Supercoherent States and Geometric Quantization of a Super Kähler Supermanifold

Abstract

The standard group theoretic construction of coherent states has recently been extended to simple Lie supergroups, yielding the so-called supercoherent states. Usual coherent states for semi-simple Lie groups are parameterized by points of a symplectic homogeneous space, which is moreover a Kahler manifold. Analogously, we show here that the OSp(1/2) coherent states are parameterized by an OSp(1/2) supersymplectic homogeneous superspace. This turns out to be a non-trivial example of Rothstein’s general supersymplectic supermanifolds, and leads to the definition of the notion of a super Kahler supermanifold. This new subcategory of supermanifolds is well suited for the super extension of geometric quantization. Indeed, super Kahler supermanifolds are naturally equipped with a super Kahler polarization. The full geometric quantization procedure is here extended to the super Kahler homogeneous superspace underlying the OSp(1/2) coherent states. The present talk is based on results obtained in Refs. 1 and 2.