Dimension ◽  
Conformal Killing ◽  
Infinitely Many Solutions ◽  
Four Dimensions ◽  
Conformal Field ◽  
Killing Equation ◽  
Associated Operators ◽  
Line Elements

We consider the conformal properties of geometries described by higher-rank line elements. A crucial role is played by the conformal Killing equation (CKE). We introduce the concept of null-flat spaces in which the line element can be written as dsr=r!dζ1 … dζr. We then show that, for null-flat spaces, the critical dimension, for which the CKE has infinitely many solutions, is equal to the rank of the metric. Therefore, in order to construct an integrable conformal field theory in four dimensions we need to rely on fourth-rank geometry. We consider the simple model ℒ = ¼ Gμνλρ ∂μ ϕ ∂ν ϕ ∂λ ϕ ∂ρϕ and show that it is an integrable conformal model in four dimensions. Furthermore, the associated operators satisfy a Vir4 algebra.

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 On the Trace Anomaly of the Chaudhuri–Choi–Rabinovici Model

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 276
Author(s):  
Yu Nakayama
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Four Dimensions ◽  
Conformal Field ◽  
Trace Anomaly ◽  
Large N ◽  
Marginal Deformation

Recently a non-supersymmetric conformal field theory with an exactly marginal deformation in the large N limit was constructed by Chaudhuri–Choi–Rabinovici. On a non-supersymmetric conformal manifold, the c coefficient of the trace anomaly in four dimensions would generically change. In this model, we, however, find that it does not change in the first non-trivial order given by three-loop diagrams.

scholarly journals CONFORMAL FIELD THEORY ON R × S3 FROM QUANTIZED GRAVITY

2009 ◽  
Vol 24 (16n17) ◽  
pp. 3073-3110 ◽  
Author(s):  
KEN-JI HAMADA
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Conformal Transformation ◽  
Conformal Symmetry ◽  
Conformal Factor ◽  
Combined System ◽  
Four Dimensions ◽  
Gravitational Fields ◽  
Conformal Field ◽  
Gauge Fixing

Conformal algebra on R × S3 derived from quantized gravitational fields is examined. The model we study is a renormalizable quantum theory of gravity in four dimensions described by a combined system of the Weyl action for the traceless tensor mode and the induced Wess–Zumino action managing nonperturbative dynamics of the conformal factor in the metric field. It is shown that the residual diffeomorphism invariance in the radiation+ gauge is equal to the conformal symmetry, and the conformal transformation preserving the gauge-fixing condition that forms a closed algebra quantum mechanically is given by a combination of naive conformal transformation and a certain field-dependent gauge transformation. The unitarity issue of gravity is discussed in the context of conformal field theory. We construct physical states by solving the conformal invariance condition and calculate their scaling dimensions. It is shown that the conformal symmetry mixes the positive-metric and the negative-metric modes and thus the negative-metric mode does not appear independently as a gauge invariant state at all.

Rational Conformal Field Theory In Four Dimensions

2002 ◽  
pp. 91-104
Author(s):  
Nikolay M. Nikolov ◽  
Yassen S. Stanev ◽  
Ivan T. Todorov
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Four Dimensions ◽  
Conformal Field

scholarly journals N=1 SUPERCONFORMAL SYMMETRY IN FOUR DIMENSIONS

1998 ◽  
Vol 13 (11) ◽  
pp. 1743-1772 ◽  
Author(s):  
JEONG-HYUCK PARK
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Green Functions ◽  
Two Dimensional ◽  
Superconformal Symmetry ◽  
Four Dimensions ◽  
Conformal Field ◽  
Explicit Formulae

The N=1, d=4 superconformal group is studied and its representations are discussed. Under superconformal transformations, left invariant derivatives and some class of superfields, including supercurrents, are shown to follow these representations. In other words, these superfields are quasiprimary by analogy with two-dimensional conformal field theory. Based on these results, we find the genenal forms for the two-point and the three-point Green functions of the quasiprimary superfields in a group theoretical way. In particular, we prove that the two-point and the three-point Green functions of supercurrents are unique and present the explicit formulae of them.

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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