OnE 10 and the DDF construction

An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particularE ₁₀, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.Supported by Konrad-Adenauer-Stiftung e.V.This article was processed by the author using the Latex style filepljour1 from Springer-Verlag.     

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.     

Characteristic identities for Kac-Moody algebras

Characteristic identities are derived for the generators of a simple Kac-Moody algebra in any highest-weight unitary representation. Entries of powers of the characteristic matrix are rigorously defined on such modules. The eigenvalues of the characteristic matrix are shifted by generators of the Virasoro algebra which commutes with the diagonal action of the Kac-Moody algebra on a tensor product module. The characteristic identity can be cast as a product of a finite number of factors linear in the sine of the characteristic matrix, and the corresponding projection operators project on to modules of the diagonal Kac-Moody algebra.     

Properties of the covariant formulation of superstring theories

The covariant two-dimensional action principle that describes the dynamics of free superstrings in a Minkowski background is reviewed. Covariant gauge conditions are formulated, which simplify the equations of motion of the superspace coordinates to free equations. In this gauge there are bosonic and fermionic constraints whose generators give a supersymmetric generalization of the Virasoro algebra. As in certain supersymmetric field theories, closure of the algebra requires using the equations of motion. Covariant constrained bracket relations are obtained for the classical theory, but it is very difficult to extend them to quantum mechanical commutation relations. Interaction vertices satisfying supersymmetry and the necessary gauge conditions are constructed. They reduce in a special frame to ones found in earlier work in the light-cone gauge, and then can be interpreted quantum mechanically.     

Cohomology of the Virasoro algebra with coefficients in string fields

The first cohomology of the Virasoro algebra with coefficients in string fields are investigated. The relation between them and the Nambu-Goto action for a closed string is established.     

String branchings on complex tori and algebraic representations of generalized Krichever-Novikov algebras

The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two-and three-point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b–c systems. The defining cocycle for this central extension deforms to the well-known Virasoro cocycle for certain kinds of degenerations of the torus.     

New Bosonic Fock Representations of Affine-Virasoro Algebras

New bosonic Fock representations of the Virasoro algebra with nonzero centres are explicitly presented. Based on these, new Fock representations of affine-Virasoro algebras are constructed.     

Deformed fields and Moyal construction of deformed super Virasoro algebra

Studied is the deformation of super Virasoro algebra proposed by Belov and Chaltikian. Starting from abstract realizations in terms of the FFZ type generators, various connections of them to other realizations are shown, especially to deformed field representations, whose bosonic part generator is recently reported as a deformed string theory on a noncommutative world-sheet. The deformed Virasoro generators can also be expressed in terms of ordinary free fields in a highly nontrivial way.     

On a class of representations of the Virasoro algebra and a conjecture of Kac

We consider a class of representations of the Virasoro algebra that we call bounded admissible representations. For this class, we prove a conjecture of Victor Kac concerning the irreducibility of these representations. Results concerning the center and dimensions of weight spaces are also obtained.     

The Structure of pre-Verma Modules over the N = 1 Ramond Algebra

In this note, we determine the structure of the pre-Verma modules over the N = 1 Ramond algebra. The results of this note together with the results obtained in Iohara and Koga [Adv Math 178:1–65, 2003] completely determine the structure of Verma modules over the N = 1 super Virasoro algebra both in Neveu Schwarz and Ramond sectors.     

Dual formulation of classicalW-algebras

By extending the concept of Maurer-Cartan equations, a dual formulation of (classical) nonlinear extensions of the Virasoro algebra is introduced. This dual formulation is closely related to three-dimensional actions which are analogous to a Chern-Simons action. An explicit construction in terms of superfields of theN = 2 superW ₄-algebra is presented.     

The relationship among canonical quantisation, Polyakov's functional integral and BRST quantisation of the string

We explore the relationship among the three covariant approaches to the first quantisation of the bosonic string by using the descent equations to calculate the central extension of the Virasoro algebra and the square of the BRST charge from the Weyl anomaly.     

Dual Peierls brackets for strings

A new type of Peierls bracket which provides an additional representation of the Virasoro algebra, and which may be of relevance for a manifestly dual string field theory is constructed for the closed and open bosonic strings.     

Modified Virasoro and super-Virasoro algebra

Applying the paraquantization of order Q to an open bosonic and spining strings, modified Virasoro and super-Virasoro algebra are obtained. It is shown that the anomaly c-number term is a linear function of Q.     

ON GEOMETRIC QUANTIZATION FOR BOSONIC STRING (Ⅱ)

The Hilbert space and the representation of the generators of Virasoro algebra for bosonic string under a holomorphic polarization are given in this paper,It is shown that the contre term of Virasoro algebra may be interpreted as curvature of a holomorphic vector bundle (holomorphic Fock bundle) on coset space G¹¹=G/H where G denotes the conformal transformation group and H the one-parameter subgroup generated by the generator L₀.The condition of the conformal anomaly cancellation may be expressed as the vanishing curvature of the bundle which is obtained by the product of the holomorphic Fock bundle and the holomorphic ghost vacuum bundle.The geometric interpretations of both classical and quantized BRST operators,ghost and antighost operators are also discussed.     

Characterizations of the quantum Witt algebra

The q-analogues of some concepts in the theory of nonassociative algebras are introduced and two characterizations are given for the quantum Witt algebra.