Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants

Abstract

Extensions of the Virasoro algebra constructed from Kac-Moody algebras using higher order Casimir invariants Schoutens, C.J.M.; Bais, F.A.; Bouwknegt, P.; Surridge, M. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We discuss extensions of the Virasoro algebra obtained by generalizing the Sugawara construction to the higher order Casimir invariants of a Lie algebra g. We generalize the GKO coset construction to the dimension-3 operator for g =A N 1 and recover results of Fateev and Zamolodchikov if N = 3. Branching rules and generalizations to all simple, simply-laced g are discussed.

The large N ’t Hooft limit of Kazama-Suzuki model

“…The W cl ∞ [λ] for the bosonic case was found in [6,35] and the corresponding quantum algebra has been studied in [2]. See also [18,17]. We expect that there are extra linear terms for the nonlinear composite currents in (3.3).…”

“…For example, the spin 3-spin 3 operator product expansion in SU(N) Casimir algebra leads to other spin 4 current and its descendant in the right hand side [18,19]. The relative coefficient functions appearing in the descendant fields for given primary field are also fixed by conformal invariance.…”

Primary currents and Riemannian geometry in algebras

“…The physical symmetry constituents are currents whose O.P.E (or different recipes), generate extensions of the conformal symmetry [47] - [49]:for this reason they are usually defined as "primary fields". Their equations of motion define the dynamics of the system and, since they are defined on a Riemann surface [50] - [52], they must be promoted to the status of geometrical objects. Since any physical investigation cannot be separated from any geometrical consideration, the same kind of symmetry arises in the context of writting down welldefined differential equations on Riemann surfaces [20] - [62].…”

W-algebras and their representations

“…Using the standard method for obtaining the commutator of two modes from the double contour integral of the singular part of the operator product expansion, we recover the Virasoro algebra (2). Taking Wick contractions may be much faster for working out operator product expansions of expressions involving only free fields, but for the more complicated algebras we will have, the method outlined above is the only practical method, and is the method by which we shall work out all operator product expansions.…”

“…and further consider only those embeddings for which g decomposes into integer spins representations of sl (2). If the generators of the sl(2) are I + , I 0 , I − then we can decompose g into eigenspaces of I 0 ,…”