Singular Dimensions of the¶N= 2 Superconformal Algebras. I

Abstract

Verma modules of superconfomal algebras can have singular vector spaces with dimensions greater than 1. Following a method developed for the Virasoro algebra by Kent, we introduce the concept of adapted orderings on superconformal algebras. We prove several general results on the ordering kernels associated to the adapted orderings and show that the size of an ordering kernel implies an upper limit for the dimension of a singular vector space. We apply this method to the topological N = 2 algebra and obtain the maximal dimensions of the singular vector spaces in the topological Verma modules: 0, 1, 2 or 3 depending on the type of Verma module and the type of singular vector. As a consequence we prove the conjecture of Gato-Rivera and Rosado on the possible existing types of topological singular vectors (4 in chiral Verma modules and 29 in complete Verma modules). Interestingly, we have found two-dimensional spaces of singular vectors at level 1. Finally, by using the topological twists and the spectral flows, we also obtain the maximal dimensions of the singular vector spaces for the Neveu-Schwarz N = 2 algebra (0, 1 or 2) and for the Ramond N = 2 algebra (0, 1, 2 or 3).

Determinant formula for the topological N = 2 superconformal algebra

“…The Verma modules built on chiral primary states are called chiral Verma modules V (|0, h G,Q ) [13][12] [18]. They are not complete because the primary state being annihilated by both G 0 and Q 0 amounts to an additional constraint not required (just allowed) by the algebra.…”

“…with l = −∆ in the case of chiral and no-label singular vectors. The maximal dimensions of the corresponding singular vector spaces [18] are two, for the singular vectors of types |χ An useful observation is that chiral singular vectors |χ (q)G,Q l can be regarded as particular cases of G 0 -closed singular vectors |χ (q)G l and/or as particular cases of Q 0 -closed singular vectors |χ (q)Q l . That is, some G 0 -closed and Q 0 -closed singular vectors 'become' chiral when the conformal weight of the singular vector turns out to be zero, i.e.…”

“…The maximal dimensions of the corresponding singular vector spaces [18] are three 5 , for the singular vectors of types |χ The maximal dimension n for a given singular vector space puts a theoretical limit on the number of linearly independent singular vectors of the corresponding type that one can write down at the same level in a given Verma module [18]. For the singular vectors given in tables (2.5) and (2.6) the corresponding maximal dimensions have been verified by the low level computations.…”

“…The possible existing types of topological singular vectors in chiral Verma modules are the following four types [12][18] built on chiral primaries |0, h G,Q : The maximal dimension of the corresponding singular vector spaces is one [18]. These singular vectors can also be organized into families [13][12] with a unique pattern: the box diagram consisting of four singular vectors, one of each type, connected by G 0 , Q 0 and the spectral flow automorphism A (see footnote 6).…”

“…However, we know that the only possible types of such singular vectors are again the four types given in table (2.8) since we have proved, in ref. [18], that the space associated to any other type of 'would-be' singular vector in chiral Verma modules has maximal dimension zero (i.e. no singular vectors of such types exist).…”