First order formalism of holographic Wilsonian renormalization group: Langevin equation

TL;DR

This paper establishes a precise mathematical relationship between holographic Wilsonian renormalization group flow and stochastic quantization, extending previous free theory results to interacting theories, with detailed analysis of scalar models in AdS space.

Contribution

It extends the holographic Wilsonian RG and stochastic quantization correspondence to interacting theories, including multi-trace operators and multi-point functions.

Findings

3-point functions from Langevin dynamics match holographic RG evolution

Map between stochastic time and radial coordinate in AdS

Extension of free theory results to interacting scalar models

Abstract

We study a mathematical relationship between holographic Wilsonian renormalization group and stochastic quantization framework. We extend the original proposal given in arXiv:1209.2242 to interacting theories. The original proposal suggests that fictitious time(or stochastic time) evolution of stochastic 2-point correlation function will be identical to the radial evolution of the double trace operator of certain classes of holographic models, which are free theories in AdS space. We study holographic gravity models with interactions in AdS space and establish a map between the holographic renormalization flow of multi-trace operators and stochastic $n$-point functions. To give precise examples, we extensively study conformally coupled scalar theory in AdS$_6$. What we have found is that the stochastic time $t$ dependent 3-point function obtained from Langevin equation with its Euclidean action being given by $S_E=2I_{os}$ is identical to holographic renormalization group evolution of holographic triple trace operator as its energy scale $r$ changes once an identification of $t=r$ is made. $I_{os}$ is the on-shell action of holographic model of conformally coupled scalar theory at the AdS boundary. We argue that this can be fully extended to mathematical relationship between multi point functions and multi trace operators in each framework.