Holographic interpretations of the renormalization group

TL;DR

The paper tackles how to define running couplings in holography beyond the classical, fixed-boundary-limit by adopting a Wilsonian bulk path-integral approach. It develops a framework with $\Psi_{UV}$ and $\Psi_{IR}$ that encodes the effects of integrating out UV modes, and analyzes how renormalization schemes affect the interpretation of single-trace and multi-trace couplings, showing that in the semiclassical limit there exist schemes where single-trace couplings satisfy bulk equations of motion for operators with $\Delta \notin \frac{d}{2}+\mathbb{Z}$ at sufficiently low momenta. The work clarifies the relation between the saddle-point evaluation of the Wilsonian action and the boundary conditions at the cutoff surface, and interprets non-local multi-trace operators as reflecting transient UV-region propagation that becomes local upon coarse-graining. It also provides a physical picture of UV/IR entanglement in terms of mixed boundary conditions and boundary terms, with implications for Lorentzian AdS/CFT and the density-matrix evolution of infrared degrees of freedom.

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 $Δ\notin \frac{d}{2} + \ZZ$, 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.