The $$ \mathcal{N} $$ = 2 supersymmetric w1+∞ symmetry in the two-dimensional SYK models

Abstract

Abstract We identify the rank (qsyk + 1) of the interaction of the two-dimensional $$ \mathcal{N} $$ 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 $$ \lambda =\frac{1}{2\left({q}_{\mathrm{syk}}+1\right)} $$ λ = 1 2 q syk + 1 by using a matrix generalization. At the vanishing λ (or the infinity limit of qsyk), the $$ \mathcal{N} $$ N = 2 supersymmetric linear $$ {W}_{\infty}^{N,N} $$ W ∞ N , N [λ = 0] algebra contains the matrix version of known $$ \mathcal{N} $$ 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 $$ {W}_{\infty}^{N,N} $$ W ∞ N , N [λ = 0] algebra. For the nonzero λ, we determine the complete $$ \mathcal{N} $$ N = 2 supersymmetric linear $$ {W}_{\infty}^{N,N} $$ 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,$$ \frac{1}{2} $$ 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 = $$ \frac{1}{4} $$ 1 4 (or qsyk = 1) case in the truncated subalgebras by calculating the vanishing structure constants.