The symplectic origin of conformal and Minkowski superspaces

TL;DR

This paper reveals a symplectic structure underlying conformal and Minkowski superspaces, offering a unified geometric framework that extends to higher dimensions and connects supersymmetry with division algebras.

Contribution

It introduces a novel realization of conformal and Minkowski superspaces as Lagrangian supermanifolds linked to symplectic transformations over division algebras, generalizing to higher dimensions.

Findings

Conformal superspaces can be described as Lagrangian supermanifolds.

The approach unifies 4D, 6D, and potentially 10D superspaces.

Provides a symplectic geometric perspective on supersymmetry.

Abstract

Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.