Analytic torsion of (2,3,5) distributions

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

Summary

The sub-Riemannian limit technique will be used to compare the aforementioned analytic torsion with the Ray-Singer torsion, and to relate the eta invariant of the Rumin differential in middle degrees to the eta invariant of the odd signature operator. The local invariants appearing in these comparison results will be expressed in terms of geometric data associated with the canonical Cartan connection and the Weyl structure provided by the sub-Riemannian metric.

For (2,3,5) nilmanifolds we will relate the whole the analytic torsion function of its Rumin complex to number theoretic properties of the corresponding lattice. Motivated by the fact that much of this analysis readily generalizes to filtered manifolds whose osculating algebras have pure cohomology, we will attempt to classify all graded nilpotent Lie algebras with pure cohomology. These appear to be very rare.

Publications

• On the eta invariant of (2,3,5) distributions. Preprint: arXiv:2504.18417 [math.DG]

  Abstract. We consider the Rumin complex associated with a generic rank two distribution on a closed 5-manifold. The Rumin differential in middle degrees gives rise to a self-adjoint differential operator of Heisenberg order two. We study the eta function and the eta invariant of said operator, twisted by unitary flat vector bundles. For (2,3,5) nilmanifolds this eta invariant vanishes but the eta function is nontrivial, in general. We establish a formula expressing the eta function of (2,3,5) nilmanifolds in terms of more elementary functions.