ADE SUBALGEBRAS OF THE TRIPLET VERTEX ALGEBRA $\mathcal{W}(p)$: A-SERIES

2013 ◽  
Vol 15 (06) ◽  
pp. 1350028 ◽  
Author(s):  
DRAŽEN ADAMOVIĆ ◽  
XIANZU LIN ◽  
ANTUN MILAS
Keyword(s):  
Central Charge ◽  
Strong Support ◽  
Vertex Algebra ◽  
Finite Subgroup ◽  
Irreducible Representations ◽  
Finite Subgroups ◽  
Complete Set ◽  
Ade Classification ◽  
Charge Formula

Motivated by [On the triplet vertex algebra [Formula: see text], Adv. Math.217 (2008) 2664–2699], for every finite subgroup Γ ⊂ PSL(2, ℂ) we investigate the fixed point subalgebra [Formula: see text] of the triplet vertex [Formula: see text], of central charge [Formula: see text], p ≥ 2. This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, ℂ). First, we prove the C2-cofiniteness of the Am-fixed subalgebra [Formula: see text]. Then we construct a family of [Formula: see text]-modules, which are expected to form a complete set of irreducible representations. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m = 2. We also present a rigorous proof of the fact that the full automorphism group of [Formula: see text] is PSL(2, ℂ).

On parafermion vertex algebras of 𝔰𝔩(2) and 𝔰𝔩(3) at level −3 2

2020 ◽  
pp. 2050086
Author(s):  
Dražen Adamović ◽  
Antun Milas ◽  
Qing Wang
Keyword(s):  
Lie Algebra ◽  
Central Charge ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Algebraic Proof ◽  
Direct Sums ◽  
Direct Sum ◽  
Irreducible Modules ◽  
Level 3 ◽  
Charge Formula

We study parafermion vertex algebras [Formula: see text] and [Formula: see text]. Using the isomorphism between [Formula: see text] and the logarithmic vertex algebra [Formula: see text] from [D. Adamović, A realization of certain modules for the [Formula: see text] superconformal algebra and the affine Lie algebra [Formula: see text], Transform. Groups 21(2) (2016) 299–327], we show that these parafermion vertex algebras are infinite direct sums of irreducible modules for the Zamolodchikov algebra [Formula: see text] of central charge [Formula: see text], and that [Formula: see text] is a direct sum of irreducible [Formula: see text]-modules. As a byproduct, we prove certain conjectures about the vertex algebra [Formula: see text]. We also obtain a vertex-algebraic proof of the irreducibility of a family of [Formula: see text] modules at [Formula: see text].

scholarly journals REPRESENTATIONS OF N=1 EXTENDED SUPERCONFORMAL ALGEBRAS WITH TWO GENERATORS

1993 ◽  
Vol 08 (08) ◽  
pp. 725-738 ◽  
Author(s):  
W. EHOLZER ◽  
A. HONECKER ◽  
R. HÜBEL
Keyword(s):  
Central Charge ◽  
Mixed Structure ◽  
Rational Models ◽  
Highest Weight ◽  
Continuous Branch ◽  
Highest Weight Representations ◽  
Exceptional Algebras ◽  
Discrete Values ◽  
Ade Classification ◽  
Charge Formula

In this paper we consider the representation theory of N=1 Super-W-algebras with two generators for conformal dimension of the additional superprimary field between two and six. In the superminimal case our results coincide with the expectation from the ADE-classification. For the parabolic algebras we find a finite number of highest weight representations and an effective central charge [Formula: see text]. Furthermore we show that most of the exceptional algebras lead to new rational models with [Formula: see text]. The remaining exceptional cases show a new ‘mixed’ structure. Besides a continuous branch of representations discrete values of the highest weight also exist.

ON SMOOTHNESS OF MINIMAL MODELS OF QUOTIENT SINGULARITIES BY FINITE SUBGROUPS OF SLn(ℂ)

2018 ◽  
Vol 60 (3) ◽  
pp. 603-634
Author(s):  
RYO YAMAGISHI
Keyword(s):  
Minimal Model ◽  
Finite Subgroup ◽  
Quotient Singularity ◽  
Crepant Resolution ◽  
Minimal Models ◽  
Cox Ring ◽  
Quotient Singularities ◽  
Finite Subgroups ◽  
Symplectic Resolutions

AbstractWe prove that a quotient singularity ℂn/G by a finite subgroup G ⊂ SLn(ℂ) has a crepant resolution only if G is generated by junior elements. This is a generalization of the result of Verbitsky (Asian J. Math.4(3) (2000), 553–563). We also give a procedure to compute the Cox ring of a minimal model of a given ℂn/G explicitly from information of G. As an application, we investigate the smoothness of minimal models of some quotient singularities. Together with work of Bellamy and Schedler, this completes the classification of symplectically imprimitive quotient singularities that admit projective symplectic resolutions.

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.

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.

Tracing the origins of hybrids through history: monstrous cultivars and Napoléon Bonaparte’s exiled paper daisies (Asteraceae; Gnaphalieae)

2021 ◽  
Author(s):  
Timothy L Collins ◽  
Jeremy J Bruhl ◽  
Alexander N Schmidt-Lebuhn ◽  
Ian R H Telford ◽  
Rose L Andrew
Keyword(s):  
Gene Flow ◽  
Phylogenetic Analyses ◽  
Strong Support ◽  
Napoleon Bonaparte ◽  
Snp Data ◽  
Cast Doubt ◽  
Molecular Phylogenetic ◽  
St Helena ◽  
New Evidence

Abstract Golden everlasting paper daisies (Xerochrysum, Gnaphalieae, Asteraceae) were some of the earliest Australian native plants to be cultivated in Europe. Reputedly a favourite of Napoléon Bonaparte and Empress Joséphine, X. bracteatum is thought to have been introduced to the island of St Helena in the South Atlantic during Napoléon’s exile there. Colourful cultivars were developed in the 1850s, and there is a widely held view that these were produced by crossing Xerochrysum with African or Asian Helichrysum spp. Recent molecular phylogenetic analyses and subtribal classification of Gnaphalieae cast doubt on this idea. Using single-nucleotide polymorphism (SNP) data, we looked for evidence of gene flow between modern cultivars, naturalized paper daisies from St Helena and four Xerochrysum spp. recorded in Europe in the 1800s. There was strong support for gene flow between cultivars and X. macranthum. Paper daisies from St Helena were genotypically congruent with X. bracteatum and showed no indications of ancestry from other species or from the cultivars, consistent with the continuous occurrence of naturalized paper daisies introduced by Joséphine and Napoléon. We also present new evidence for the origin of colourful Xerochrysum cultivars and hybridization of congeners in Europe from Australian collections.

scholarly journals REFINING THE CLASSIFICATION OF THE IRREPS OF THE 1D N-EXTENDED SUPERSYMMETRY

2008 ◽  
Vol 23 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHANNA KUZNETSOVA ◽  
FRANCESCO TOPPAN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Supersymmetric Quantum Mechanics ◽  
Irreducible Representations

The linear finite irreducible representations of the algebra of the 1D N-Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to N ≤ 8.

SUPERCONFORMAL ORBIFOLDS INVOLVING THE FERMION NUMBER OPERATOR

2004 ◽  
Vol 19 (22) ◽  
pp. 3637-3667 ◽  
Author(s):  
KATRIN WENDLAND
Keyword(s):  
Central Charge ◽  
Space Time ◽  
Field Theories ◽  
Two Dimensional ◽  
Number Operator ◽  
Superconformal Field Theories ◽  
Fermion Number ◽  
Multicritical Points ◽  
Arbitrary Dimensions ◽  
Charge Formula

We consider orbifolds of two-dimensional unitary toroidal superconformal field theories with target spaces of arbitrary dimensions, where the orbifold group involves the space–time fermion number operator. We construct all so-called superaffine, orbifold prime and super-M-orbifold models by generalizing the constructions of Dixon, Ginsparg and Harvey. We also correct claims made by Dixon, Ginsparg and Harvey about multicritical points among those models with central charge [Formula: see text].

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