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

TL;DR

The paper constructs a two-parameter quantum deformation SVir_{q,k} of the ${\mathcal{N}}=2$ superconformal algebra, establishing quartic (and lower) relations and a q->1 limit yielding the classical algebra with central charge \(c=\frac{3k}{k+2}\). It develops a generating-function formalism, builds Verma modules, and conjectures a factorizable Kac determinant with supporting NS/R-sector examples and screening-based evidence. A Wakimoto-type free-field realization is developed by twisting a U(1) boson and gluing two deformed Y-algebras, revealing deep links to deformed parafermions and the Ding-Iohara-Miki algebra, and suggesting a route to a quantum toroidal uplift. The conformal limit analysis and the Wakimoto construction together provide a nontrivial, explicit representation theory for SVir_{q,k}, with potential connections to DIM/quiver algebras and refined topological-strings related structures.

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$\check{\mathrm{c}}$ák.

Figures (6)

• Figure 1: $5$-brane junction with $D3$ brane configuration (see also the table \ref{['D5-D3']}). Our convention of the ordering is counterclockwise. The orientation of the edges is outgoing.
• Figure 2: Gluing of two $Y$-algebras $Y_{K,M,L}[\Psi]$ and $Y_{K,N,M}[\Psi-1]$ according to the toric diagram of ALE space of type $A_1$.
• Figure 3: $\mathcal{N}=2$ superconformal algebra $\times U(1)$ (left) vs. $\widehat{\mathfrak{sl}}_2 \times U(1)$ (right) by the gluing of two $Y$-algebras.
• Figure 4: Gluing for $U_q(\widehat{\mathfrak{sl}}_2)$
• Figure 5: Gluing for $\mathcal{N}=2$ superconformal algebra

Theorems & Definitions (162)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Remark 1.5
• Proposition 1.6
• proof
• Definition 1.7
• Proposition 1.8
• proof