[Paper Review] A Variational Principle for Radial Flows in Holographic theories

Ratindranath Akhoury|ArXiv.org|Jul 6, 2000

Black Holes and Theoretical Physics20 references3 citations

TL;DR

This paper formulates a coordinate-independent variational principle for radial flows in holographic theories, treating the radial coordinate ρ as a renormalization group parameter. It derives flow equations for the effective action on d-dimensional surfaces, reproduces known results in AdS supergravity, and establishes generalized junction conditions for domain walls, enabling a systematic study of brane-world scenarios and boundary conditions in holography.

ABSTRACT

We develop furthur the correspondence between a d+1 dimensional theory and a d dimensional one with the "radial" (d+1)th corodinate ρplaying the role of an evolution parameter. We discuss the evolution of an effective action defined on a d dimensional surface charactarized by ρby means of a new variational principle. The conditions under which the flow equations are valid are discussed in detail as is the choice of boundary conditions. It is explained how domain walls may be incorporated in the framework and some generalized junction conditions are obtained. The general principles are illustrated on the example of a supergravity theory on AdS_{d+1}.

Motivation & Objective

• To develop a general, coordinate-independent variational principle for the evolution of effective actions along the radial coordinate ρ in holographic theories.
• To reformulate the radial evolution of bulk fields as a minimization problem, linking it to Hamilton-Jacobi formalism and renormalization group flow.
• To derive and analyze boundary conditions for flow equations, particularly in the presence of domain walls or branes.
• To establish criteria for the validity of the classical approximation in flow equations by studying unbounded second derivatives of the action.
• To generalize junction conditions at domain walls and connect them to induced gravity and cosmological constants on branes.

Proposed method

• Formulates a multi-stage optimization procedure where the effective action S on a d-dimensional surface at fixed ρ is minimized under variation of fields.
• Derives flow equations by requiring the action integral to be stationary under variations of the fields, leading to a first-order system in ρ.
• Applies the variational principle to supergravity on AdS_{d+1}, reproducing known flow equations from previous works.
• Introduces a criterion for the validity of the classical approximation by analyzing when second partial derivatives of S become unbounded, leading to a Jacobi-type differential equation.
• Derives generalized junction conditions at domain walls by matching the normal derivatives of the effective action across the wall, using the variational principle.
• Uses the framework to compute induced gravity terms on branes, including Newton's constant and higher-order curvature corrections, matching Randall-Sundrum results.

Experimental results

Research questions

• RQ1 How can a variational principle be formulated to describe the radial evolution of effective actions in holographic theories?
• RQ2 What are the conditions under which the classical approximation in radial flow equations remains valid, particularly when second derivatives of the action become significant?
• RQ3 How can domain walls or branes be consistently incorporated into the radial flow framework using variational principles?
• RQ4 What are the generalized junction conditions at domain walls, and how do they relate to induced gravity and cosmological constants on branes?
• RQ5 How does the variational approach allow for a novel formulation of boundary conditions and symmetry constraints in holographic RG flows?

Key findings

• The variational principle successfully reproduces the flow equations derived in earlier works, such as those in the Hamilton-Jacobi framework, and provides a coordinate-independent formulation.
• The classical approximation for flow equations breaks down when solutions to the derived Jacobi-type equation become unbounded, signaling the need to include second-derivative terms in the action.
• The junction condition across a domain wall is derived as $ S_{,g_{\mu\nu}}^{+} - S_{,g_{\mu\nu}}^{-} = T \sqrt{g} \left( \frac{1}{2} g^{\mu\nu} \right) $, which leads to the Randall-Sundrum brane tension condition $ T = -4(d-1)/r $.
• The induced gravity on the brane matches the Randall-Sundrum result, with the Newton constant given by $ G_d = (d-2)/(2r) $, and higher-order curvature corrections computed from $ \Phi_1, \Phi_2, \Phi_3 $.
• The effective action on any foliation of AdS_{d+1} at fixed ρ yields the same local structure, showing that the induced gravity on a brane at ρ is equivalent to the effective action at that radial slice.
• The formalism allows for a systematic derivation of Ward-like identities and symmetry constraints, opening avenues for studying boundary CFTs through the flow equations.