Irreducible Modules over the Mirror Heisenberg-Virasoro Algebra

2021 ◽  
Author(s):  
Dong Liu ◽  
Yufeng Pei ◽  
Limeng Xia ◽  
Kaiming Zhao
Keyword(s):  
Virasoro Algebra ◽  
Irreducible Modules

scholarly journals Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

2012 ◽  
Vol 55 (3) ◽  
pp. 697-709 ◽  
Author(s):  
Xiangqian Guo ◽  
Rencai Lu ◽  
Kaiming Zhao
Keyword(s):  
Direct Summand ◽  
Virasoro Algebra ◽  
Additive Subgroup ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Intermediate Series ◽  
Complex Number Field ◽  
Weight Modules

AbstractLet G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

scholarly journals Unitary Modules for the Twisted Heisenberg–Virasoro Algebra

2019 ◽  
Vol 26 (03) ◽  
pp. 529-540
Author(s):  
Xiufu Zhang ◽  
Shaobin Tan ◽  
Haifeng Lian
Keyword(s):  
Virasoro Algebra ◽  
Irreducible Module ◽  
Irreducible Modules ◽  
Intermediate Series

The conjugate-linear anti-involutions and unitary irreducible modules of the intermediate series over the twisted Heisenberg–Virasoro algebra are classified, respectively. We prove that any unitary irreducible module of the intermediate series over the twisted Heisenberg–Virasoro algebra is of the form [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text].

Quasi-particles in conformal field theoryThis paper was presented at the Theory CANADA 4 conference, held at the Centre de Recherches Mathématiques at the Université de Montréal, Québec, Canada on 4–7 June 2008.

2009 ◽  
Vol 87 (3) ◽  
pp. 205-211
Author(s):  
P. Mathieu
Keyword(s):  
Virasoro Algebra ◽  
Positive Definite ◽  
Exclusion Principle ◽  
Theoretical Interpretation ◽  
Minimal Models ◽  
Quasi Particle ◽  
Conformal Field ◽  
Irreducible Modules ◽  
Alternating Sums ◽  
State Content

After recalling some basing features of conformal field theory, we present an elementary introduction to the description of the states in the irreducible modules of the minimal models. The characters, which encode the state content of each module, are easily constructed from the representation theory of the Virasoro algebra and they take the form of infinite alternating sums. On the other hand, the relation between the minimal models and particular statistical models has led to the discovery of alternative expressions for the characters, which are positive definite. These entail a (yet to be fully worked out) quasi-particle description of the space of states, via a filling process with restrictions akin to the exclusion principle. The simplest class of positive characters, pertaining to the ℳ(2,p) models, is presented, and its underlying combinatorial aspects are spelled out. A natural conformal field theoretical interpretation of its quasi-particles is presented.

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.

TENSOR PRODUCT DECOMPOSITIONS FOR AFFINE KAC-MOODY ALGEBRAS

1994 ◽  
Vol 06 (06) ◽  
pp. 1269-1299 ◽  
Author(s):  
M. D. GOULD
Keyword(s):  
Tensor Product ◽  
Product Space ◽  
Virasoro Algebra ◽  
Moody Algebra ◽  
Product Decomposition ◽  
Highest Weight ◽  
Tensor Product Space ◽  
Irreducible Modules ◽  
Coset Construction ◽  
Highest Weight Modules

The decomposition into irreducible modules is determined, for the tensor product of two arbitrary irreducible integrable highest weight modules, for an (untwisted) affine Kac-Moody algebra L. The result is applied to investigate full multiplicity regions and cyclic modules within the tensor product space for an affine Kac-Moody algebra. The tensor product decomposition into irreducible modules over L ⊕ Vir (Coset construction), Vir the Virasoro algebra, is also briefly investigated.

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".  

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

scholarly journals On local and integrated stress-tensor commutators

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Mert Besken ◽  
Jan de Boer ◽  
Grégoire Mathys
Keyword(s):  
Stress Tensor ◽  
Free Field ◽  
Virasoro Algebra ◽  
Light Sheet ◽  
Operator Product ◽  
Local Form ◽  
Two Dimensional ◽  
Conformal Embedding ◽  
General Aspects ◽  
The Right

Abstract We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.

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