Emergent geometry from field theory: Wilson's renormalization group revisited

TL;DR

This paper proposes a geometric interpretation of Wilson's renormalization group in real space, revealing an emergent curved space metric that resembles Einstein's equations, and connects to AdS/CFT duality through a Landau-Ginzburg framework.

Contribution

It introduces a novel geometrical description of RG flow, linking it to emergent gravity and providing a new perspective on AdS/CFT duality via Landau-Ginzburg theory.

Findings

RG equations encode the metric of an emergent curved space

Emergent metric satisfies an Einstein-like equation

Connection established between RG, emergent geometry, and AdS/CFT duality

Abstract

We find a geometrical description from a field theoretical setup based on Wilson's renormalization group in real space. We show that renormalization group equations of coupling parameters encode the metric structure of an emergent curved space, regarded to be an Einstein equation for the emergent gravity. Self-consistent equations of local order-parameter fields with an emergent metric turn out to describe low energy dynamics of a strongly coupled field theory, analogous to the Maxwell equation of the Einstein-Maxwell theory in the AdS$_{d+2}$/CFT$_{d+1}$ duality conjecture. We claim that the AdS$_{3}$/CFT$_{2}$ duality may be interpreted as Landau-Ginzburg theory combined with Wilson's renormalization group, which introduces vertex corrections into the Landau-Ginzburg theory in the large$-N_{s}$ limit, where $N_{s}$ is the number of fermion flavors.