Highest weight representations of the N=1 Ramond algebra

TL;DR

This paper explores the complex structure of highest weight representations of the N=1 Ramond algebra, revealing richer features like degenerate and subsingular vectors, and providing explicit examples and conjectures for their classification.

Contribution

It introduces a detailed analysis of the Ramond algebra's Verma modules, including the computation of the ordering kernel and explicit construction of singular vectors and embedding diagrams.

Findings

Verma modules contain degenerate (2D) singular vector spaces

Supersymmetric cases can include subsingular vectors

Explicit examples of singular vectors up to level 3

Abstract

We analyse the highest weight representations of the N=1 Ramond algebra and show that their structure is richer than previously suggested in the literature. In particular, we show that certain Verma modules over the N=1 Ramond algebra contain degenerate (2-dimensional) singular vector spaces and that in the supersymmetric case they can even contain subsingular vectors. After choosing a suitable ordering for the N=1 Ramond algebra generators we compute the ordering kernel, which turns out to be two-dimensional for complete Verma modules and one-dimensional for G-closed Verma modules. These two-dimensional ordering kernels allow us to derive multiplication rules for singular vector operators and lead to expressions for degenerate singular vectors. Using these multiplication rules we study descendant singular vectors and derive the Ramond embedding diagrams for the rational models. We give all explicit examples for singular vectors, degenerate singular vectors, and subsingular vectors until level 3. We conjecture the ordering kernel coefficients of all (primitive) singular vectors and therefore identify these vectors uniquely.