Large-$N$ Carrollian Thermodynamics from AdS Black-Hole Phase-Space Contractions

Abstract

We develop a boundary large-$N$ interpretation of finite Carrollian black-hole thermodynamics and formulate its thermal Carrollian correlators in a celestial conformal-primary basis. The bulk input is a phase-space contraction, developed in earlier work, in which the time generator and Newton constant are scaled so that the extended AdS first law has a finite Carrollian limit. We show that this finite sector is naturally realized holographically as a double-scaled low-temperature, large-$N$ ensemble: the Carrollian temperature decreases while the effective number of boundary degrees of freedom grows, leaving the thermodynamic products in the first law finite. The large-$N$ dictionary is anchored by the standard $\mathrm{AdS}_5/\mathrm{CFT}_4$ normalization and by the Brown--Henneaux central charge in $\mathrm{AdS}_3/\mathrm{CFT}_2$. We construct the finite Carrollian Brown--York stress tensor on the contracted AdS boundary and show that its global energy charge reproduces the contracted bulk Hamiltonian. The boundary first law rewrites the bulk pressure term as a combined variation of spatial volume and holographic normalization, and identifies the Hawking--Page locus with the zero of the corresponding chemical potential conjugate to the holographic normalization. The same energy is the constant mode of a Carrollian time-shift charge, so the first law is the thermal zero-mode sector of a Carrollian time-shift Ward identity. Finally, we formulate the thermal celestial-basis correlators as positive/negative-frequency Mellin transforms of Carrollian spectral kernels.