# A geometric viewpoint on generalized hydrodynamics

## TL;DR

This paper presents a geometric reformulation of generalized hydrodynamics (GHD), revealing a new perspective that simplifies solving the equations and enhances understanding of many-body integrable systems.

## Contribution

It recasts GHD equations into a geometric framework with state-dependent metrics, providing a novel solution method for initial value problems in integrable systems.

## Key findings

- Derived a geometric interpretation of GHD equations.
- Developed a new integral equation-based solution algorithm.
- Demonstrated the efficiency of the geometric approach for solving GHD.

## Abstract

Generalized hydrodynamics (GHD) is a large-scale theory for the dynamics of many-body integrable systems. It consists of an infinite set of conservation laws for quasi-particles traveling with effective ("dressed") velocities that depend on the local state. We show that these equations can be recast into a geometric dynamical problem. They are conservation equations with state-independent quasi-particle velocities, in a space equipped with a family of metrics, parametrized by the quasi-particles' type and speed, that depend on the local state. In the classical hard rod or soliton gas picture, these metrics measure the free length of space as perceived by quasi-particles, in the quantum picture, they weigh space with the density of states available to them. Using this geometric construction, we find a general solution to the initial value problem of GHD, in terms of a set of integral equations where time appears explicitly. These integral equations are solvable by iteration and provide an extremely efficient solution algorithm for GHD.