# Schwarzian mechanics via nonlinear realizations

## TL;DR

This paper uses nonlinear realizations to clarify the conceptual foundations of Schwarzian mechanics, linking the Schwarzian derivative to group invariance and providing a geometric Lagrangian formulation.

## Contribution

It demonstrates how the Schwarzian derivative naturally emerges from SL(2,R) group considerations and offers a geometric Lagrangian approach for a variant of Schwarzian mechanics.

## Key findings

- Schwarzian derivative linked to Maurer-Cartan forms
- Invariant formulation under SL(2,R) group
- Geometric description via 4D ultrahyperbolic metric

## Abstract

The method of nonlinear realizations is used to clarify some conceptual and technical issues related to the Schwarzian mechanics. It is shown that the Schwarzian derivative arises naturally, if one applies the method to SL(2,R) times R group and decides to keep the number of the independent Goldstone fields to a minimum. The Schwarzian derivative is linked to the invariant Maurer-Cartan one-forms, which make its SL(2,R)-invariance manifest. A Lagrangian formulation for a variant of the Schwarzian mechanics studied recently in [Nucl. Phys. B 936 (2018) 661] is built and its geometric description in terms of 4d metric of the ultrahyperbolic signature is given.