Higher Spin Fronsdal Equations from the Exact Renormalization Group

TL;DR

This paper demonstrates that linearized exact renormalization group equations of free vector models reproduce the higher spin Fronsdal equations in AdS space, establishing a canonical equivalence between boundary RG and bulk dynamics.

Contribution

It shows that the linearized RG equations for free vector models are equivalent to the higher spin Fronsdal equations in AdS, linking boundary RG flow to bulk higher spin gravity.

Findings

RG equations reproduce Fronsdal equations on AdS

Bulk equations correspond to quadratic Casimir of conformal modules

Boundary RG dynamics is canonically equivalent to bulk equations

Abstract

We show that truncating the exact renormalization group equations of free $U(N)$ vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on $AdS_{d+1}$ for all higher spin fields, with the correct boundary conditions. More precisely, we establish canonical equivalence between the linearized RG equations and the familiar local, second order differential equations on $AdS_{d+1}$, namely the higher spin Fronsdal equations. This result is natural because the second-order bulk equations of motion on $AdS$ simply report the value of the quadratic Casimir of the corresponding conformal modules in the CFT. We thus see that the bulk Hamiltonian dynamics given by the boundary exact RG is in a different but equivalent canonical frame than that which is most natural from the bulk point of view.