Holographic Krylov complexity in the Coulomb branch of ${\cal N}=4$ SYM

TL;DR

This paper investigates holographic Krylov complexity in the Coulomb branch of ${ m extbf{N}=4}$ SYM, revealing oscillatory behavior linked to the Coulomb scale and comparing results with field theory.

Contribution

It provides the first holographic analysis of Krylov complexity in the Coulomb branch, including exact solutions and behavior near singularities.

Findings

Oscillatory Krylov complexity when avoiding singularity

Amplitude depends on UV cutoff, Coulomb scale, and angular momentum

Qualitative agreement with field-theoretic calculations

Abstract

We study holographic Krylov complexity in the Coulomb branch of ${\cal N}=4$ SYM. Adopting the proposal that the time derivative of the Krylov complexity is dual to the proper radial momentum of a massive particle, we investigate two probe geodesics within this geometry. For one of the radial trajectories we obtain exact analytic results, even when additional motion in the internal space is included. In cases where the geodesic avoids the interior curvature singularity, the Krylov complexity exhibits oscillatory behavior, with a frequency governed by the Coulomb scale and an amplitude determined by the UV cutoff, the Coulomb scale, and the angular momentum. This oscillatory pattern is lost, when the radial trajectory is approaching the singularity. Finally, we compare our holographic results with field-theoretic calculations, finding qualitative agreement.