Stochastic quantization and holographic Wilsonian renormalization group of conformally coupled scalar in AdS$_{4}$

TL;DR

This paper investigates the connection between holographic Wilsonian renormalization groups and stochastic quantization for a conformally coupled scalar in AdS4, revealing how stochastic four-point functions relate to holographic deformations under specific boundary conditions.

Contribution

It demonstrates that stochastic four-point functions are fully captured by holographic quadruple trace deformations in AdS4 with conformally coupled scalars, extending the framework to interacting theories.

Findings

Stochastic four-point functions match holographic quadruple trace deformations.

Dirichlet boundary conditions lead to vanishing stochastic three-point functions.

The Euclidean action relates to the holographic on-shell action as $S_E = -2 I_{os}$.

Abstract

In this paper, we explore the relationship between holographic Wilsonian renormalization groups and stochastic quantization in conformally coupled scalar theory in AdS$_{4}$. The relationship between these two different frameworks is firstly proposed in arXiv:1209.2242 and tested in various free theories. However, research on the theory with interactions has recently begun. In this paper, we show that the stochastic four-point function obtained by the Langevin equation is completely captured by the holographic quadruple trace deformation when the Euclidean action $S_{E}$ is given by $S_{E}=-2I_{os}$ where $I_{os}$ is the holographic on-shell action in the conformally coupled scalar theory in AdS$_{4},$ together with a condition that the stochastic fictitious time $t$ is also identified with AdS radial variable $r$. We extensively explore a case that the boundary condition on the conformal boundary is Dirichlet boundary condition, and in that case, the stochastic three-point function trivially vanishes. This agrees with that the holographic triple trace deformation vanishes when Dirichlet boundary condition is applied on the conformal boundary.