scholarly journals On the Real Part of a Conformal Field Theory

Universe ◽  
2018 ◽  
Vol 4 (9) ◽  
pp. 97
Author(s):  
Doron Gepner ◽  
Hervé Partouche
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Coulomb Gas ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Exceptional Holonomy ◽  
General Method

Every conformal field theory has the symmetry of taking each field to its adjoint. We consider here the quotient (orbifold) conformal field theory obtained by twisting with respect to this symmetry. A general method for computing such quotients is developed using the Coulomb gas representation. Examples of parafermions, S U ( 2 ) current algebra and the N = 2 minimal models are described explicitly. The partition functions and the dimensions of the disordered fields are given. This result is a tool for finding new theories. For instance, it is of importance in analyzing the conformal field theories of exceptional holonomy manifolds.

scholarly journals BITS AND PIECES IN LOGARITHMIC CONFORMAL FIELD THEORY

2003 ◽  
Vol 18 (25) ◽  
pp. 4497-4591 ◽  
Author(s):  
MICHAEL A. I. FLOHR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Hall ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
Logarithmic Conformal Field Theory ◽  
Verlinde Formula

These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. The two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.

scholarly journals JORDAN CELLS IN LOGARITHMIC LIMITS OF CONFORMAL FIELD THEORY

2007 ◽  
Vol 22 (01) ◽  
pp. 67-82 ◽  
Author(s):  
JØRGEN RASMUSSEN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Field Theories ◽  
General Construction ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Arbitrary Rank ◽  
Jordan Cells ◽  
Virasoro Characters

It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of minimal models in conformal field theory. Characters of quasirational representations are found to emerge as the limits of the associated irreducible Virasoro characters.

scholarly journals Wronskian indices and rational conformal field theories

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Arpit Das ◽  
Chethan N. Gowdigere ◽  
Jagannath Santara
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Partial Classification ◽  
Level 1 ◽  
Linear Differential ◽  
Full Classification

Abstract The classification scheme for rational conformal field theories, given by the Mathur-Mukhi-Sen (MMS) program, identifies a rational conformal field theory by two numbers: (n, l). n is the number of characters of the rational conformal field theory. The characters form linearly independent solutions to a modular linear differential equation (which is also labelled by (n, l)); the Wronskian index l is a non-negative integer associated to the structure of zeroes of the Wronskian.In this paper, we compute the (n, l) values for three classes of well-known CFTs viz. the WZW CFTs, the Virasoro minimal models and the $$ \mathcal{N} $$ N = 1 super-Virasoro minimal models. For the latter two, we obtain exact formulae for the Wronskian indices. For WZW CFTs, we get exact formulae for small ranks (upto 2) and all levels and for all ranks and small levels (upto 2) and for the rest we compute using a computer program. We find that any WZW CFT at level 1 has a vanishing Wronskian index as does the $$ {\hat{\mathbf{A}}}_{\mathbf{1}} $$ A ̂ 1 CFT at all levels. We find intriguing coincidences such as: (i) for the same level CFTs with $$ {\hat{\mathbf{A}}}_{\mathbf{2}} $$ A ̂ 2 and $$ {\hat{\mathbf{G}}}_{\mathbf{2}} $$ G ̂ 2 have the same (n, l) values, (ii) for the same level CFTs with $$ {\hat{\mathbf{B}}}_{\mathbf{r}} $$ B ̂ r and $$ {\hat{\mathbf{D}}}_{\mathbf{r}} $$ D ̂ r have the same (n, l) values for all r ≥ 5.Classifying all rational conformal field theories for a given (n, l) is one of the aims of the MMS program. We can use our computations to provide partial classifications. For the famous (2, 0) case, our partial classification turns out to be the full classification (achieved by MMS three decades ago). For the (3, 0) case, our partial classification includes two infinite series of CFTs as well as fifteen “discrete” CFTs; except three all others have Kac-Moody symmetry.

scholarly journals Rational conformal field theory with matrix level and strings on a torus

2014 ◽  
Vol 92 (1) ◽  
pp. 65-70 ◽  
Author(s):  
Ali Nassar ◽  
Mark A. Walton
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Complex Multiplication ◽  
Field Theories ◽  
Partition Functions ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Fractional Quantum Hall ◽  
Conformal Field ◽  
The Matrix

Study of the matrix-level affine algebra Um,K is motivated by conformal field theory and the fractional quantum Hall effect. Gannon completed the classification of Um,K modular-invariant partition functions. Here we connect the algebra U2,K to strings on 2-tori describable by rational conformal field theories. As Gukov and Vafa proved, rationality selects the complex-multiplication tori. We point out that the rational conformal field theories describing strings on complex-multiplication tori have characters and partition functions identical to those of the matrix-level algebra Um,K. This connection makes it obvious that the rational theories are dense in the moduli space of strings on Tm, and may prove useful in other ways.

Conformal Field Theories with Extended Symmetries

2020 ◽  
pp. 476-517
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Lie Algebras ◽  
Lattice Models ◽  
Field Theories ◽  
Conformal Transformations ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Field ◽  
Conformal Models

The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, Z N transformations and current algebras. It also covers superconformal models, the Neveu–Schwarz and Ramond sectors, irreducible representations and minimal models, additional symmetry, the supersymmetric Landau–Ginzburg theory, parafermion models, the relation to lattice models, Kac–Moody algebras, Virasoro operators, the Sugawara Formula, maximal weights and conformal models as cosets. The appendix provides for the interested reader a self-contained discussion on the Lie algebras, include the dual Coxeter numbers, properties of weight vectors and roots/simple roots.

scholarly journals Parafermionization, bosonization, and critical parafermionic theories

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Yuan Yao ◽  
Akira Furusaki
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Degrees Of Freedom ◽  
Lattice Models ◽  
Partition Functions ◽  
Minimal Models ◽  
Fermionic Model ◽  
Conformal Field ◽  
Critical Theories ◽  
Wigner Transformation

AbstractWe formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.

SOME LANDAU-GINZBURG MODELS FROM CONFORMAL FIELD THEORY

1990 ◽  
Vol 05 (12) ◽  
pp. 2343-2358 ◽  
Author(s):  
KEKE LI
Keyword(s):  
Field Theory ◽  
Fixed Point ◽  
Conformal Field Theory ◽  
Current Algebra ◽  
Operator Algebra ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Coset Models ◽  
Marginal Deformation

A method of constructing critical (fixed point) Landau-Ginzburg action from operator algebra is applied to several classes of conformal field theories, including lines of c = 1 models and the coset models based on SU(2) current algebra. For the c = 1 models, the Landau-Ginzberg potential is argued to be physically consistent, and it resembles a modality-one singularity with modal deformation representing exactly the marginal deformation. The potentials for the coset models manifestly possess correct discrete symmetries.

scholarly journals TWO-DIMENSIONAL FRACTIONAL SUPERSYMMETRIC CONFORMAL FIELD THEORIES AND THE TWO-POINT FUNCTIONS

2001 ◽  
Vol 16 (12) ◽  
pp. 2165-2173 ◽  
Author(s):  
FARDIN KHEIRANDISH ◽  
MOHAMMAD KHORRAMI
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Closed Subalgebra

A general two-dimensional fractional supersymmetric conformal field theory is investigated. The structure of the symmetries of the theory is studied. Then, applying the generators of the closed subalgebra generated by (L-1,L0,G-1/3) and [Formula: see text], the two-point functions of the component fields of supermultiplets are calculated.

scholarly journals CHARACTERS FOR COSET CONFORMAL FIELD THEORIES AND MAVERICK EXAMPLES

1993 ◽  
Vol 08 (23) ◽  
pp. 4103-4121 ◽  
Author(s):  
DAVID C. DUNBAR ◽  
KEITH G. JOSHI
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Current Method ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field

We present an example of a coset conformal field theory which cannot be described by the identification current method. To study such examples we determine formulae for the characters of coset conformal field theories.

scholarly journals Logarithmic Operators in Conformal Field Theory and the ${\mathcal W}_\infty$-Algebra

1997 ◽  
Vol 12 (21) ◽  
pp. 3723-3738 ◽  
Author(s):  
A. Shafiekhani ◽  
M. R. Rahimi Tabar
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Operator Product Expansion ◽  
Field Theories ◽  
Operator Product ◽  
Tensor Operator ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Point Correlation

It is shown explicitly that the correlation functions of conformal field theories (CFT) with the logarithmic operators are invariant under the differential realization of Borel subalgebra of [Formula: see text]-algebra. This algebra is constructed by tensor-operator algebra of differential representation of ordinary [Formula: see text]. This method allows us to write differential equations which can be used to find general expression for three- and four-point correlation functions possessing logarithmic operators. The operator product expansion (OPE) coefficients of general logarithmic CFT are given up to third level.

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