The N $$ \mathcal{N} $$ = 2 supersymmetric w 1+∞ symmetry in the two-dimensional SYK models

Abstract

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

Keywords

• Conformal and W Symmetry
• Higher Spin Symmetry