Holographic Derivation of BPZ-Type Null State Equations in Higher Dimensional CFTs

A set of linear differential equations was recently put forward as higher-dimensional generalizations of the BPZ null-state equations in two-dimensional CFTs at large central charge. In this work, we derive these higher-dimensional equations from gravity, based on the AdS/CFT correspondence. A near-boundary expansion is employed to analyze a light scalar field equation in a black hole background. There is a decoupling mechanism in the bulk perturbative series at certain conformal dimensions, resulting in isolated lower-order equations. We find that the results agree with the previously proposed four-dimensional CFT equations, which capture the resummed contributions from minimal-twist multi-stress tensor operators. The holographic calculation also allows one to obtain additional CFT differential equations that extend beyond the near-lightcone regime.