$q$-Virasoro/W Algebra at Root of Unity and Parafermions

TL;DR

This paper explores the emergence of parafermions in the root of unity limit of $q$-Virasoro/$W_n$ algebra, linking it to specific coset models and their central charges.

Contribution

It reveals the appearance of parafermions in the root of unity limit of $q$-Virasoro/$W_n$ algebra and determines the associated central charge of related coset models.

Findings

Parafermions appear in the root of unity limit of $q$-Virasoro/$W_n$ algebra.

The central charge of a specific coset model is derived from parafermion construction.

The work connects algebraic limits to conformal field theory structures.

Abstract

We demonstrate that the parafermions appear in the $r$-th root of unity limit of $q$-Virasoro/$W_n$ algebra. The proper value of the central charge of the coset model $ \frac{\widehat{\mathfrak{sl}}(n)_r \oplus \widehat{\mathfrak{sl}}(n)_{m-n}}{\widehat{\mathfrak{sl}}(n)_{m-n+r}}$ is given from the parafermion construction of the block in the limit.