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

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.