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.
Categories: hep-th