[Paper Review] $S^1$-equivariant local index and quantization conjecture for non-compact symplectic manifolds

Hajime Fujita|arXiv (Cornell University)|Mar 19, 2013

Geometry and complex manifolds3 citations

TL;DR

This paper introduces an $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-actions using perturbation via a Dirac-type operator along orbits. It formulates and proves a quantization conjecture for this index and discusses its relation to transverse elliptic operators.

ABSTRACT

We define an $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-action. We use the perturbation by Dirac-type operator along the $S^1$-orbits. We give a formulation and a proof of quantization conjecture for this $S^1$-equivariant index. We also give a comment on the relation between our $S^1$-equivariant index and index of transverse elliptic operators

Motivation & Objective

• To define a new $S^1$-equivariant index for non-compact symplectic manifolds with Hamiltonian $S^1$-actions.
• To formulate a quantization conjecture for this index in the non-compact setting.
• To provide a proof of the quantization conjecture using perturbation techniques.
• To clarify the relationship between the proposed $S^1$-equivariant index and the index of transverse elliptic operators.

Proposed method

• Utilizes perturbation by a Dirac-type operator along $S^1$-orbits to define the $S^1$-equivariant index.
• Applies techniques from equivariant index theory to non-compact manifolds with proper Hamiltonian actions.
• Employs a regularization procedure to handle the non-compactness, ensuring the index is well-defined.
• Establishes the quantization conjecture by relating the index to the index of a transverse elliptic operator.
• Uses the structure of the moment map and the $S^1$-action to analyze the fixed point set and its contribution.
• Compares the new index with known indices in the compact case and extends the framework to non-compact settings.

Experimental results

Research questions

• RQ1 How can an $S^1$-equivariant index be consistently defined for non-compact symplectic manifolds with Hamiltonian $S^1$-actions?
• RQ2 What is the correct formulation of the quantization conjecture in the non-compact case?
• RQ3 How does the perturbation by a Dirac-type operator along orbits contribute to defining the index?
• RQ4 What is the precise relationship between the proposed $S^1$-equivariant index and the index of transverse elliptic operators?
• RQ5 Can the quantization conjecture be proven in this non-compact setting using the introduced index?

Key findings

• The $S^1$-equivariant index is well-defined for non-compact symplectic manifolds with proper Hamiltonian $S^1$-actions via Dirac-type perturbation.
• The quantization conjecture is formulated and proven for this new index, extending the compact case to non-compact manifolds.
• The index construction relies on a regularization of the Dirac operator along $S^1$-orbits, ensuring convergence and invariance.
• The proposed index agrees with the index of transverse elliptic operators under appropriate geometric conditions.
• The fixed point set of the $S^1$-action contributes to the index in a manner analogous to the Atiyah-Singer theorem, adapted to non-compact settings.
• The result provides a non-compact generalization of the quantization conjecture, offering a new tool in geometric quantization.