Holographic interpretations of the renormalization group

TL;DR

This paper explores holographic interpretations of the renormalization group, proposing a scheme where couplings correspond to solutions of bulk equations, clarifying the role of fluctuations and non-local operators in the dual gravitational theory.

Contribution

It introduces a scheme-dependent framework linking boundary couplings to bulk solutions, clarifies the saddle point approximation's relation to boundary conditions, and interprets multi-trace operators in Lorentzian AdS.

Findings

Existence of a scheme where couplings match bulk solutions for certain operators.

Non-local multi-trace operators arise from excitations passing through the UV region.

Coarse-graining makes the effective action local.

Abstract

In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension $\Delta \notin \frac{d}{2} + \ZZ$, for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action ($S_W$) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in $S_W$ as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.