Verma Modules, Extremal Vectors, and Singular Vectors on the Non-Critical N=2 String Worldsheet

Abstract

We formulate the general construction for singular vectors in Verma modules of the affine sl(2|1) superalgebra. We then construct sl(2|1) representations out of the fields of the non-critical N=2 string. This allows us to extend naturally to sl(2|1) several crucial properties of the N=2 superconformal algebra, first of all the construction of extremal states (an analogue of different pictures for non-free fermions) and the spectral flow transform (which then affects the Liouville sector). We further evaluate the affine sl(2|1) singular vectors in the realization of sl(2|1) provided by the N=2 string. We establish that, with a notable exception, the respective singular vectors are in a 2:1 correspondence, namely two different sl(2|1) singular vectors evaluate as an N=2 superconformal singular vector (however, those singular vectors that are labelled by a pair of positive integers get these integers transposed under the reduction). We also analyse the `exceptional' cases, which amount to a series of sl(2|1) singular vectors, labelled by r>=1, which do not have an N=2 counterpart, and discuss the mechanism by which the multiplicity of singular vectors becomes equal to two at certain points in the weight spaces of both algebras.

arXiv commentLaTeX2.09, 34p

References in corpus (4)

• Fusion of twisted representations
• All Singular Vectors of the N=2 Superconformal Algebra via the Algebraic Continuation Approach
• Geometrical Structures of M-Theory
• The String Measure and Spectral Flow of Critical N=2 Strings

Cited by in corpus (7)

• Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras
• Higher-Level Appell Functions, Modular Transformations, and Characters
• Affine sl(2|1) and D(2|1;alpha) as Vertex Operator Extensions of Dual Affine sl(2) Algebras
• d=2, N=2 Superconformally Covariant Operators and Super W-Algebras
• A Note concerning Subsingular Vectors and Embedding Diagrams of the N=2 Superconformal Algebras
• A Semi-Infinite Construction of Unitary N=2 Modules
• Untwisting Topological Field Theories