The ${\cal N}=4$ Supersymmetric Linear $W_{\infty}[\lambda]$ Algebra

TL;DR

This paper constructs an ${ m N}=4$ supersymmetric extension of the linear $W_{ m ext{infinity}}[ ext{lambda}]$ algebra from a known ${ m N}=2$ algebra, analyzing its operator product expansions and relation to coset models.

Contribution

It explicitly constructs the ${ m N}=4$ supersymmetric algebra and its multiplets, and relates its OPEs to those in the ${ m N}=4$ coset model in a large N, k limit.

Findings

OPEs match those of the ${ m N}=4$ coset model in a specific limit.

Constructed the ${ m N}=4$ stress energy tensor and multiplets.

Established relations between parameters lambda and lambda_co.

Abstract

From the recently known ${\cal N}=2$ supersymmetric linear $W_{\infty}^{K,K}[\lambda]$ algebra where $K$ is the dimension of fundamental (or antifundamental) representation of bifundamental $\beta \, \gamma$ and $b \, c$ ghost system, we determine its ${\cal N}=4$ supersymmetric enhancement at $K=2$. We construct the ${\cal N}=4$ stress energy tensor, the first ${\cal N}=4$ multiplet and their operator product expansions (OPEs) in terms of above bifundamentals. We show that the OPEs between the first ${\cal N}=4$ multiplet and itself are the same as the corresponding ones in the ${\cal N}=4$ coset $\frac{SU(N+2)}{SU(N)}$ model under the large $(N,k)$ 't Hooft-like limit with fixed $\lambda_{co} \equiv \frac{(N+1)}{(k+N+2)}$, up to two central terms. The two parameters are related to each other $\lambda =\frac{1}{2}\, \lambda_{co}$. We also provide other OPEs by considering the second, the third and the fourth ${\cal N}=4$ multiplets in the ${\cal N}=4$ supersymmetric linear $W_{\infty}[\lambda]$ algebra.