AdS/CFT and the geometry of an energy gap

TL;DR

This paper investigates how the energy gap in a holographic conformal field theory depends on the boundary space's geometry, showing it is minimized for spherical geometries through geometric and holographic arguments.

Contribution

It extends the understanding of the energy gap dependence on geometry from free scalars to holographic CFTs using geometric and holographic methods.

Findings

The energy gap normalized by the minimum Ricci scalar is minimized for spherical boundary geometries.

Holographic black hole states exhibit bounds on energy and entropy ratios similar to scalar fluctuations.

The results hold for any smooth Einstein boundary geometry, not just spheres.

Abstract

We consider a CFT defined on a static metric that is the product of time with a smooth closed space of positive scalar curvature. We expect the theory to exhibit an energy gap and our aim is to investigate how that gap depends on the geometry of the space. For a free conformal scalar it is straightforward to show the gap normalised by the minimum value of the Ricci scalar of the space is minimised when the space is a sphere. Our main result is then to show using geometric arguments that precisely the same result holds for fluctuations of a scalar operator in any holographic CFT. We prove this under the assumption that the dual vacuum geometry is a smooth Einstein metric ending only on the conformal boundary, and then consider fluctuations of a minimally coupled massive scalar field about this. We also argue the holographic CFT will have states dual to small bulk black holes whose existence is related to the energy gap. We show the thermodynamic properties of these black holes obey a bound of a similar nature to that of the scalar fluctuations - the ratio of CFT energy to a power of entropy for the states dual to black holes in the 'small' limit is bounded from below when appropriately normalised by the minimum Ricci scalar of the boundary space. Again the bound is saturated for a sphere.