Analytic torsion of filtered manifolds

Abstract: 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 new 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 of this complex 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. 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 intend to compute the analytic torsion for these geometries, and use the computations to address global questions like: Which closed 5-manifolds admit a rank two distribution of Cartan type? Are there, beyond the rather obvious topological obstructions, any geometrical obstructions to the existence of rank two distributions 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 local normal forms. That is, locally, any two Engel structures look alike. Recently, significant progress has been made in the construction of Engel structures. However, few tools are available to distinguish different Engel structures. The proposed analytic torsion provides a global invariant which we will use to tackle this problem.

Within the framework of this research project, we studied spectral properties of differential operators naturally associated with geometric structures on closed manifolds. We extracted spectral invariants and related them to global features of the underlying geometry. The kind of spectral invariants we considered is known as analytic torsion. A prototypical example is the classical Ray-Singer analytic 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 extended the definition of the analytic torsion to a large class of filtered manifolds, and we established basic properties of this invariant, including the dependence on auxiliary geometrical choices. In particular, this can be applied to the Rumin complex of an intriguing geometry in five dimensions with the awkward name ``generic rank two distributions in dimension five'' also known as (2,3,5) distributions. This geometry has first been studied by Cartan in a celebrated paper from 1910 where he described a close connection with the exceptional Lie group G2. The analytic torsion provides a new global invariant that can be used to study (2,3,5) distributions. For (2,3,5) distributions on nilmanifolds we have been able to compute the analytic torsion of the Rumin complex. For these manifolds, the zeta function associated with the Rumin complex can be expressed in terms of classical Epstein zeta functions and another function that encodes counts of solutions to quadratic congruences. This unexpected connection with number theory yet has to be explored more thoroughly. Our computations for nilmanifolds also revealed a connection to mathematical physics: the Rumin complex gives rise to quite exceptional generalizations of the quantum harmonic and quartic oscillators. We obtained nontrivial results about their spectra and their spectral determinants.