The Dirac operator on cusped hyperbolic manifolds

TL;DR

This paper investigates the behavior of spin structures on cusped hyperbolic manifolds and how they influence the spectrum of the Dirac operator, revealing examples with both continuous and discrete spectra across various dimensions.

Contribution

It classifies spin structures on cusps of hyperbolic manifolds and constructs examples with specific spectral properties of the Dirac operator in all dimensions.

Findings

Existence of cusped hyperbolic manifolds with Lie cusp structures in all dimensions.

Existence of manifolds with all bounding cusp structures in dimensions up to 8.

Spectrum of the Dirac operator is R in some cases and discrete in others, depending on cusp structures.

Abstract

We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in all dimensions, and whose spectrum is discrete in all dimensions n <= 8.