On the Supersymplectic Homogeneous Superspace Underlying the OSp(1/2) Coherent States

TL;DR

This paper extends coherent states methods to $OSp(1/2)$, revealing their parametrization by a supersymplectic supermanifold linked to supercoadjoint orbits, and explores its geometric structure as a Rothstein supersymplectic supermanifold.

Contribution

It identifies the supersymplectic structure of $OSp(1/2)/U(1)$ as a Rothstein supermanifold and links it to supercoadjoint orbits, introducing superK"ahler geometry concepts.

Findings

Parametrization of $OSp(1/2)$ coherent states by a supersymplectic supermanifold.

Identification of the supermanifold as a Rothstein supersymplectic supermanifold.

Connection of the supersymplectic structure to $SU(1,1)$-invariant K"ahler forms.

Abstract

In this work we extend Onofri and Perelomov's coherent states methods to the recently introduced $OSp(1/2)$ coherent states. These latter are shown to be parametrized by points of a supersymplectic supermanifold, namely the homogeneous superspace $OSp(1/2)/U(1)$, which is clearly identified with a supercoadjoint orbit of $OSp(1/2)$ by exhibiting the corresponding equivariant supermoment map. Moreover, this supermanifold is shown to be a nontrivial example of Rothstein's supersymplectic supermanifolds. More precisely, we show that its supersymplectic structure is completely determined in terms of $SU(1,1)$-invariant (but unrelated) K\"ahler $2$-form and K\"ahler metric on the unit disc. This result allows us to define the notions of a superK\"ahler supermanifold and a superK\"ahler superpotential, the geometric structure of the former being encoded into the latter.