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

• Published in: arXiv.org 2024-07
• Main Authors: Awata, H, Harada, K, Kanno, H, Shiraishi, J
• Format: Article
• Language: English
• Subjects: Algebra; Deformation; Operators (mathematics); Representations
• ISSN: 2331-8422

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.