Vector Models in the Singlet Sector at Finite Temperature

TL;DR

This work analyzes the thermal behavior of the $O(N)$ vector model in the $O(N)$ singlet sector in 2+1 dimensions, testing its conjectured AdS/CFT duality to Vasiliev higher spin gravity. Using a matrix-model approach that enforces singlet constraints, the authors compute the boundary partition function and contrast it with the bulk higher-spin gas, revealing a large-$N$ phase transition at $T \sim \sqrt{N}$ (Planck-scale bulk energy) where the effective DOF count is reduced due to relations among bilinear invariants. The results indicate no thermodynamically stable large AdS black hole at moderate temperatures and show that nonperturbative corrections are too small to correspond to black-hole configurations, with the transition persisting in the interacting, critical vector model at $T \approx 0.581068\sqrt{N}$. Finite-$N$ relations among invariants further constrain the high-temperature behavior, highlighting the role of operator relations in shaping thermodynamics. Overall, the paper clarifies how singlet-projected vector models align with higher-spin bulk expectations and where conventional bulk black holes may be absent or subdominant.

Abstract

We study the thermal properties of the O(N) vector-like scalar theory in the singlet sector in 2+1 dimensions. This theory is conjectured to be the AdS/CFT dual of Vasiliev higher spin gravity. We find that a large N transition occurs but only at a very high temperature of order \sqrt{N}. This corresponds to the bulk Planck energy. The transition signals a decrease in the number of degrees of freedom from that expected in the simple higher spin gas, due to relations among the O(N) bilinear invariants.