Stochastic quantization and holographic Wilsonian renormalization group of conformally coupled scalar in AdS4

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 first 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.