Geometric Quantization of Coadjoint Superorbits and Schrödinger-Fock Representations of the Heisenberg Supergroup

Abstract

In the framework of Kirillov's orbit method for nilpotent supergroups, coadjoint superorbits of the Heisenberg supergroup naturally carry supersymplectic structures arising from the Kirillov-Kostant-Souriau form. Their geometric quantizationprovides a geometric realization of the irreducible unitary representations of the Heisenberg supergroup. In particular, after choosing an appropriate polarization, the resulting quantumstate space is identified with \(L^2(\mathbb R^m)\otimes \Lambda^\bullet(\mathbb C^n),\) combining the bosonic Schrödinger representation with the fermionic Fock representation. Thus $\mathfrak h^{m|n}$ provides the simplest geometric model combining bosonic and fermionic quantization. The bosonic sector reproduces the canonical commutation relations, while the fermionic sector generates a Clifford algebra through anticommutation relations. This construction shows that Schrödinger-Fock representations arise naturally as the geometric quantization of coadjoint superorbits, thereby providing a deep connection between supergeometry, supersymplectic structures, and the representation theory of nilpotent supergroup. We begin by explicit calculations for the case $m=n=1$ and generalize to the arbitrary values of $m$ and $n$.