Eur. Phys. J. C (2023) 83: 615
https://doi.org/10.1140/epjc/s10052-023-11752-z

Regular Article - Theoretical Physics

The structure of the $\mathcal{N}=4$ supersymmetric linear $W_{\infty }[\lambda ]$ algebra

¹ Department of Physics, Kyungpook National University, 41566, Taegu, South Korea
² Department of Physics, New York University, 726 Broadway, 10003, New York, NY, USA

^a ahn@knu.ac.kr

Received: 18 September 2022
Accepted: 24 June 2023
Published online: 14 July 2023

Abstract

For the vanishing deformation parameter $\lambda$, the full structure of the (anti)commutator relations in the $\mathcal{N}=4$ supersymmetric linear $W_{\infty }[\lambda =0]$ algebra is obtained for arbitrary weights $h_1$ and $h_2$ of the currents appearing on the left hand sides in these (anti)commutators. The $w_{1+\infty }$ algebra can be seen from this by taking the vanishing limit of other deformation parameter q with the proper contractions of the currents. For the nonzero $\lambda$, the complete structure of the $\mathcal{N}=4$ supersymmetric linear $W_{\infty }[\lambda ]$ algebra is determined for the arbitrary weight $h_1$ together with the constraint $h_1-3\le h_2\le h_1+1$. The additional structures on the right hand sides in the (anti)commutators, compared to the above $\lambda =0$ case, arise for the arbitrary weights $h_1$ and $h_2$ where the weight $h_2$ is outside of above region.