A quantum deformation of the ${\mathcal N}=2$ superconformal algebra

TL;DR

This paper introduces a quantum deformation of the ${\mathcal N}=2$ superconformal algebra, exploring its structure, representations, and conjectured properties, and demonstrating how it reduces to the classical algebra in a certain limit.

Contribution

The paper constructs a new quantum algebra ${\mathcal{SV}ir}_{q,k}$, analyzes its properties, and connects it to known structures like the ${\mathcal N}=2$ superconformal algebra and quantum affine algebras.

Findings

Conjecture on the factorization of Kac determinants.

Evidence supporting the Kac determinant conjecture from screening operators.

Recovery of the classical ${\mathcal N}=2$ superconformal algebra in the limit $q\to 1$.

Abstract

We introduce a unital associative algebra ${\mathcal{SV}ir\!}_{q,k}$, having $q$ and $k$ as complex parameters, generated by the elements $K^\pm_m$ ($\pm m\geq 0$), $T_m$ ($m\in \mathbb{Z}$), and $G^\pm_m$ ($m\in \mathbb{Z}+{1\over 2}$ in the Neveu-Schwarz sector, $m\in \mathbb{Z}$ in the Ramond sector), satisfying relations which are at most quartic. Calculations of some low-lying Kac determinants are made, providing us with a conjecture for the factorization property of the Kac determinants. The analysis of the screening operators gives a supporting evidence for our conjecture. It is shown that by taking the limit $q\rightarrow 1$ of ${\mathcal{SV}ir\!}_{q,k}$ we recover the ordinary ${\mathcal N}=2$ superconformal algebra. We also give a nontrivial Heisenberg representation of the algebra ${\mathcal{SV}ir\!}_{q,k}$, making a twist of the $U(1)$ boson in the Wakimoto representation of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$, which naturally follows from the construction of ${\mathcal{SV}ir\!}_{q,k}$ by gluing the deformed $Y$-algebras of Gaiotto and Rap$\check{\mathrm{c}}$\'{a}k.