[Paper Review] Geometric quantization for proper moment maps

Xiaonan Ma, Weiping Zhang|arXiv (Cornell University)|Dec 20, 2008

Homotopy and Cohomology in Algebraic Topology7 citations

TL;DR

This paper establishes a geometric quantization formula for Hamiltonian actions of compact Lie groups on non-compact symplectic manifolds when the moment map is proper, thereby resolving Vergne's conjecture in the non-compact setting. It extends the Guillemin-Sternberg conjecture on 'quantization commuting with reduction' to non-compact manifolds using analytic and geometric techniques, proving the formula holds under properness conditions.

ABSTRACT

Abstract. We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we resolve the conjecture of Vergne in this non-compact setting. The famous geometric quantization conjecture of Guillemin and Sternberg [9] states that for a compact pre-quantizable symplectic manifold admitting a Hamiltonian action of a compact Lie group, the principle of “quantization commutes with reduction ” holds. This conjecture was first proved independently by Meinrenken [14] and Vergne [23] for

Motivation & Objective

• To extend the geometric quantization conjecture of Guillemin and Sternberg to non-compact symplectic manifolds with proper moment maps.
• To resolve Vergne's conjecture regarding geometric quantization in the non-compact case.
• To establish a formula for geometric quantization that remains valid under properness of the moment map.
• To generalize the principle of 'quantization commuting with reduction' beyond compact manifolds.

Proposed method

• Utilizes analytic methods in geometric quantization, particularly the theory of L2-harmonic forms and L2-index theorems.
• Applies the theory of proper moment maps to control the behavior of the quantization on non-compact spaces.
• Employs the Kostant pre-quantization line bundle and constructs a canonical L2-space of sections over the manifold.
• Uses the Duistermaat-Heckman measure and localization techniques to analyze the quantization formula.
• Applies the Atiyah-Singer L2-index theorem to compute the quantization in the presence of group actions.
• Establishes a correspondence between the quantization of the original manifold and the reduced space via the proper moment map.

Experimental results

Research questions

• RQ1 Does the principle of 'quantization commuting with reduction' hold for non-compact symplectic manifolds with proper moment maps?
• RQ2 Can Vergne's conjecture on geometric quantization be extended to the non-compact setting?
• RQ3 How does the properness of the moment map affect the construction and validity of the geometric quantization formula?
• RQ4 What analytic tools are required to define and compute geometric quantization on non-compact manifolds with group actions?
• RQ5 Is the L2-index theorem applicable in this non-compact, proper moment map setting to recover the quantization formula?

Key findings

• The geometric quantization formula holds for Hamiltonian actions on non-compact symplectic manifolds when the moment map is proper.
• The conjecture of Vergne is resolved in the non-compact case, confirming the validity of the quantization formula under properness.
• The principle of 'quantization commuting with reduction' is established for non-compact manifolds with proper moment maps.
• The L2-index theorem provides a key analytic tool to compute the quantization in this setting.
• The construction of the quantization space via L2-harmonic forms is well-defined and finite-dimensional under the properness condition.
• The reduced space inherits a well-defined quantization that matches the quantization of the original space, confirming the commutativity.