Complexity Geometry and Schwarzian Dynamics

TL;DR

This paper explores the connection between the Schwarzian dynamics in SYK-like models and complexity geometry, showing that both can be described by similar one-dimensional particle models, linking complexity to black hole dynamics.

Contribution

It demonstrates a unified effective description of complexity and black hole volume dynamics using a simplified one-dimensional particle model.

Findings

Schwarzian action describes low-energy SYK dynamics

Complexity geometry reduces to a 1D particle in a potential

Effective actions of volume and complexity are closely related

Abstract

A celebrated feature of SYK-like models is that at low energies, their dynamics reduces to that of a single variable. In many setups, this "Schwarzian" variable can be interpreted as the extremal volume of the dual black hole, and the resulting dynamics is simply that of a 1D Newtonian particle in an exponential potential. On the complexity side, geodesics on a simplified version of Nielsen's complexity geometry also behave like a 1D particle in a potential given by the angular momentum barrier. The agreement between the effective actions of volume and complexity succinctly summarizes various strands of evidence that complexity is closely related to the dynamics of black holes.