A Supersymmetric $w_{1+\mathrm{\infty }}$ Symmetry, the Extended Supergravity and the Celestial Holography

We determine the $\mathcal{N}=4$ supersymmetric $W_{1+\mathrm{\infty }}^{2,2}[\lambda =\frac{1}{4}]$ algebra which is an extension of $\mathcal{N}=4$ $SO(4)$ superconformal algebra with vanishing central charge. We identify the soft current algebra between the graviton, the gravitinos, the vectors, the Majorana fermions, the scalar or the pseudoscalar, from the $\mathcal{N}=4$ supersymmetric $w_{1+\mathrm{\infty }}^{2,2}[\lambda =\frac{1}{4}]$ algebra, in two dimensions with the $\mathcal{N}=4$ supergravity theory with $SO(4)$ global symmetry in four dimensions found by Das (at Stony Brook in 1977), via celestial holography. Furthermore, the truncations of $\mathcal{N}=4$ supersymmetric soft current algebra provide the soft current algebras for the $\mathcal{N}=2,3$ supergravity theories, the $\mathcal{N}=2$ supergravity coupled to its Abelian vector multiplet and the $\mathcal{N}=1$ supersymmetric Maxwell Einstein theory. For the $\mathcal{N}=2$ supergravity theory, the soft current algebra can be also realized from the $\mathcal{N}=2$ supersymmetric $w_{1+\mathrm{\infty }}^{K,K}[\lambda =0]$ algebra.