The ${\cal N}=2$ Supersymmetric $w_{1+\infty}$ Symmetry in the Two-Dimensional SYK Models

TL;DR

This paper identifies a connection between the two-dimensional ${ m N}=2$ SYK model's interaction rank and a deformation parameter in a linear $W_{ infty}[ lambda]$ algebra, revealing new algebraic structures and their dependence on model parameters.

Contribution

It establishes a detailed correspondence between the ${ m N}=2$ SYK model and a deformed linear $W_{ infty}[ lambda]$ algebra, including explicit structure constants and special cases.

Findings

The rank of the interaction relates to the deformation parameter lambda.

Complete algebraic structure constants are expressed via hypergeometric functions.

Special case at lambda=1/4 shows vanishing structure constants.

Abstract

We identify the rank $(q_{syk}+1)$ of the interaction of the two-dimensional ${\cal N}=(2,2)$ SYK model with the deformation parameter $\lambda$ in the Bergshoeff, de Wit and Vasiliev(in 1991)'s linear $W_{\infty}[\lambda]$ algebra via $\lambda =\frac{1}{2(q_{syk}+1)}$ by using a matrix generalization. At the vanishing $\lambda$ (or the infinity limit of $q_{syk}$), the ${\cal N}=2$ supersymmetric linear $W_{\infty}^{N,N}[\lambda=0]$ algebra contains the matrix version of known ${\cal N}=2$ $W_{\infty}$ algebra, as a subalgebra, by realizing that the $N$-chiral multiplets and the $N$-Fermi multiplets in the above SYK models play the role of the same number of $\beta\, \gamma$ and $b\, c$ ghost systems in the linear $W_{\infty}^{N,N}[\lambda=0]$ algebra. For the nonzero $\lambda$, we determine the complete ${\cal N}=2$ supersymmetric linear $W_{\infty}^{N,N}[\lambda]$ algebra where the structure constants are given by the linear combinations of two different generalized hypergeometric functions having the $\lambda$ dependence. The weight-$1, \frac{1}{2}$ currents occur in the right hand sides of this algebra and their structure constants have the $\lambda$ factors. We also describe the $\lambda =\frac{1}{4}$ (or $q_{syk}=1$) case in the truncated subalgebras by calculating the vanishing structure constants.