A Supersymmetric Extension of $w_{1+\mathrm{\infty }}$ Algebra in the Celestial Holography

We determine the $\mathcal{N}=1$ supersymmetric topological $W_{\mathrm{\infty }}$ algebra by using the $\lambda$ deformed bosons $(\beta ,\gamma )$ and fermions $(b,c)$ ghost system. By considering the real bosons and the real fermions at $\lambda =0$ (or $\lambda =\frac{1}{2}$), the $\mathcal{N}=1$ supersymmetric $W_{\frac{\mathrm{\infty }}{2}}$ algebra is obtained. At $\lambda =\frac{1}{4}$, other $\mathcal{N}=1$ supersymmetric $W_{1+\mathrm{\infty }}[\lambda =\frac{1}{4}]$ algebra is determined. We also obtain the extension of Lie superalgebra $PSU(2,2\mathrm{\mid }\mathcal{N}=4)$ appearing in the worldsheet theory by using the symplectic bosons and the fermions. We identify the soft current algebra between the graviton, the gravitino, the photon (the gluon), the photino (the gluino) or the scalars, equivalent to $\mathcal{N}=1$ supersymmetric $W_{1+\mathrm{\infty }}[\lambda ]$ algebra, in two dimensions with the $\mathcal{N}=1$ supergravity theory in four dimensions discovered by Freedman, van Nieuwenhuizen and Ferrara in 1976 and its matter coupled theories, via celestial holography.