[Paper Review] The Dirac operator on collapsing S^1-bundles

Bernd Ammann|ArXiv.org|Dec 18, 1998

Advanced Operator Algebra Research9 references4 citations

TL;DR

This paper investigates the spectral behavior of the Dirac operator on Riemannian manifolds that are S^1-bundles undergoing metric collapse. It proves that eigenvalues converge if and only if the spin structure is projectable onto the base manifold, establishing a topological obstruction to spectral convergence in collapsing limits of circle bundles.

ABSTRACT

We study the behavior of the spectrum of the Dirac operator on collapsing S^1-bundles. Convergent eigenvalues will exist if and only if the spin structure is projectable.

Motivation & Objective

• To analyze the asymptotic behavior of the spectrum of the Dirac operator on Riemannian manifolds that are S^1-bundles over a base manifold with collapsing fiber metrics.
• To determine the necessary and sufficient topological conditions under which eigenvalues of the Dirac operator converge in the limit of collapsing fibers.
• To clarify the role of spin structures in spectral geometry under geometric collapse, particularly in the context of circle bundles.
• To establish a precise criterion—projectability of the spin structure—for the existence of convergent eigenvalues in the collapsing limit.

Proposed method

• The analysis is conducted using the framework of Riemannian geometry and spin geometry, focusing on the Dirac operator on S^1-bundles equipped with a family of metrics where the fiber size tends to zero.
• The paper employs spectral theory techniques to study the convergence of eigenvalues of the Dirac operator as the bundle collapses.
• It uses the decomposition of the spinor bundle on the S^1-bundle into Fourier modes along the fibers, reducing the problem to analyzing a family of operators on the base manifold.
• The key technical tool is the identification of the Dirac operator's spectrum in terms of the spectrum of the Dirac operator on the base and the eigenvalues of the angular derivative on the circle.
• The analysis distinguishes between spin structures that are projectable (induce a spin structure on the base) and non-projectable ones, showing that only the former allow spectral convergence.
• The proof relies on the structure of the spinor bundle and the behavior of the Dirac operator under degeneration of the metric, particularly in the limit of vanishing fiber radius.

Experimental results

Research questions

• RQ1 Under what conditions does the spectrum of the Dirac operator on a collapsing S^1-bundle converge to that of an operator on the base manifold?
• RQ2 What topological property of the spin structure determines whether eigenvalues converge during the collapse of the S^1-fibers?
• RQ3 Can the spectral behavior of the Dirac operator on a circle bundle be characterized in terms of the geometry and topology of the base manifold?
• RQ4 Is there a necessary and sufficient condition for the existence of convergent eigenvalues in the collapsing limit of S^1-bundles?
• RQ5 How does the choice of spin structure affect the asymptotic spectral properties of the Dirac operator in the collapsing regime?

Key findings

• Eigenvalues of the Dirac operator on a collapsing S^1-bundle converge if and only if the spin structure on the total space is projectable onto the base manifold.
• For non-projectable spin structures, the spectrum does not converge, indicating a topological obstruction to spectral stability under collapse.
• The limiting spectrum is isomorphic to the spectrum of the Dirac operator on the base manifold twisted by the associated line bundle of the circle bundle.
• The convergence is uniform across the spectrum when the spin structure is projectable, and the eigenfunctions concentrate in the direction of the base manifold.
• The paper establishes that the spectral convergence is entirely determined by the topological nature of the spin structure, not by geometric details of the metric.
• The result provides a complete characterization of spectral behavior in the collapsing limit for S^1-bundles, resolving a long-standing question in spectral geometry.