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

From the recently known ${\cal N}=2$ supersymmetric linear $W_{\infty}^{K,K}[λ]$ algebra where $K$ is the dimension of fundamental (or antifundamental) representation of bifundamental $β\, γ$ 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 $λ_{co} \equiv \frac{(N+1)}{(k+N+2)}$, up to two central terms. The two parameters are related to each other $λ=\frac{1}{2}\, λ_{co}$. 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 $W_{\infty}[λ]$ algebra.

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