Analytic torsion of filtered manifolds

Within the framework of this research project, we will study spectral properties of differential operators associated with geometric structures on closed manifolds. We will extract spectral invariants and relate them to global features of the underlying geometry.

The kind of spectral invariants we will consider is known as analytic torsion. A prototypical example is the classical Ray-Singer torsion in Riemannian geometry. This invariant is defined in terms of the (discrete) spectrum of Laplace type operators, and is known to capture a subtle topological (global) invariant called the Reidemeister torsion. These Laplacians are elliptic operators constructed using the de Rham complex of differential forms. Recently, Rumin and Seshadri have proposed a similar invariant for contact manifolds which is based on hypoelliptic operators associated with the Rumin complex.

In this project we will extend the study of analytic torsion to a large class of filtered manifolds, including all regular parabolic geometries. Each of these geometries gives rise to a Rumin type complex, computing the de Rham cohomology. Formally, it is quite clear what the definition of the analytic torsion should be. The analysis involved, however, is quite subtle and not yet fully developed.

For some geometries we already have a rigorous definition of the analytic torsion, and we intend to study its dependence on auxiliary geometrical choices next. Among these is an intriguing geometry in five dimensions with the awkward name ``rank two distributions of Cartan type in five dimensions’’ a.k.a. ``generic rank two distributions in dimension five’’. This geometry is intimately related to the exceptional Lie group G2, and has been thoroughly studied (locally) for quite some time. We will use the analytic torsion to address questions like: Which closed 5-manifolds admit a rank two distribution of Cartan type?

For many geometries a rigorous definition of the analytic torsion will require some new analysis, to be worked out within this project. Engel structures provide a well studied 4-dimensional example of such a geometry. Like contact structures, Engel structures admit normal forms. That is, locally, any two Engel structures look alike. We hope that the analytic torsion will be able to tell them apart, globally.