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

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