Supersymmetric String Configurations in Ad$\mathbb{S}_5\times \mathbb{S}^5

In this work, we investigate off-shell supersymmetric string configurations in Euclidean Ad$\mathbb{S}_5\times \mathbb{S}^5$ background. By extending the Green-Schwarz supersymmetric equations using the pure spinor formalism, we demonstrate that while the solutions to the Green-Schwarz equations are necessarily minimal surfaces, the solutions to the extended equations form infinite-dimensional families, even with fixed boundary conditions, thus preventing direct derivations of exact results via supersymmetric localization. For concreteness, we have selected a supercharge that localizes the off-shell supersymmetric configurations to an Ad$\mathbb{S}_4 \times \mathbb{S}^2$ slice and observed that the corresponding action functional reduces to an integral of a two-form.