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On the eta invariant of (2,3,5) distributions

Stefan Haller

TL;DR

This paper analyzes the eta invariant of the self-adjoint operator S arising from the Rumin complex for generic rank two (2,3,5) distributions on closed 5-manifolds. By situating S within the Rockland framework, it establishes meromorphic continuation of η_S(s) and shows η(S) is invariant under metric and distribution deformations; it then leverages representation theory of the nilpotent Lie group G to decompose η_S(s) and obtain explicit formulas on (2,3,5) nilmanifolds. For nilmanifolds, η_S(s) is shown to be entire and often vanishes at s = 0, while nontrivial contributions are described via Hurwitz zeta and polylogarithm functions, with concrete residues and special values computed. These results illuminate the spectral asymmetry of the Rumin operator in sub-Riemannian geometry and connect analytic invariants to the underlying parabolic and representation-theoretic structure.

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.

On the eta invariant of (2,3,5) distributions

TL;DR

This paper analyzes the eta invariant of the self-adjoint operator S arising from the Rumin complex for generic rank two (2,3,5) distributions on closed 5-manifolds. By situating S within the Rockland framework, it establishes meromorphic continuation of η_S(s) and shows η(S) is invariant under metric and distribution deformations; it then leverages representation theory of the nilpotent Lie group G to decompose η_S(s) and obtain explicit formulas on (2,3,5) nilmanifolds. For nilmanifolds, η_S(s) is shown to be entire and often vanishes at s = 0, while nontrivial contributions are described via Hurwitz zeta and polylogarithm functions, with concrete residues and special values computed. These results illuminate the spectral asymmetry of the Rumin operator in sub-Riemannian geometry and connect analytic invariants to the underlying parabolic and representation-theoretic structure.

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.

Paper Structure

This paper contains 5 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. The eta invariant on filtered manifolds
  3. The eta invariant of a (2,3,5) distribution
  4. The eta invariant in irreducible unitary representations
  5. The eta invariant of (2,3,5) nilmanifolds

Key Result

Theorem 1

Let $\mathcal{D}$ be a generic rank two distribution on a closed 5-manifold $M$, let $g$ be a fiberwise graded Euclidean inner product on $\mathfrak tM$, let $\mathfrak o$ be an orientation of $M$, and let $F$ be a unitary flat vector bundle over $M$. Then the associated operator $S=S_{\mathcal{D},g converges absolutely for $\Re s>5$. This function extends to an entire function on the complex plan

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Theorem 3
  • Corollary 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 21 more