Singular vectors in Verma modules over Kac?Moody algebras

The WZW model and the βγ system

“…This approach is also used extensively in the construction of new W algebras in [28], for example. 7 The singular vectors for the admissible representations were obtained long ago by Malikov, Feigin and Fuchs [29], and are referred to as MFF singular vectors. Their theorem 3.2 [29] expresses a singular vector as a monomial involving fractional powers: for the vacuum representation of the k = −1/2 model, the MFF singular vector reads…”

Virasoro singular vectors via quantum DS reduction

“…This enables us in particular to prove the conjecture in [5] exploiting the explicit formula for the Vir singular vectors written by Kent [9], thus demonstrating that the formulae in [6] and [9] are not just analogous in form, but up to BRST trivial terms, simply identical. In a forthcoming paper [10] the approach developed here will be generalised for the W 3 -algebras.…”

“…The construction in [9] was inspired by the analogous expressions for the general singular vectors in the Verma modules of A (1) 1 in [6]. Let V 0 = V {J ;ν} , for k = 2 , be a sl(2) k Verma module h.w.…”

“…is a singular vector in the module generated on V {J ;ν} [6]. It can be cast in a canonical way into an integral powers polynomial of X − 0 and X a −n .…”

“…It is given [6] by an expression like (31) but everywhere f 1 is replaced by f 0 and vice versa, while the powers are kept the same as in (31) with only m ′ changed to m ′ + 1 and the h.w. state V {J ;ν} replaced by V {1/ν−J−1;ν} .…”