Krylov Correlators in $\mathfrak{sl}(2,\mathbb R)$ Models: Exact Results and Holographic Complexity

TL;DR

This paper derives exact results for Krylov correlators in systems with $rak{sl}(2,R)$ symmetry and explores their connection to holographic complexity and black hole dynamics, extending the complexity--momentum correspondence.

Contribution

It introduces exact calculations of Krylov correlators in $rak{sl}(2,R)$ systems and links them to holographic complexity and black hole radial momentum.

Findings

Out-of-time-ordered Krylov correlators relate to black hole radial momentum.

Derived exact results for Krylov correlators in $rak{sl}(2,R)$ and Heisenberg-Weyl symmetric systems.

First step towards generalizing the complexity--momentum relation in holography.

Abstract

In holography, the complexity--momentum correspondence relates the increasing momentum of a point particle falling into an eternal black hole to the rate of growth of the Krylov complexity of the dual boundary state, a conjecture established exactly for the BTZ black hole in AdS$_{3}$ at the semiclassical level. We examine possible extensions of the correspondence by considering boundary higher Krylov complexities and Krylov correlators encoding fluctuations and temporal correlations of the spreading quantum state. To this end, we derive exact results for Krylov correlators in quantum systems with $\mathfrak{sl}(2,\mathbb{R})$ or Heisenberg-Weyl symmetry and apply them to the complexity--momentum correspondence. We show that certain out-of-time-ordered correlators of two or more Krylov speed operators at different times are proportional to combinations of the proper radial momenta of a particle falling into the BTZ black hole in AdS$_{3}$, evaluated at those times. This represents a first step in the generalization of the original complexity--momentum relation.