Holographic Krylov complexity in confining gauge theories

TL;DR

This work probes Krylov (spread) complexity in a confining, top-down holographic setting by studying a probe geodesic in the Anabalón–Ross background, which realizes a twisted-circle compactification of $\mathcal{N}=4$ SYM flowing to a gapped 3d theory. The authors derive exact analytic geodesic solutions in terms of Jacobi elliptic functions (for the SUSY case $\mu=0$), enabling precise computation of the proper radial momentum and, hence, the Krylov complexity. They show that the momentum and complexity exhibit oscillations due to reflections from the IR end-of-space, interpreted as a holographic signature of a finite Hilbert space induced by the UV cutoff and IR confinement. This top-down result extends the momentum–complexity correspondence beyond conformal theories, highlights qualitative differences between confining and conformal dynamics, and opens avenues for a field-theory computation of Lanczos data and for comparisons with lattice or tensor-network models. The work thus provides analytic control over Krylov complexity in a realistic holographic confining system and suggests concrete future directions to connect bulk diagnostics with dual QFT observables.

Abstract

We study holographic Krylov complexity in the Anabalon-Ross solitonic background, a top-down Type IIB solution describing a twisted-circle compactification of ${\cal N}=4$ SYM that flows to a confining, gapped three-dimensional theory. Following the proposal that the time derivative of Krylov complexity is dual to the proper radial momentum of a falling bulk particle, we analyze probe geodesics in this geometry. We obtain exact analytic solutions for the radial trajectory in terms of elliptic functions, confirming and extending UV and IR asymptotic expansions. The proper momentum and resulting complexity exhibit oscillatory behaviour, which we interpret as a holographic signature of the finite Hilbert-space truncation induced by the UV cutoff together with the IR end-of-space. Our results provide a controlled top-down test of the spread-momentum correspondence and highlight qualitative differences between conformal and confining holographic dynamics.

Figures (5)

• Figure 1: The function $z(t)$ for choices of $(Ql)=1/10, 1, 2$ and $\mathcal{H}=10$ (Upper panel) and the function $z(t)$ for pure AdS background ($Q=0$) (Lower panel).
• Figure 2: $P_{\bar{\rho}} (t)$ for $Ql=1$ and $\mathcal{H}=10$. The particle reaches to the end of space at $t_e\simeq 1.21$.
• Figure 3: $\epsilon \;\mathcal{C}(t)$, for $Ql=1$ and $\mathcal{H}=10,20$.
• Figure 4: $z(t)$ for various values of Q and $\mu=1.$
• Figure 5: $\epsilon \;\mathcal{C}(t)$, for $Q=1, \mu=1$ and $l=1$ and $\mathcal{H}=10,20.$