[Paper Review] Families index for manifolds with hyperbolic cusp singularities

Pierre Albin, Frédéric Rochon|ArXiv.org|Jan 14, 2008

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper establishes a families index theorem for Dirac-type operators on manifolds with fibered hyperbolic cusp (φ-hc) metrics, extending Vaillant’s work by incorporating Fredholm perturbations and generalizing the index formula to families. It proves that the Chern character of the index class equals a topological integral involving the A-hat class and η-invariants, extending results from single operators to families and linking to fibered cusp metrics via conformal equivalence.

ABSTRACT

Manifolds with fibered hyperbolic cusp metrics include hyperbolic manifolds with cusps and locally symmetric spaces of Q-rank one. We extend Vaillant's treatment of Dirac-type operators associated to these metrics by weaking the hypotheses on the boundary families through the use of Fredholm perturbations as in the family index theorem of Melrose and Piazza and by treating the index of families of such operators. We also extend the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators associated to fibered cusp metrics (sometimes known as fibered boundary metrics).

Motivation & Objective

• To extend Vaillant’s index theorem for single operators on manifolds with fibered hyperbolic cusp metrics to families of such operators.
• To weaken boundary hypotheses using Fredholm perturbations, adapting Melrose and Piazza’s families index theorem framework.
• To establish a families index formula for φ-hc metrics that includes η-invariants and Chern characters of kernel and cokernel data.
• To generalize the index theorem of Moroianu and Leichtnam-Mazzeo-Piazza to families of perturbed Dirac-type operators on φ-hc metrics.
• To show that the φ-c (fibered cusp) families index theorem follows from the φ-hc result via conformal equivalence, under kernel and cokernel rank constancy.

Proposed method

• Uses a modified version of the Melrose-Piazza families index theorem proof for asymptotically cylindrical manifolds to handle φ-hc metrics.
• Applies Fredholm perturbations to the vertical Dirac operator to ensure the existence of spectral sections and stabilize the index construction.
• Constructs a φ-hc Bismut superconnection and rescales it to analyze the curvature and Chern character of the index bundle.
• Employs the η-invariant and its perturbed version (η⁺_P) to account for spectral asymmetry in the index formula.
• Reduces the φ-c index problem to the φ-hc case via conformal equivalence, using Moroianu’s method adapted to families.
• Applies the Agranovich-Dynin formula to relate η-invariants associated with different spectral projections, enabling relative index theorems.

Experimental results

Research questions

• RQ1 How can the families index theorem be extended from single operators to families of Dirac-type operators on manifolds with fibered hyperbolic cusp metrics?
• RQ2 What role do Fredholm perturbations play in stabilizing the index and enabling the construction of spectral sections for families?
• RQ3 How do η-invariants and Chern characters of kernels and cokernels enter the families index formula for φ-hc metrics?
• RQ4 Can the φ-c (fibered cusp) families index theorem be derived from the φ-hc result via conformal equivalence and kernel rank assumptions?
• RQ5 What is the precise relationship between the Chern character of the index bundle and the topological invariants in the φ-hc index formula?

Key findings

• The Chern character of the index bundle for a family of φ-hc Dirac-type operators is given by a topological integral involving the A-hat class of the fibration and the η-invariant of the vertical operator.
• The index formula includes a correction term involving the η-invariant of the vertical Dirac operator, which depends on the choice of spectral projection.
• When the kernel and cokernel of the horizontal operator have constant rank, the index formula simplifies and includes a term involving the Chern character of the kernel bundle.
• The relative index formula holds in cohomology, with differences in η-invariants corresponding to the Chern character of the Fredholm pair of spectral projections.
• The φ-c families index theorem is deduced from the φ-hc result via conformal equivalence, provided the kernel and cokernel dimensions are constant across the base.
• The formula generalizes the single-operator index theorems of Vaillant, Moroianu, and Leichtnam-Mazzeo-Piazza to the families setting, with explicit cohomological expressions.