Unitary representations of the Virasoro and super-Virasoro algebras

It is shown that a method previously given for constructing representations of the Virasoro algebra out of representations of affine Kac-Moody algebras yields the full discrete series of highest weight irreducible representations of the Virasoro algebra. The corresponding method for the super-Virasoro algebras (i.e. the Neveu-Schwarz and Ramond algebras) is described in detail and shown to yield the full discrete series of irreducible highest weight representations.     

A presentation for the Virasoro and super-Virasoro algebras

It is shown that the entire Virasoro, Ramond and Neveu-Schwarz algebras can each be constructed from a finite number of well-chosen generators satisfying a small number of conditions. The most economical sets consist of just two starting generators in all cases, subject to eight conditions for the Virasoro case, five conditions for the Ramond case, and nine conditions for the Neveu-Schwarz case.Work supported by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38     

The Virasoro algebra

Three theroems are proved. With suitable hypotheses in each case, characterizations are found for the Virasoro algebra, for some of its representations, and for the Ramond-Neveu-Schwarz superalgebra built around the Virasoro algebra.     

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

Yangian construction of the Virasoro algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algera.     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.     

Singular vectors of the Virasoro algebra

We present an explicit construction of the singular (or null) vectors in highest weight Verma modules.     

Vertex operators for deformed Virasoro algebra

Vertex operators for the deformed Virasoro algebra are defined, their bosonic representation is constructed, and a difference equation for the simplest vertex operators is described. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 11, 876–881 (10 June 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

? → Neveu-Schwarz-Ramond superalgebra → Virasoro algebra

There exists a unique object which stands to the Neveu-Schwarz-Ramond (NSR) superalgebra exactly in the same relation as the NSR superalgebra stands to the Virasoro algebra.     

Redundancy of conditions for a Virasoro algebra

I show that the Fairlie, Nuyts, Zachos construction of Virasoro algebra contains redundant conditions.Work supported in part by the U.S. Department of Energy, Division of High Energy Physics, Contract W-31-109-ENG-38     

Hawking radiation and near horizon universality of chiral Virasoro algebra

We show that the diffeomorphism anomaly together with the trace anomaly reveal a chiral Virasoro algebra near the event horizon of a black hole. This algebra is the same irrespective of whether the anomaly is covariant or consistent, thereby manifesting its universal character and the fact that only the outgoing modes are relevant near the horizon. Our analysis therefore clarifies the role of the trace anomaly in the diffeomorphism anomaly approach [Robinson and Wilczek in Phys. Rev. Lett. 95:011303, 2005; Iso et al. in Phys. Rev. Lett. 96:151302, 2006; Banerjee and Kulkarni in Phys. Rev. D 77:024018, 2008; Gangopadhyay and Kulkarni in Phys. Rev. D 77:024038, 2008] to the Hawking radiation.     

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello–Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta–gamma system using the method of effective BV quantization.