Realization of Lie superalgebras G(3) and F(4) as symmetries of supergeometries

TL;DR

This paper demonstrates that the exceptional Lie superalgebras G(3) and F(4) can be realized as symmetries of specific supergeometries, expanding understanding of their geometric and algebraic structures.

Contribution

It provides explicit realizations of G(3) and F(4) as symmetry algebras of supergeometries via Tanaka-Weisfeiler prolongations, identifying numerous inequivalent supergeometries.

Findings

19 inequivalent G(3)-supergeometries identified

55 inequivalent F(4)-supergeometries identified

Explicit supersymmetry realizations in some cases

Abstract

For every parabolic subgroup $P$ of a Lie supergroup $G$ the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the quesiton whether $\mathfrak{g}=\operatorname{Lie}(G)$ is the maximal symmetry of this supergeometry in the case of exceptional Lie superalgebras $G(3)$ and $F(4)$. Our approach is to consider the negatively graded Lie superalgebras for every choice of parabolic, and to compute the Tanaka-Weisfeiler prolongations, with reduction of the structure group when required (2 resp 3 cases), thus realizing $G(3)$ and $F(4)$ as symmetries of supergeometries. This gives 19 inequivalent $G(3)$-supergeometries and 55 inequivalent $F(4)$-supergeometries, in majority of cases (17 resp 52 cases) those being encoded as vector superdistributions. We describe those supergeometries and realize supersymmetry explicitly in some cases.