Vacuum energy is non-positive for (2+1)-dimensional holographic CFTs

TL;DR

This paper proves that the vacuum energy of (2+1)-dimensional holographic CFTs on static spacetimes is non-positive, with the vacuum energy being negative unless the boundary metric is conformally a product of time and a constant curvature space.

Contribution

It provides a geometric argument establishing a non-positivity bound on vacuum energy for these CFTs, extending previous understanding of holographic energy bounds.

Findings

Vacuum energy is non-positive for (2+1)-dimensional holographic CFTs.

Vacuum energy is negative unless the boundary metric is conformally a product of time and constant curvature space.

The argument applies even with certain bulk singularities, as long as they are hidden by horizons.

Abstract

We consider a (2+1)-dimensional holographic CFT on a static spacetime with globally timelike Killing vector. Taking the spatial geometry to be closed but otherwise general we expect a non-trivial vacuum energy at zero temperature due to the Casimir effect. We assume a thermal state has an AdS/CFT dual description as a static smooth solution to gravity with a negative cosmological constant, which ends only on the conformal boundary or horizons. A bulk geometric argument then provides an upper bound on the ratio of CFT free energy to temperature. Considering the zero temperature limit of this bound implies the vacuum energy of the CFT is non-positive. Furthermore the vacuum energy must be negative unless the boundary metric is locally conformal to a product of time with a constant curvature space. We emphasise the argument does not require the zero temperature bulk geometry to be smooth, but only that singularities are `good' so are hidden by horizons at finite temperature.