Conformal maps in higher dimensions and derived geometry

TL;DR

This paper introduces a derived, infinite-dimensional dg-Lie algebra framework for conformal transformations in higher dimensions, extending classical symmetry groups by including deformations of conformal structures.

Contribution

It proposes a novel derived enhancement of the conformal Lie algebra using deformation theory of ambitwistor space, expanding the understanding of conformal symmetries in higher dimensions.

Findings

Develops a derived dg-Lie algebra model for conformal symmetries

Incorporates deformations of conformal structures into the symmetry algebra

Extends classical conformal symmetry analysis to an infinite-dimensional setting

Abstract

By Liouville's theorem, in dimensions 3 or more conformal transformations form a finite-dimensional group, an apparent drastic departure from the 2-dimensional case. We propose a derived enhancement of the conformal Lie algebra which is an infinite-dimensional dg-Lie algebra incorporating not only symmetries but also deformations of the conformal structure. Our approach is based on (derived) deformation theory of the ambitwistor space of complex null-geodesics.