ON THE CLASSIFICATION OF W-ALGEBRAS

1995 ◽  
Vol 10 (10) ◽  
pp. 1413-1448
Author(s):  
DIRK VERSTEGEN
Keyword(s):  
Central Charge ◽  
Virasoro Algebra ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Bootstrap Approach ◽  
Finite Set

We review and extend the conformal bootstrap approach to the classification of quantum W-algebras. These are extensions of the Virasoro algebra by a finite set of primary fields. Explicit forms are given for the most general crossing-symmetric four-point functions. Together with a large c expansion of the conformal blocks, this gives a powerful tool for finding all W-algebras that are associative for generic values of the central charge c.

THE CONFORMAL BOOTSTRAP AND SUPER W ALGEBRAS

1992 ◽  
Vol 07 (03) ◽  
pp. 591-617 ◽  
Author(s):  
JOSÉ M. FIGUEROA-O'FARRILL ◽  
STANY SCHRANS
Keyword(s):  
Systematic Study ◽  
Minimal Model ◽  
Central Charge ◽  
Virasoro Algebra ◽  
Symmetry Algebra ◽  
Modular Invariant ◽  
Conformal Bootstrap ◽  
Nonlinear Algebra

We undertake a systematic study of the possible extensions of the N = 1 super Virasoro algebra by a superprimary field of spin [Formula: see text]. Besides new extensions, which exist only for specific values of the central charge, we find a new nonlinear algebra (super W2) generated by a spin 2 superprimary which is associative for all values of the central charge. Furthermore, the spin 3 extension is argued to be the symmetry algebra of the m = 6 super Virasoro unitary minimal model, by exhibiting the (A7, D4)-type modular invariant as diagonal in terms of extended characters.

scholarly journals A conformal bootstrap approach to critical percolation in two dimensions

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Marco Picco ◽  
Sylvain Ribault ◽  
Raoul Santachiara
Keyword(s):  
Monte Carlo ◽  
Potts Model ◽  
Virasoro Algebra ◽  
Two Dimensions ◽  
Critical Percolation ◽  
Conformal Bootstrap ◽  
Bootstrap Approach

We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.

scholarly journals Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

2021 ◽  
Vol 11 (5) ◽  
Author(s):  
Nikita Nemkov ◽  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Theta Functions ◽  
Conformal Dimension ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Crossing Symmetry ◽  
Point Correlation ◽  
Structure Constants ◽  
Twist Fields

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

scholarly journals Towards the classification of rank-r $$ \mathcal{N} $$ = 2 SCFTs. Part I. Twisted partition function and central charge formulae

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Mario Martone
Keyword(s):  
Partition Function ◽  
Central Charge ◽  
Singular Locus ◽  
Companion Paper ◽  
Coulomb Branch ◽  
Energy Limit ◽  
Superconformal Field Theories ◽  
Rank 2 ◽  
Explicit Formulae

Abstract We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

Analysis of statistical coefficients and autoregressive parameters over intrinsic mode functions (IMFs) for epileptic seizure detection

2020 ◽  
Vol 65 (6) ◽  
pp. 693-704
Author(s):  
Rafik Djemili
Keyword(s):  
Involuntary Movement ◽  
Epileptic Seizures ◽  
Intrinsic Mode Functions ◽  
Eeg Signals ◽  
Time Segment ◽  
Feature Extraction Method ◽  
Mode Decomposition ◽  
Finite Set ◽  
Student’S T

AbstractEpilepsy is a persistent neurological disorder impacting over 50 million people around the world. It is characterized by repeated seizures defined as brief episodes of involuntary movement that might entail the human body. Electroencephalography (EEG) signals are usually used for the detection of epileptic seizures. This paper introduces a new feature extraction method for the classification of seizure and seizure-free EEG time segments. The proposed method relies on the empirical mode decomposition (EMD), statistics and autoregressive (AR) parameters. The EMD method decomposes an EEG time segment into a finite set of intrinsic mode functions (IMFs) from which statistical coefficients and autoregressive parameters are computed. Nevertheless, the calculated features could be of high dimension as the number of IMFs increases, the Student’s t-test and the Mann–Whitney U test were thus employed for features ranking in order to withdraw lower significant features. The obtained features have been used for the classification of seizure and seizure-free EEG signals by the application of a feed-forward multilayer perceptron neural network (MLPNN) classifier. Experimental results carried out on the EEG database provided by the University of Bonn, Germany, demonstrated the effectiveness of the proposed method which performance assessed by the classification accuracy (CA) is compared to other existing performances reported in the literature.

scholarly journals Geometrical four-point functions in the two-dimensional critical Q-state Potts model: the interchiral conformal bootstrap

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yifei He ◽  
Jesper Lykke Jacobsen ◽  
Hubert Saleur
Keyword(s):  
Potts Model ◽  
Correlation Functions ◽  
Liouville Theory ◽  
Analytical Determination ◽  
Conformal Weight ◽  
Two Dimensional ◽  
Conformal Blocks ◽  
Conformal Bootstrap ◽  
Point Correlation

Abstract Based on the spectrum identified in our earlier work [1], we numerically solve the bootstrap to determine four-point correlation functions of the geometrical connectivities in the Q-state Potts model. Crucial in our approach is the existence of “interchiral conformal blocks”, which arise from the degeneracy of fields with conformal weight hr,1, with r ∈ ℕ*, and are related to the underlying presence of the “interchiral algebra” introduced in [2]. We also find evidence for the existence of “renormalized” recursions, replacing those that follow from the degeneracy of the field $$ {\Phi}_{12}^D $$ Φ 12 D in Liouville theory, and obtain the first few such recursions in closed form. This hints at the possibility of the full analytical determination of correlation functions in this model.

scholarly journals A FOUR-DIMENSIONAL SUPERSTRING FROM THE BOSONIC STRING WITH SOME APPLICATIONS

2005 ◽  
Vol 20 (01) ◽  
pp. 99-128 ◽  
Author(s):  
B. B. DEO ◽  
L. MAHARANA
Keyword(s):  
Standard Model ◽  
Central Charge ◽  
Virasoro Algebra ◽  
Low Energy ◽  
Majorana Fermions ◽  
Modular Invariance ◽  
Model Group ◽  
Supersymmetry Algebra ◽  
Four Dimensions ◽  
Bosonic Representation

A string in four dimensions is constructed by supplementing it with 44 Majorana fermions. The later are represented by 11 vectors in the bosonic representation SO (D-1,1). The central charge is 26. The fermions are grouped in such a way that the resulting action is worldsheet supersymmetric. The energy–momentum and current generators satisfy the super-Virasoro algebra. GSO projections are necessary for proving modular invariance. Space–time supersymmetry algebra is deduced and is substantiated for specific modes of zero mass. The symmetry group of the model can descend to the low energy standard model group SU (3)× SU L(2)× U Y(1) through the Pati–Salam group.

scholarly journals Construction and classification of holomorphic vertex operator algebras

2020 ◽  
Vol 2020 (759) ◽  
pp. 61-99 ◽  
Author(s):  
Jethro van Ekeren ◽  
Sven Möller ◽  
Nils R. Scheithauer
Keyword(s):  
Vertex Operator ◽  
Operator Algebras ◽  
Central Charge ◽  
Field Theories ◽  
Vertex Operator Algebras ◽  
Conformal Field Theories ◽  
Cyclic Groups ◽  
Conformal Field

AbstractWe develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens’ classification of {V_{1}}-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally, we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

scholarly journals On 2d CFTs that interpolate between minimal models

2019 ◽  
Vol 6 (6) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Field Theories ◽  
Two Dimensional ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Exactly Solvable ◽  
Structure Constants

We investigate exactly solvable two-dimensional conformal field theories that exist at generic values of the central charge, and that interpolate between A-series or D-series minimal models. When the central charge becomes rational, correlation functions of these CFTs may tend to correlation functions of minimal models, or diverge, or have finite limits which can be logarithmic. These results are based on analytic relations between four-point structure constants and residues of conformal blocks.

scholarly journals CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS

2011 ◽  
Vol 22 (02) ◽  
pp. 201-222 ◽  
Author(s):  
XIAOLI KONG ◽  
HONGJIA CHEN ◽  
CHENGMING BAI
Keyword(s):  
Virasoro Algebra ◽  
Indecomposable Module ◽  
Multiplication Operators ◽  
Algebraic Structures ◽  
Witt Algebra ◽  
Left Multiplication ◽  
Symmetric Algebras ◽  
Nonlinear Math ◽  
Math Phys

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

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