On the extended W-algebra of type sl_2 at positive rational level

TL;DR

This paper constructs and analyzes the extended W-algebra of type sl_2 at positive rational levels, establishing its structure, module classification, and properties relevant to logarithmic conformal field theories.

Contribution

It provides a new construction of the algebra using screening operators and proves c_2-cofiniteness, Zhu algebra structure, and module classification.

Findings

Proved c_2-cofiniteness of M_{p_+,p_-}

Calculated the Zhu algebra structure

Classified all simple modules

Abstract

The extended W-algebra of type sl_2 at positive rational level, denoted by M_{p_+,p_-}, is a vertex operator algebra that was originally proposed in [1]. This vertex operator algebra is an extension of the minimal model vertex operator algebra and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_{p_+,p_-} in terms of screening operators and use this construction to prove that M_{p_+,p_-} satisfies Zhu's c_2-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra) and classify all simple M_{p_+,p_-}-modules.