Degenerating Kahler structures and geometric quantization

作者：Nunes Joao P^*

来源：Reviews in Mathematical Physics, 2014, 26(9): 1430009.

DOI：10.1142/S0129055X1430009X

• Geometric quantization
• coherent state transforms
• theta functions
• toric varieties

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kahler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

• 出版日期2014-10