The N = 2 supersymmetric w _1+∞ symmetry in the two-dimensional SYK models

We identify the rank (q_syk + 1) of the interaction of the two-dimensional N = (2, 2) SYK model with the deformation parameter λ in the Bergshoeff, de Wit and Vasiliev (in 1991)’s linear W_∞[λ] algebra via λ=12(qsyk+1) by using a matrix generalization. At the vanishing λ (or the infinity limit of q_syk), the N = 2 supersymmetric linear W∞N,N[λ = 0] algebra contains the matrix version of known N = 2 W_∞ 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 βγ and bc ghost systems in the linear W∞N,N[λ = 0] algebra. For the nonzero λ, we determine the complete N = 2 supersymmetric linear W∞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-1,12 currents occur in the right hand sides of this algebra and their structure constants have the λ factors. We also describe the λ = 14 (or q_syk = 1) case in the truncated subalgebras by calculating the vanishing structure constants.