Analytic torsion of filtered manifolds

Stefan Haller

This was a four year (July 2019 -- July 2023) research project funded by the Austrian Science Fund (FWF), Grant DOI: 10.55776/P31663

The project was hosted at the Department of Mathematics at the University of Vienna and led by the principal investigator Stefan Haller.

Summary

In this research project we studied analytic torsion of differential complexes naturally associated with filtered manifolds. In particular, we studied the analytic torsion of the Rumin complex on filtered manifolds whose osculating algebras have pure cohomology. As the Rumin complex of such a filtered manifold is a Rockland complex, the analogue of an elliptic complex in the Heisenberg calculus, the necessary subelliptic analysis is readily available. Besides trivially filtered manifolds which give rise to the Ray-Singer analytic torsion, and contact manifolds which give rise to the Rumin-Seshadri analytic torsion, this class includes generic rank two distributions in dimension five, also known as (2,3,5) distributions. These distributions have first been studied by Cartan in a celebrated paper form 1910 where he shows that they can be described equivalently as regular normal parabolic geometries associated with a particular parabolic subgroup of the split real form of the exceptional Lie group G₂.

We expect that the analytic torsion of the Rumin complex captures global features of the underlying filtered geometry. We hope that it will shed new light on questions related to the h-principle on closed manifolds: Is every formal structure (filtration) homotopic to a genuine filtered manifold? Are two formally homotopic filtered manifolds actually homotopic? More specifically: Are there, beyond the rather obvious topological restrictions, any geometrical obstructions to the existence of (2,3,5) distributions on closed 5-manifolds?

Publications

Graded hypoellipticity of BGG sequences (with Shantanu Dave). Ann. Global Anal. Geom. 62(2022), 721-789. Preprint: arXiv:1705.01659 [math.DG]
Abstract. This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein-Gelfand-Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.
Analytic torsion of generic rank two distributions in dimension five. J. Geom. Anal. 32(2022), article 248. Preprint: arXiv:2107.02062 [math.DG]
Abstract. We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincaré duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds, we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.
Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group. Preprint: arXiv:2309.11159 [math.SP]
Abstract. We study the Rumin differentials of the 5-dimensional graded nilpotent Lie group that appears as osculating group of generic rank two distributions in dimension five. In irreducible unitary representations of this group, the Rumin differentials provide intriguing generalizations of the classical quantum harmonic oscillator. For the Schrödinger representations, we compute the spectrum and the zeta regularized determinant of each Rumin differential. In the generic representations, we evaluate their alternating product, i.e., the analytic torsion of the Rumin complex.
Analytic torsion of nilmanifolds with (2,3,5) distributions. Preprint: arXiv:2311.16647 [math.DG]
Abstract. We consider generic rank two distributions on 5-dimensional nilmanifolds, and show that the analytic torsion of their Rumin complex coincides with the Ray-Singer torsion.