The Hamilton-Jacobi equation and holographic renormalization group flows on sphere

Abstract

We study the Hamilton-Jacobi formulation of effective mechanical actions associated with holographic renormalization group flows when the field theory is put on the sphere and mass terms are turned on. Although the system is supersymmetric and it is described by a superpotential, Hamilton’s characteristic function is not readily given by the superpotential when the boundary of AdS is curved. We propose a method to construct the solution as a series expansion in scalar field degrees of freedom. The coefficients are functions of the warp factor to be determined by a differential equation one obtains when the ansatz is substituted into the Hamilton-Jacobi equation. We also show how the solution can be derived from the BPS equations without having to solve differential equations. The characteristic function readily provides information on holographic counterterms which cancel divergences of the on-shell action near the boundary of AdS.

Black hole superpotential as a unifying entropy function and BPS thermodynamics

“…However, first order equations can also be obtained for non-supersymmetric solutions, as was first pointed out in the context of 'fake supergravity' [79]. Hamilton-Jacobi theory provides a general and systematic way for deriving genuine first order flow equations for both supersymmetric and non-supersymmetric systems, and has been applied not only to domain wall type solutions in different settings [80][81][82][83][84][85][86][87], but also to black holes [41,85,[88][89][90][91][92].…”

“…It should be emphasized that although the focus of the present work is on solutions of the STU model in five dimensions, the effective superpotential approach is applicable to any supergravity theory in any dimension and has already been used in several different contexts to describe domain wall [79][80][81][82][83][84][85][86][87] and static black holes [40,41,85,[88][89][90][91][92]. To our knowledge, the exact superpotential (3.65) is the first example for a rotating solution.…”

Black hole superpotential as a unifying entropy function and BPS thermodynamics

“…However, first order equations can also be obtained for non-supersymmetric solutions, as was first pointed out in the context of 'fake supergravity' [78]. Hamilton-Jacobi theory provides a general and systematic way for deriving genuine first order flow equations for both supersymmetric and non-supersymmetric systems, and has been applied not only to domain wall type solutions in different settings [79][80][81][82][83][84][85][86], but also to black holes [41,84,[87][88][89][90][91]. While the Hamilton-Jacobi approach is applicable to any system, alternative approaches to deriving first order flow equations may exist in certain special cases [38,40,[92][93][94][95][96].…”