Melonic ladder holography with a single free bulk field in AdS$_{d+1}$

Abstract

We derive a cylinder from the pair Casimir of a CFT$_d$: at zero centre momentum it depends only on relative dilations and rotations. Relative-coordinate radial quantisation then encodes exchanged family labels as cylinder energies. For a melonic CFT$_d$ in the normalisable and stable range, the reduced ladder sum is the Robin boundary-to-boundary propagator of a single free scalar field in rigid AdS$_{d+1}$, with the cylinder as its conformal boundary. The Robin boundary condition and bulk regularity (or parity) determine the holographic radial extension of the boundary data. A nonzero Robin boundary coefficient in global AdS breaks $2d$ AdS generators, yet the scalar's boundary propagator still encodes the conformal data of the melonic ladder. We demonstrate the correspondence for the Sachdev--Ye--Kitaev and the $d=2$ disorder-free long-range Amit--Roginsky melonic models, which correspond to a single free local scalar in AdS$_2$ and AdS$_3$, respectively.