Coset construction for extended Virasoro algebras

Abstract

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.

Spinor field realizations of non-critical strings

“…The BRST charge for W 3 string was first constructed in [6], and the detailed studies of it can be found in [6][7][8][9][10][11][12]. A natural generalization of the W 3 string, i.e.…”

“…where f 4 [6], f 4 [7], f 4 [8], f 4 [10], f 4 [14], f 4 [15], f 4 [18], f 4 [20] are arbitrary constants but do not vanish at the same time.…”

W-gravity from Chern-Simons theory

“…T (z)T (w) ∼ c/2 (z − w) 4 The above algebra can be generalized to the case of a W N algebra, which contains fields W (k) of spin k, with k running from 2 to N. In [2] it was shown that this algebra is intimately related to the affine Kac-Moody algebra associated to su(N), in the sense that the fields W (k) can be constructed from the Kac-Moody currents by using the higher-order Casimir invariants of the underlying finite dimensional Lie algebra. Besides this connection with affine Kac-Moody algebras, it is now clear that these non-linear extensions of the Virasoro algebra play a central role in many other areas of two dimensional physics.…”

symmetry of quantum W∞ gravity

“…In principle, a similar procedure should be possible for W ∞ or W 1+∞ gravity. Indeed a generalisation of the Sugawara construction, known as Casimir algebras, has been given for W -extended conformal algebras [6]. Essentially, one builds the higher-spin currents by using symmetric invariant tensors of the Lie algebra underlying the W algebra.…”