# The complex life of hydrodynamic modes

## TL;DR

This paper explores the analytic structure of hydrodynamic dispersion relations using complex analysis, revealing how critical points relate to spectral properties in holographic theories and their implications for thermal correlation functions.

## Contribution

It introduces a novel complex-analytic approach to study hydrodynamic modes and their spectral properties, especially in holographic models with broken symmetries.

## Key findings

- Critical points determine convergence radii of dispersion series.
- Level-crossings in quasinormal spectra relate to critical points.
- Pole-skipping occurs in various thermal correlators, not just energy density.

## Abstract

We study analytic properties of the dispersion relations in classical hydrodynamics by treating them as Puiseux series in complex momentum. The radii of convergence of the series are determined by the critical points of the associated complex spectral curves. For theories that admit a dual gravitational description through holography, the critical points correspond to level-crossings in the quasinormal spectrum of the dual black hole. We illustrate these methods in ${\cal N}=4$ supersymmetric Yang-Mills theory in 3+1 dimensions, in a holographic model with broken translation symmetry in 2+1 dimensions, and in conformal field theory in 1+1 dimensions. We comment on the pole-skipping phenomenon in thermal correlation functions, and show that it is not specific to energy density correlations.