The Hamilton-Jacobi Equation and Holographic Renormalization Group Flows on Sphere

TL;DR

This paper develops a method to solve the Hamilton-Jacobi equation for holographic RG flows on a sphere with mass deformations, providing insights into counterterms and boundary divergences in curved spacetime.

Contribution

It introduces a series expansion approach to construct solutions for the Hamilton-Jacobi equation in curved boundary settings, linking to BPS equations without solving differential equations.

Findings

Series expansion method for Hamilton-Jacobi solutions

Relation between characteristic function and holographic counterterms

Solution derivation from BPS equations

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.