Remarks on higher Schwarzians

TL;DR

This paper explores higher-order generalizations of the Schwarzian derivative, linking them to the l-conformal Galilei group and discussing their properties and potential physical applications.

Contribution

It connects Bertilsson's higher Schwarzian derivative to the l-conformal Galilei group and analyzes its properties, including recurrence relations and symmetries.

Findings

Established a recurrence relation for higher Schwarzians

Linked higher Schwarzians to the l-conformal Galilei group

Discussed symmetry transformations and composition laws

Abstract

The Schwarzian derivative has recently received renewed attention in connection with the study of the Sachdev-Ye-Kitaev model. In mathematics literature, various higher order generalizations of the Schwarzian derivative are known due to Aharonov, Bertilsson, and Schippers. Physical applications of the higher Schwarzian derivatives have not yet been discussed in any detail. In this work, we link Bertilsson's variant to the l-conformal Galilei group, as well as discuss some of its interesting peculiarities. These include a recurrence relation, which allows one to construct the higher Schwarzians iteratively, a composition law, and symmetry transformations.