A Matrix View of Holographic Null-State Equations in $d\geq 2$

Abstract

We show that the null-state-type partial differential equations recently derived from AdS gravity with a black hole can be efficiently organized using finite-dimensional matrices with systematic off-diagonal structures. This matrix formulation generates an infinite tower of CFT differential equations at arbitrary levels. The $d=2$ matrix reproduces a known result based on Virasoro null-state vectors at large central charge and admits an algebraic representation in terms of a linear combination of the identity matrix and $\mathfrak{sl}_2$ generators, with a heavy-state deformation encoding the central charge contribution. The $d=4$ governing matrix exhibits a similar yet richer structure involving products of $\mathfrak{sl}_2$ generators. We also discuss recursion relations and factorized product structures associated with these holographic differential equations.