Incomplete RG: Hawking-Page transition, C-theorem and relevant scalar deformations of global AdS

TL;DR

This paper explores how relevant scalar deformations affect the RG flow, entanglement entropy, and phase transitions in holographic theories with compact boundaries, revealing incomplete RG flows and modified Hawking-Page transitions.

Contribution

It introduces the concept of incomplete RG flows and an incomplete C-theorem in holographic theories with compact boundaries, analyzing their effects on phase transitions and geometric quantities.

Findings

Incomplete RG flow terminates at finite scale due to boundary size.

Deformation influences entanglement entropy and scalar curvature monotonically.

Hawking-Page transition temperature increases, but black hole size decreases at transition.

Abstract

We discuss relevant scalar deformations of a holographic theory with a compact boundary. An example of such a theory would be the global AdS$_4$ with its spatially compact boundary $S^2$. To introduce a relevant deformation, we choose to turn on a time-independent and spatially homogeneous non-normalizable scalar operator with $m^2 = -2$. The finite size of a compact boundary cuts down the RG flow at a finite length scale leading to an incomplete RG flow to IR. We discuss a version of {\it incomplete} C-theorem and an {\it incomplete} attractor like mechanism. We discuss the implication of our results for entanglement entropy and geometric quantities like scalar curvature, volume and mass scale of fundamental excitation of the how these quantities increase or decrease (often monotonically) with the strength of the deformation. Thermal physics of a holographic theory defined on a compact boundary is more interesting than its non-compact counterpart. It is well known that with a compact boundary, there is a possibility of a first order Hawking-Page transition dual to a de-confinement phase transition. From a gravity perspective, a relevant deformation dumps negative energy inside the bulk, increasing the effective cosmological constant ($\Lambda$) of the AdS. Dumping more negative energy in the bulk would make the HP transition harder and the corresponding HP transition temperature would increase. However, we have found the size of the BH at the transition temperature decreases.