G(3)-supergeometry and a supersymmetric extension of the Hilbert-Cartan equation

TL;DR

This paper explores the realization of the Lie superalgebra G(3) as supersymmetry in geometric structures, including super-versions of the Hilbert-Cartan equation, and analyzes their symmetries and invariants.

Contribution

It introduces super-extensions of classical geometric PDEs, computes their symmetry algebras, and reveals new supersymmetry phenomena and invariants.

Findings

Realization of G(3) as supersymmetry of super-versions of Hilbert-Cartan equation.

Explicit symmetry computations and Spencer cohomology analysis.

Identification of a supersymmetric binary quadratic form as an invariant.

Abstract

We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka-Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector (2|4,1|2,2|0) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby providing a "square-root" of Cartan's classical binary quartic invariant for generic rank 2 distributions in a 5-dimensional space. Finally, we obtain super-extensions of Cartan's classical submaximally symmetric models, compute their symmetries and observe a supersymmetry dimension gap phenomenon.