Geometric realizations of the Lie superalgebra D(2,1;a)

TL;DR

This paper explores the geometric realizations of the exceptional Lie superalgebra D(2,1;a) through various supergeometries derived from parabolic subgroups, demonstrating explicit realizations of its supersymmetry in six distinct cases.

Contribution

It classifies and explicitly constructs six inequivalent supergeometries associated with D(2,1;a), including novel higher order structure reductions, expanding understanding of its geometric realizations.

Findings

Six inequivalent supergeometries constructed

Explicit realizations of D(2,1;a) supersymmetry provided

Introduction of higher order structure reductions as a new feature

Abstract

For every parabolic subgroup $P$ of a Lie supergroup $G$, the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the question whether $\mathfrak{g}=\text{Lie}(G)$ is the maximal supersymmetry of this supergeometry in the case of the exceptional Lie superalgebra $D(2,1;a)$. For each choice of parabolic $\mathfrak{p}\subset\mathfrak{g}$, we consider the corresponding negatively graded Lie subalgebra $\mathfrak{m}\subset\mathfrak{g}$, and compute its Tanaka--Weisfeiler prolongations, with reduction of the structure group when required, thus realizing $D(2,1;a)$ via symmetries of supergeometries. This gives 6 inequivalent supergeometries: one of these is a vector superdistribution, two are given by cone fields of supervarieties, and the remaining three are higher order structure reductions (a novel feature). We describe those supergeometries and realize $D(2,1;a)$ supersymmetry explicitly in each case.