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