Holographic and Wilsonian Renormalization Groups

TL;DR

The authors establish a structured correspondence between the holographic renormalization group and the Wilsonian renormalization group, highlighting the central role of multi-trace operators and the necessity of gauge-fixed bulk path integrals. Through concrete examples—free scalars, bulk gauge fields, domain-wall flows, and backreaction—they demonstrate how UV/IR amplitudes generate single- and double-trace couplings, and how the Wilsonian action emerges from holographic data. They discuss how different RG formalisms relate, clarify the distinction between full Wilsonian flows and projections, and reveal the implications for field theory in the large-N limit, including potential links to string-field theory. The work points toward a more local holographic framework, where radial evolution encodes a deeper, regulator-aware understanding of bulk locality and subregion dynamics.

Abstract

We develop parallels between the holographic renormalization group in the bulk and the Wilsonian renormalization group in the dual field theory. Our philosophy differs from most previous work on the holographic RG; the most notable feature is the key role of multi-trace operators. We work out the forms of various single- and double-trace flows. The key question, `what cutoff on the field theory corresponds to a radial cutoff in the bulk?' is left unanswered, but by sharpening the analogy between the two sides we identify possible directions.

Figures (3)

• Figure 1: The scale $\delta$ on which the Wilsonian action is localized.
• Figure 2: Terms in the Wilson RG. a) Tree level term quadratic in the interaction. b) One loop term linear in the interaction.
• Figure 3: The Wilson RG in double line notation.