Dirichlet gravitons in AdS$_4$ and $\mathcal{L}_{\Lambda}w_{1+\infty}$ wedge algebra

Abstract

We derive the AdS deformation of the flat-space soft-graviton algebra with Dirichlet boundary conditions. Starting from the AdS$_4$ spinor-helicity representation, we Mellin-transform linearized graviton solutions and construct the AdS wedge, including its Laurent completion. The Dirichlet boundary condition pairs opposite helicities, while AdS covariance fixes their relative normalization. Assuming linear closure and normalizing the global modes to reproduce the geometric AdS isometries, covariance and the Jacobi identities uniquely determine the soft-mode bracket. The result is the wedge of $\mathcal L_\Lambda w_{1+\infty}$, where $\Lambda=-\ell^{-2}$. This provides a bulk derivation of the cosmological-constant deformation that incorporates the AdS boundary condition, without relying on collinear splitting functions or a self-dual truncation.