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

Abstract

We identify the rank $(q_{syk}+1)$ of the interaction of the two-dimensional $\mathcal{N}=(2,2)$ 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 $\mathcal{N}=2$ supersymmetric linear $W_{\infty }^{N,N}[\lambda =0]$ algebra contains the matrix version of known $\mathcal{N}=2$ $W_{\infty }$ 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 $bc$ ghost systems in the linear $W_{\infty }^{N,N}[\lambda =0]$ algebra. For the nonzero $\lambda$, we determine the complete $\mathcal{N}=2$ 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.

arXiv comment48 pages

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Cited by in corpus (3)

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• The Structure of the $\mathcal{N}=4$ Supersymmetric Linear $W_{\infty }[\lambda ]$ Algebra
• A route from maximal chaoticity to integrability