GEOMETRIC QUANTIZATION OF A SEMI-GLOBAL MODEL OF A FOCUS-FOCUS SINGULARITY

Abstract

Quantization of a classical mechanical system is an old problem in physics. In classical mechanics, the evolution of the system is given by a Hamiltonian vector field on a symplectic manifold (``phase space''). Geometric quantization is a procedure to construct a quantum system using the geometry of the classical phase space. A completely integrable system is a symplectic manifold with a moment map. If the moment map has singularities, the geometric quantization of such system becomes difficult to construct. In such case one needs to use tools from algebraic geometry (sheaves, cohomologies, etc.) to quantize such a system. The non-degenerate singularities of moment maps have been completely classified. In this dissertation we study a 4-dimensional symplectic manifold with a moment map that has a non-degenerate singularity of the so-called focus-focus type. A simple mechanical system with such a singularity is the spherical pendulum (a point mass moving without resistance on the surface of a sphere under the influence of the Earth's gravity field). We compute the geometric quantization of a focus-focus singularity by constructing a fine resolution and computing the corresponding sheaf cohomology groups.