# Modularity of logarithmic parafermion vertex algebras

## TL;DR

This paper investigates the structure and modular properties of logarithmic parafermion vertex algebras, specifically the cosets and their extensions at negative admissible levels, revealing finite irreducible modules and modular group representations.

## Contribution

It provides a detailed analysis of the modularity and irreducible modules of logarithmic parafermion vertex algebras and their extensions, assuming tensor category applicability.

## Key findings

- Finite irreducible modules for $C_k$ and $B_k$ are classified.
- Characters of $B_k$ modules form a finite-dimensional modular group representation.
- Conjecture that $B_k$ are $C_2$-cofinite vertex operator algebras.

## Abstract

The parafermionic cosets $C_k = \mathrm{Com} (H, L_k(\mathfrak{sl}_2) )$ are studied for negative admissible levels $k$, as are certain infinite-order simple current extensions $B_k$ of $C_k$. Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to $C_k$, all irreducible $C_k$- and $B_k$-modules are obtained from those of $L_k(\mathfrak{sl}_2)$, as are the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible $B_k$-modules. The irreducible $C_k$- and $B_k$-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the $B_k$ are $C_2$-cofinite vertex operator algebras.