Holography and Renormalization in Lorentzian Signature

TL;DR

This paper explores the holographic correspondence in Lorentzian signature, linking supergravity equations to field theory renormalization group equations, and constructs a gravitational dual for correlation functions between initial and final states.

Contribution

It extends the holographic duality framework to Lorentzian signature, constructing a gravitational dual of the generating function for correlation functions with specified initial and final states.

Findings

Identifies how to specify both couplings and states in the dual field theory.

Constructs a classical supergravity limit for certain states.

Shows the data for solutions are encoded in the field theory despite first-order RG equations.

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.