Holography and Renormalization in Lorentzian Signature
Albion Lawrence, Amit Sever
TL;DR
This work resolves the tension between second-order bulk dynamics and first-order RG flows in Lorentzian AdS/CFT by showing that the bulk evolution is fully specified not just by couplings but also by the quantum state, encoded as constants of motion in the Hamilton-Jacobi framework. The generating functional for time-ordered correlators is given, in the large-N limit, by the classical bulk action evaluated on saddles interpolating prescribed initial and final states, with state dependence mirroring the Callan-Symanzik evolution of correlators. It derives holographic beta functions from near-boundary data, establishes a Lorentzian CS equation for matrix elements, and clarifies the roles of state, IR physics, and potential IR walls in the holographic RG. The results bridge Hamilton-Jacobi and CS formalisms in a Lorentzian setting and illuminate how holographic RG encodes both coupling flows and state-dependent dynamics, with implications for Wilsonian interpretations and IR completions.
Abstract
De Boer et. al. have found an asymptotic equivalence between the Hamilton-Jacobi equations for supergravity in (d+1)-dimensional asymptotic anti-de Sitter space, and the Callan-Symanzik equations for the dual d-dimensional perturbed conformal field theory. We discuss this correspondence in Lorentzian signature. We construct a gravitational dual of the generating function of correlation functions between initial and final states, in accordance with the construction of Marolf, and find a class of states for which the result has a classical supergravity limit. We show how the data specifying the full set of solutions to the second-order supergravity equations of motion are described in the field theory, despite the first-order nature of the renormalization group equations for the running couplings: one must specify both the couplings and the states, and the latter affects the solutions to the Callan-Symanzik equations.