# Elliptic algebra, Frenkel-Kac construction and root of unity limit

## TL;DR

This paper explores the elliptic algebra's role as a dynamical symmetry in 2d/5d correspondence, revealing new connections to parafermions, free bosons, and 2d coset CFTs at roots of unity.

## Contribution

It introduces a realization of the level-1 elliptic algebra using an elliptic Frenkel-Kac construction and analyzes its root of unity limit revealing connections to parafermions and coset CFTs.

## Key findings

- Realization of level-1 elliptic algebra via elliptic Frenkel-Kac construction.
- Emergence of parafermions and free bosons at roots of unity.
- Matching of central charge with 2d coset CFT with para-Virasoro symmetry.

## Abstract

We argue that the level-$1$ elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is a dynamical symmetry realized as a part of 2d/5d correspondence where the Drinfeld currents are the screening currents to the $q$-Virasoro/W block in the 2d side. For the case of $U_{q,p}(\widehat{\mathfrak{sl}}(2))$, the level-$1$ module has a realization by an elliptic version of the Frenkel-Kac construction. The module admits the action of the deformed Virasoro algebra. In a $r$-th root of unity limit of $p$ with $q^2 \rightarrow 1$, the $\mathbb{Z}_r$-parafermions and a free boson appear and the value of the central charge that we obtain agrees with that of the 2d coset CFT with para-Virasoro symmetry, which corresponds to the 4d $\mathcal{N}=2$ $SU(2)$ gauge theory on $\mathbb{R}^4/\mathbb{Z}_r$.