Segal-Bargmann transforms for Lie supergroups

The Segal-Bargmann (SB) transform is a well-understood integral transform, which plays an important role in quantum mechanics by mapping the Schrödinger space to the Fock space. More recently, this operator has been reinterpreted as an intertwining operator between different realisations of the minimal representation of the symplectic Lie algebra. There exist many variations on this transform, for instance as intertwining operators for the minimal representations of other Lie algebras.

In this project we will develop a theory of SB transforms on supermanifolds. This is motivated, on the one hand, by the link to the physical theory of super quantum mechanics and, on the other hand, by its relevance in the development of representation theory of Lie superalgebras. Supermanifolds are spaces where, aside from the ordinary commuting bosonic variables, anticommuting variables representing fermionic fields are introduced. Schrödinger equations on such manifolds provide an important method to incorporate symmetries between fermions and bosons.

The first concrete objective of the project is to construct the SB transform on flat superspace. This transform will not only be connected to the super harmonic oscillator, but also to the recently obtained minimal representation of the orthosymplectic superalgebra. In a further level of abstraction we want to develop a general theory for minimal representations and SB transforms for Lie superalgebras, as recently achieved for Lie algebras.