Representation Theory of the Affine Lie Superalgebra $\hslc$ at Fractional Level

Abstract

N=2 noncritical strings are closely related to the SL(2/1; R)/SL(2/1; R) Wess-ZuminoNovikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the latter. An important ingredient is the precise knowledge of theŝl(2/1; C) representation theory at fractional level. In this paper, the embedding diagrams of singular vectors appearing inŝl(2/1; C) Verma modules for fractional values of the level (k = p/q − 1, p and q coprime) are derived analytically.

Characters of admissible representations of the affine superalgebra

“…Suppose now that h and t are the quantum numbers of an N = 4 hws in a unitary representation, i.e., and let c be its U(1) charge. The g-product in (A.1) corresponds, in theŝl(2|1; C) k Kac-Kazhdan determinant formula [23,11], to the factors, φ (0) n ((ᾱ 1 +ᾱ 2 ), 0, m), φ (0) n (−(ᾱ 1 +ᾱ 2 ), 0, 1 + m), m ∈ Z + , n ∈ N, (A.4) in the notations of [5], when theŝl(2|1; C) k hws has isospin 1 2 h − and hypercharge 1 2 h + . So, whenever a factor of the g-product vanishes, i.e.…”

“…When the level is fractional and of the form (1.2), admissible representations do occur for the embedding diagrams of class IV (in the notations of [5]), which are identical in structure to the embedding diagrams for the minimal representations of the N = 2 superconformal algebra as found in [12] or in [15]. This analogy between embedding diagrams is reminiscent of the analogy betweenŝl(2, C) (resp.ô sp(1|2; C)) and Virasoro (resp.…”

“…The Lie superalgebra A(1, 0) ≡ sl(2|1; C) has rank two and its simple roots can be chosen to be fermionic. The root diagram ( Fig.1) can be represented in a 2-dimensional Minkowski space with the fermionic roots ±α 1 and ±α 2 in the lightlike directions (see [5] for notations),…”

“…In the notation of [5], Z ′ 0 is the bosonic hws. Z ′ a+1 and T ′ a+1 are bosonic singular vectors while T + a , Z + a+1 , T − a and Z − a+1 are fermionic singular vectors.…”

Parafermionic representation of the affine algebra at fractional level

“…At level k = 1 u − 1, the vacuum representation is in the Ramond sector of the theory and is labeled by the two quantum numbers h R − = 0 and h R + = 0, which correspond to the isospin and U(1) charges of the highest weight state [6,8]. The associated character is given by,…”