Renormalization Group Flows from Gravity in Anti-de Sitter Space versus Black Hole No-Hair Theorems

TL;DR

This paper explores the inequalities governing holographic renormalization group flows in AdS spaces, revealing that generic flows tend to produce singular geometries and discussing implications for the black hole no-hair theorems.

Contribution

It extends black hole no-hair theorem techniques to holographic RG flows, analyzing inequalities and their implications for the geometry of dual gravitational solutions.

Findings

Generic relevant or irrelevant operator flows lead to singular geometries.

Potential conflict with Polchinski's decoupling theorem for irrelevant operators.

Proposes resolutions to reconcile these issues.

Abstract

Black hole no-hair theorems are proven using inequalities that govern the radial dependence of spherically symmetric configurations of matter fields. In this paper, we analyze the analogous inequalities for geometries dual to renormalization group flows via the AdS/CFT correspondence. These inequalities give much useful information about the qualitative properties of such flows. For Poincare invariant flows, we show that generic flows of relevant or irrelevant operators lead to singular geometries. For the case of irrelevant operators, this leads to an apparent conflict with Polchinski's decoupling theorem, and we offer two possible resolutions to this problem.