Holographic renormalisation group flows and renormalisation from a Wilsonian perspective

Abstract

From the Wilsonian point of view, renormalisable theories are understood as submanifolds in theory space emanating from a particular fixed point under renormalisation group evolution. We show how this picture precisely applies to their gravity duals. We investigate the Hamilton-Jacobi equation satisfied by the Wilson action and find the corresponding fixed points and their eigendeformations, which have a diagonal evolution close to the fixed points. The relevant eigendeformations are used to construct renormalised theories. We explore the relation of this formalism with holographic renormalisation. We also discuss different renormalisation schemes and show that the solutions to the gravity equations of motion can be used as renormalised couplings that parametrise the renormalised theories. This provides a transparent connection between holographic renormalisation group flows in the Wilsonian and non-Wilsonian approaches. The general results are illustrated by explicit calculations in an interacting scalar theory in AdS space.

Wilson action for the O(N) model

“…In the AdS/CFT correspondence the radial direction can be interpreted as the scale of the boundary field theory. Thus, a radial evolution can be thought of as an RG evolution and has been dubbed "holgraphic RG" [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The precise connection between the boundary RG and holographic RG is, however, still an open question.…”

A holographic form for Wilson's RG

“…In the bulk this corresponds to radial evolution using the bulk "radial" Hamiltonian. This evolution has been termed the "holographic RG" for the boundary theory and should logically be related to the actual RG of the boundary theory [17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Renormalised 3-point functions of stress tensors and conserved currents in CFT

“…A Wilsonian approach to renormalization of composite operators in CFT can be found in[49,50] 12. This dependence can be seen explicitly in the regulated theory, where a power of the renormalisation scale is required to bring the counterterm dimension to match the regulated spacetime dimensiond 13.…”