A supersymmetric extension of w_1+∞ algebra in the celestial holography

We determine the $\mathcal{N}$ \( \mathcal{N} \) = 1 supersymmetric topological W_∞ algebra by using the λ deformed bosons (β, γ) and fermions (b, c) ghost system. By considering the real bosons and the real fermions at λ = 0 (or λ = $\frac{1}{2}$ \( \frac{1}{2} \) ), the $\mathcal{N}$ \( \mathcal{N} \) = 1 supersymmetric $W_{\frac{\infty }{2}}$ \( {W}_{\frac{\infty }{2}} \) algebra is obtained. At λ = $\frac{1}{4}$ \( \frac{1}{4} \) , other $\mathcal{N}$ \( \mathcal{N} \) = 1 supersymmetric W_1+∞[λ = $\frac{1}{4}$ \( \frac{1}{4} \) ] algebra is determined. We also obtain the extension of Lie superalgebra PSU(2, 2| $\mathcal{N}$ \( \mathcal{N} \) = 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 $\mathcal{N}$ \( \mathcal{N} \) = 1 supersymmetric W_1+∞[λ] algebra, in two dimensions with the $\mathcal{N}$ \( \mathcal{N} \) = 1 supergravity theory in four dimensions discovered by Freedman, van Nieuwenhuizen and Ferrara in 1976 and its matter coupled theories, via celestial holography.