[Paper Review] Singular Reduction and Quantization

Eckhard Meinrenken, Reyer Sjamaar|ArXiv.org|Jul 30, 1997

Geometry and complex manifolds21 references4 citations

TL;DR

This paper extends the 'quantization commutes with reduction' theorem to singular symplectic quotients by introducing partial desingularizations to define the Riemann-Roch number of the quotient. Using this framework, the authors prove that the equivariant index of a prequantum bundle and its dual, as well as the arithmetic genus of a Hamiltonian manifold, are invariant under symplectic reduction, even when the quotient is singular.

ABSTRACT

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.

Motivation & Objective

• To generalize the 'quantization commutes with reduction' theorem to cases where the symplectic quotient is singular, rather than just nonsingular.
• To define a meaningful Riemann-Roch number for singular symplectic quotients, which is essential for extending index-theoretic results.
• To compute multiplicities for the equivariant index of the dual of a prequantum bundle in the singular setting.
• To establish the invariance of the arithmetic genus of a Hamiltonian G-manifold under symplectic reduction, even when the quotient is singular.
• To provide a rigorous framework using partial desingularizations to handle singular reduction in geometric quantization.

Proposed method

• Introduce partial desingularizations of singular symplectic quotients to construct a well-defined Riemann-Roch number for the quotient space.
• Apply index theory to the prequantum line bundle and its dual on the original manifold, focusing on the G-invariant part of the equivariant index.
• Use the theory of equivariant K-theory and the Atiyah-Segal completion theorem to relate the index of the original manifold to that of the desingularized quotient.
• Construct a resolution of singularities that preserves the symplectic structure in a controlled way, enabling the application of standard quantization techniques.
• Leverage the invariance of the arithmetic genus under symplectic reduction by showing it is preserved through the desingularization process.
• Utilize the Riemann-Roch theorem for orbifolds and singular spaces, adapted via desingularization, to compute the Riemann-Roch number of the quotient.

Experimental results

Research questions

• RQ1 Can the 'quantization commutes with reduction' theorem be extended to singular symplectic quotients?
• RQ2 How can the Riemann-Roch number be meaningfully defined for a singular symplectic quotient?
• RQ3 What is the behavior of the equivariant index of the dual of a prequantum bundle under singular reduction?
• RQ4 Is the arithmetic genus of a Hamiltonian G-manifold invariant under symplectic reduction when the quotient is singular?
• RQ5 Can partial desingularizations serve as a viable tool to generalize index-theoretic results in geometric quantization to singular settings?

Key findings

• The paper establishes that the G-invariant part of the equivariant index of a prequantizable Hamiltonian G-manifold equals the Riemann-Roch number of its singular symplectic quotient, when the latter is defined via partial desingularization.
• The authors prove that the multiplicities of the equivariant index of the dual of a prequantum bundle are preserved under singular reduction, extending known results to singular cases.
• The arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction, even when the quotient is singular, as shown through the desingularization method.
• The Riemann-Roch number of a singular symplectic quotient is well-defined and computable via partial desingularizations, providing a consistent generalization of the nonsingular case.
• The framework successfully generalizes the 'quantization commutes with reduction' principle to singular quotients, resolving a long-standing gap in geometric quantization theory.
• The results are formally published in Topology, Volume 38, Issue 4, 1999, confirming the mathematical rigor and acceptance of the approach.