First-order formalism of holographic Wilsonian renormalization group: Langevin equation

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.