The N=1 triplet vertex operator superalgebras

TL;DR

This paper introduces a new family of N=1 vertex operator superalgebras called SW(m), classifies their modules, computes characters, and explores connections to quantum groups, advancing understanding in logarithmic conformal field theory.

Contribution

It presents the first classification of irreducible modules for SW(m), computes their characters, and investigates links to quantum group representations.

Findings

SW(m) are C_2-cofinite superalgebras analogous to triplet algebras

Complete classification of irreducible SW(m)-modules

Computed bosonic and fermionic characters of modules

Abstract

We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We classify irreducible SW(m)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible SW(m) characters. Finally, we contemplate possible connections between the category of SW(m)-modules and the category of modules for the quantum group U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on properties of characters and the Zhu's algebra A(SW(m)). This paper is a continuation of arXiv:0707.1857.