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

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.

Figures (8)

• Figure 1: The left panel plots the function $z(t)$ for different values of $L_{\phi}$ (blue for $L_{\phi}=0$, red for $L_{\phi}=2$, orange for $L_{\phi}=4$), with $\mathcal{H}=10$ and $\ell=1$. The right panel shows the corresponding $\phi(t)$ evolution using the same parameter values.
• Figure 2: The left panel plots the function $z(t)$ for different values of $L_{\phi}$ (blue for $L_{\phi}=0$, red for $L_{\phi}=2$, orange for $L_{\phi}=4$), with $\mathcal{H}=10$ and $\ell=1$. The right panel shows the corresponding $\phi(t)$ evolution using the same parameter values.
• Figure 3: The left panel plots $P_{\bar{\rho}}$ and the right panel ${\cal C}(t)$ for different values of $L_{\phi}$ (blue for $L_{\phi}=0$, red for $L_{\phi}=2$, orange for $L_{\phi}=4$), with $\mathcal{H}=10$ and $\ell=1$.
• Figure 4: The left panel plots the radial $R_z$ (solid lines) and angular $R_{\phi}$ (dotted lines) contributions for different values of $L_{\phi}$ (blue for $L_{\phi}=1$, orange for $L_{\phi}=2$), with $\mathcal{H}=10$ and $\ell=1$. The right panel plots $R_z$ and $R_{\phi}$ for different values of $\ell$ (blue for $\ell=1$, orange for $\ell=2$), with $\mathcal{H}=10$ and $L_{\phi}=1$.
• Figure 5: The left panel plots ${\cal C}(t)$ for $\ell=1$ and two different values of $\mathcal{H}$ (blue for $\mathcal{H}=10$, orange for $\mathcal{H}=20$) and the right panel ${\cal C}(t)$ for $\mathcal{H}=10$ and two different values of $\ell$ (blue for $\ell=1$, orange for $\ell=2$).