scholarly journals The N = 1 super Heisenberg–Virasoro vertex algebra at level zero

2021 ◽  
Author(s):  
Dražen Adamović ◽  
Berislav Jandrić ◽  
Gordan Radobolja
Keyword(s):  
Fock Space ◽  
Free Field ◽  
Virasoro Algebra ◽  
Vertex Algebra ◽  
Verma Modules ◽  
Minimal Models ◽  
Highest Weight ◽  
Level Zero ◽  
Free Field Realization ◽  
Highest Weight Representations

We study the representation theory of the [Formula: see text] super Heisenberg–Virasoro vertex algebra at level zero, which extends the previous work on the Heisenberg–Virasoro vertex algebra [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342; D. Adamović and G. Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero, Commun. Contemp. Math. 21(2) (2019) 1850008; Y. Billig, Representations of the twisted Heisenberg–Virasoro algebra at level zero, Can. Math. Bull. 46(4) (2003) 529–537] to the super case. We calculated all characters of irreducible highest weight representations by investigating certain Fock space representations. Quite surprisingly, we found that the maximal submodules of certain Verma modules are generated by subsingular vectors. The formulas for singular and subsingular vectors are obtained using screening operators appearing in a study of certain logarithmic vertex algebras [D. Adamović and A. Milas, On W-algebras associated to [Formula: see text] minimal models and their representations, Int. Math. Res. Notices 2010(20) (2010) 3896–3934].

scholarly journals Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro algebra at level zero

2019 ◽  
Vol 21 (02) ◽  
pp. 1850008 ◽  
Author(s):  
Dražen Adamović ◽  
Gordan Radobolja
Keyword(s):  
Free Field ◽  
Virasoro Algebra ◽  
Large Family ◽  
Verma Modules ◽  
Irreducible Modules ◽  
Level Zero ◽  
Free Field Realization ◽  
Dual Module ◽  
Whittaker Modules ◽  
New Applications

This paper is a continuation of [D. Adamović and G. Radobolja, Free field realization of the twisted Heisenberg–Virasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219(10) (2015) 4322–4342]. We present certain new applications and generalizations of the free field realization of the twisted Heisenberg–Virasoro algebra [Formula: see text] at level zero. We find explicit formulas for singular vectors in certain Verma modules. A free field realization of self-dual modules for [Formula: see text] is presented by combining a bosonic construction of Whittaker modules from [D. Adamović, R. Lu and K. Zhao, Whittaker modules for the affine Lie algebra [Formula: see text], Adv. Math. 289 (2016) 438–479; arXiv:1409.5354] with a construction of logarithmic modules for vertex algebras. As an application, we prove that there exists a non-split self-extension of irreducible self-dual module which is a logarithmic module of rank two. We construct a large family of logarithmic modules containing different types of highest weight modules as subquotients. We believe that these logarithmic modules are related with projective covers of irreducible modules in a suitable category of [Formula: see text]-modules.

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

1992 ◽  
Vol 07 (13) ◽  
pp. 3023-3033 ◽  
Author(s):  
LOUIS BENOIT ◽  
YVAN SAINT-AUBIN
Keyword(s):  
Explicit Expression ◽  
Explicit Formula ◽  
Singular Vector ◽  
Central Charge ◽  
Discrete Series ◽  
Virasoro Algebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Highest Weight ◽  
Highest Weight Representations

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

scholarly journals UNITARY REPRESENTATIONS OF W INFINITY ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6339-6355 ◽  
Author(s):  
SATORU ODAKE
Keyword(s):  
Central Charge ◽  
Discrete Series ◽  
Free Field ◽  
Unitary Representations ◽  
Continuous Series ◽  
Highest Weight ◽  
Free Field Realization ◽  
Highest Weight Representations

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras [Formula: see text] with central charges (2, 1, 3, 2M, N, 2M+N). The characters of these representations are computed. We construct a new extended superalgebra [Formula: see text], whose bosonic sector is [Formula: see text]. Its representations obtained from a free field realization with central charge 2M+N, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for M=N.

FREE FIELD APPROACH TO D-BRANES IN N=2 SUPERCONFORMAL MINIMAL MODELS

2004 ◽  
Vol 19 (supp02) ◽  
pp. 294-310
Author(s):  
S. E. PARKHOMENKO
Keyword(s):  
Free Field ◽  
Virasoro Algebra ◽  
Minimal Models ◽  
Field Approach ◽  
Free Field Realization

The approach to construction of D-branes in the N=2 superconformal minimal models based on a free-field realization of the N=2 super-Virasoro algebra unitary modules is represented.

DIFFERENTIAL EQUATIONS IN g≥2 CONFORMAL FIELD THEORY

1989 ◽  
Vol 04 (18) ◽  
pp. 1773-1782
Author(s):  
AKISHI KATO ◽  
TOMOKI NAKANISHI
Keyword(s):  
Differential Equations ◽  
Riemann Surfaces ◽  
Virasoro Algebra ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Highest Weight ◽  
Higher Genus ◽  
Highest Weight Representations ◽  
Null State

We consider the minimal conformal field theories on Riemann surfaces of genus greater than one. We illustrate in a simple example how the null state conditions in the highest weight representations of the Virasoro algebra turn into differential equations including the moduli variables for correlators between degenerate fields. In particular, the set of an infinite number of partial differential equations satisfied by higher genus characters is obtained.

A CALOGERO–SUTHERLAND MODEL BASED ON THE APERIODIC VIRASORO ALGEBRA

2000 ◽  
Vol 15 (26) ◽  
pp. 4179-4189
Author(s):  
REIDUN TWAROCK
Keyword(s):  
Virasoro Algebra ◽  
Additional Term ◽  
Specific Data ◽  
New Model ◽  
Model Based ◽  
Highest Weight ◽  
Sutherland Model ◽  
Point Set ◽  
Highest Weight Representations ◽  
One To One

We derive a new model of Calogero–Sutherland type based on the aperiodic Virasoro algebra, which is an aperiodic analog of the Virasoro algebra with generators in a one-to-one correspondence with an aperiodic point set. It is shown that the Hamiltonian obtained in this setting contains an additional term with respect to the corresponding model based on the Virasoro algebra, and may thus be considered as a perturbation of the latter. The solvability of the model follows from the results about the highest weight representations of the aperiodic Virasoro algebra. The additional term in the Hamiltonian is shown to depend on a parameter, which is given in terms of model specific data such as a representation number, and a set of operators which appear as a consequence of the aperiodic setting in the framework of oscillator representations.

scholarly journals HIGHEST WEIGHT REPRESENTATIONS AND KAC DETERMINANTS FOR A CLASS OF CONFORMAL GALILEI ALGEBRAS WITH CENTRAL EXTENSION

2012 ◽  
Vol 23 (11) ◽  
pp. 1250118 ◽  
Author(s):  
NARUHIKO AIZAWA ◽  
PHILLIP S. ISAAC ◽  
YUTA KIMURA
Keyword(s):  
Central Extension ◽  
Spatial Dimension ◽  
Verma Modules ◽  
Infinite Dimensional ◽  
Singular Vectors ◽  
Highest Weight ◽  
Irreducible Modules ◽  
Highest Weight Representations ◽  
Quotient Modules

We investigate the representations of a class of conformal Galilei algebras in one spatial dimension with central extension. This is done by explicitly constructing all singular vectors within the Verma modules, proving their completeness and then deducing irreducibility of the associated highest weight quotient modules. A resulting classification of infinite dimensional irreducible modules is presented. It is also shown that a formula for the Kac determinant is deduced from our construction of singular vectors. Thus we prove a conjecture of Dobrev, Doebner and Mrugalla for the case of the Schrödinger algebra.

BERNSTEIN-GEL’FAND-GEL’FAND RESOLUTION OF W3 ALGEBRA IN THE LATTICE APPROACH

1992 ◽  
Vol 07 (10) ◽  
pp. 2219-2244
Author(s):  
H.A. BOUGOURZI ◽  
Y. KIKUCHI
Keyword(s):  
Free Field ◽  
Verma Modules ◽  
Field Representation ◽  
Highest Weight ◽  
Highest Weight Representation ◽  
Lattice Approach

We describe the embedding structure of the Verma modules of the W3 algebra using the free field representation (FFR) in the lattice approach. This structure is expressed by a set of intertwining diagrams. In particular, we show how these diagrams can be used to achieve the Bernstein-Gel’fand-Gel’fand (BGG) resolution of the irreducible highest weight representation (IHWR) in terms of the Verma modules. As an application of prime importance, we show how the character of the W3 IHWR can be readily derived using the Rocha-Caridi procedure.

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