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

Abstract

We introduce a unital associative algebra ${\mathcal{S}\mathcal{V}ir}_{q,k}$, having $q$ and $k$ as complex parameters, generated by the elements $K_m^{\pm }$ ($\pm m\ge 0$), $T_m$ ($m\in \mathbb{Z}$), and $G_m^{\pm }$ ($m\in \mathbb{Z}+\frac{1}{2}$ 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 $q\to 1$ of ${\mathcal{S}\mathcal{V}ir}_{q,k}$ we recover the ordinary $\mathcal{N}=2$ superconformal algebra. We also give a nontrivial Heisenberg representation of the algebra ${\mathcal{S}\mathcal{V}ir}_{q,k}$, 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 ${\mathcal{S}\mathcal{V}ir}_{q,k}$ by gluing the deformed $Y$-algebras of Gaiotto and Rap$\check{\mathrm{c}}$ák.

85 pages,(v2) several elucidations provided, typos fixed