Hints on integrability in the Wilsonian/holographic renormalization group

TL;DR

This paper demonstrates that the Wilsonian renormalization group equations for large N matrix scalar theories can be reformulated as an integrable Hamiltonian system, revealing deep connections to known integrable equations like KdV.

Contribution

It shows that the low-energy sector of the Wilsonian RG in matrix scalar theories reduces to an integrable Hamiltonian system, providing new insights into the structure of the RG flow.

Findings

Hamiltonian form of Polchinski equations at large N

Reduction to integrable KdV-type equations at low energies

Connection to effective theories in matrix quantum mechanics

Abstract

The Polchinski equations for the Wilsonian renormalization group in the $D$--dimensional matrix scalar field theory can be written at large $N$ in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the complete basis of single trace operators. We show that at low energies independently of the dimensionality $D$ the Hamiltonian system in question (for the subsector of operators without derivatives) reduces to the {\it integrable} effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger--Hopf) equation and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.