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Parabolic noncommutative geometry

Magnus Fries, Magnus Goffeng, Ada Masters

TL;DR

The paper develops a directional extension of spectral noncommutative geometry by introducing strictly tangled spectral triples (ST2), a finite collection of anticommuting unbounded operators controlled by a tropical bounding matrix to model parabolic anisotropy. It shows how ST2s assemble into higher order spectral triples (HOSTs) with a well-defined K-homology class, providing a robust framework for finite summability and equivariance, including conformal actions. The work offers a suite of concrete instances—from Rockland complexes and the Rumin complex on contact manifolds to nilpotent group C*-algebras and parabolic crossed products—demonstrating how directional spectral data encode global geometric invariants in anisotropic settings. It also notes constraints in higher rank cases and clarifies the relationship with unbounded Kasparov products, thereby broadening the applicability of noncommutative geometry to parabolic and dynamical contexts with directional spectral bounds.

Abstract

We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called ``bad Kasparov products''. Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into $K$-homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group $C^*$-algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.

Parabolic noncommutative geometry

TL;DR

The paper develops a directional extension of spectral noncommutative geometry by introducing strictly tangled spectral triples (ST2), a finite collection of anticommuting unbounded operators controlled by a tropical bounding matrix to model parabolic anisotropy. It shows how ST2s assemble into higher order spectral triples (HOSTs) with a well-defined K-homology class, providing a robust framework for finite summability and equivariance, including conformal actions. The work offers a suite of concrete instances—from Rockland complexes and the Rumin complex on contact manifolds to nilpotent group C*-algebras and parabolic crossed products—demonstrating how directional spectral data encode global geometric invariants in anisotropic settings. It also notes constraints in higher rank cases and clarifies the relationship with unbounded Kasparov products, thereby broadening the applicability of noncommutative geometry to parabolic and dynamical contexts with directional spectral bounds.

Abstract

We introduce to spectral noncommutative geometry the notion of tangled spectral triple, which encompasses the anisotropies arising in parabolic geometry as well as the parabolic commutator bounds arising in so-called ``bad Kasparov products''. Tangled spectral triples incorporate anisotropy by replacing the unbounded operator in a spectral triple that mimics a Dirac operator with several unbounded operators mimicking directional Dirac operators. We allow for varying and dependent orders in different directions, controlled by using the tools of tropical combinatorics. We study the conformal equivariance of tangled spectral triples as well as how they fit into -homology by means of producing higher order spectral triples. Our main examples are hypoelliptic spectral triples constructed from Rockland complexes on parabolic geometries; we also build spectral triples on nilpotent group -algebras from the dual Dirac element and crossed product spectral triples for parabolic dynamical systems.

Paper Structure

This paper contains 30 sections, 43 theorems, 309 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Higher order spectral triples
  4. Conformal equivariance of higher order spectral triples
  5. Strictly tangled spectral triples (ST2), definition and examples
  6. Anticommuting collections of operators
  7. Bounds and tropical combinatorics
  8. Strictly tangled spectral triples
  9. Examples
  10. The Rumin complex in three dimensions on a nilmanifold
  11. A strictly tangled spectral triple on the Heisenberg group
  12. Parabolic large diffeomorphisms on the torus
  13. Analysis of strictly tangled noncommutative geometries
  14. Assembling an ST2 into a higher order spectral triple
  15. Finite summability of strictly tangled spectral triples
  16. ...and 15 more sections

Key Result

Theorem 1

Let $(\mathcal{A},\mathpzc{H},{\boldsymbol{D}})$ be an ST2 with ${\boldsymbol{D}}=(D_j)_{j\in I}$ the finite collection of self-adjoint operators and bounding matrix ${\boldsymbol{\epsilon}} \in M_I([0,\infty))$. Consider the non-empty set For ${\boldsymbol{t}}\in \Omega({\boldsymbol{\epsilon}})$, we define the operator If ${\boldsymbol{t}}\in \Omega({\boldsymbol{\epsilon}})\cap(0,1]^n$, the tri

Theorems & Definitions (116)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 106 more