A superunitary Fock model of the exceptional Lie supergroup $\mathbb{D}(2,1;\alpha)$

TL;DR

This paper constructs a superunitary Fock model for the minimal representation of the exceptional Lie supergroup $ ext{D}(2,1; ext{α})$, providing explicit group-level actions and confirming superunitarity.

Contribution

It introduces a new Fock model for the minimal representation of $ ext{D}(2,1; ext{α})$ and demonstrates its superunitarity, extending previous algebraic constructions.

Findings

Explicit Fock model for $ ext{D}(2,1; ext{α})$

Group-level action expressions provided

Representation confirmed superunitary

Abstract

We construct a Fock model of the minimal representation of the exceptional Lie supergroup $\mathbb{D}(2,1, \alpha)$. Explicit expressions for the action are given by integrating to group level a Fock model of the Lie superalgebra $D(2,1, \alpha)$ constructed earlier by the authors. It is also shown that the representation is superunitary in the sense of de Goursac--Michel.