D-branes and nonlinear interactions in the $AdS$ pure spinor string

We develop a coordinate-independent algebraic description of zero-field boundary conditions for the pure spinor string on $AdS_5\times S^5$ in the closed-string channel. The construction is organized by involutive automorphisms of $\mathfrak{psu}(2,2|4)$ that exchange the odd sectors of the $\mathbb Z_4$ grading. Their fixed subalgebras provide candidate half-BPS, Lorentzian D-brane geometries with vanishing background world-volume flux, consistent with the standard $AdS_2$, $AdS_3\times S^1$, $AdS_4\times S^2$, and related embeddings. Together, the seven representatives realize the Lorentzian zero-flux D1, D3, D5, and D7 probe sectors considered here. We then construct a local ansatz for the matter and ghost boundary interactions and derive a system of consistency equations for the tangential superconnection, transverse embedding, and four ghost-gluing couplings. The resulting system is a set of Dirac-Born-Infeld-like equations of motion on the brane world-volume.