1. A quantum deformation of the ${\mathcal N}=2$ superconformal algebra
- Author
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Awata, H., Harada, K., Kanno, H., and Shiraishi, J.
- Subjects
Mathematics - Quantum Algebra ,High Energy Physics - Theory ,Mathematical Physics - 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., Comment: 83 pages
- Published
- 2024