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Higher Riemann-Hilbert correspondence for foliations

Qingyun Zeng

TL;DR

This work generalizes classical de Rham and Riemann–Hilbert correspondences to foliations by developing an $A_{ fty}$-enhanced framework. Using Chen's iterated integrals and Igusa-style cube-to-simplex maps, it constructs an $A_{ fty}$-quasi-isomorphism between foliated forms $(\Omega^{\bullet}(\mathcal{F}), -d, \wedge)$ and foliated cochains $(C^{\bullet}(\mathcal{F}), \delta, \cup)$. It then builds an integration-based $A_{ fty}$-functor RH from cohesive modules over the foliated de Rham dga to $\infty$-local systems, proving an $A_{ infinity}$-quasi-equivalence with representations of the monodromy $\infty$-groupoid $\operatorname{Mon}_{\infty}(\mathcal{F})$ and connecting representations of $L_{\infty}$-algebroids to Lie $\infty$-groupoids. Together, these results extend the RH correspondence to foliations, providing a higher-categorical bridge between foliated $L_{\infty}$-algebroid representations and leafwise $\infty$-groupoid representations, with potential extensions to singular foliations via the monodromy and integration framework.

Abstract

This paper explores foliated differential graded algebras (dga) and their role in extending fundamental theorems of differential geometry to foliations. We establish an $A_{\infty}$ de Rham theorem for foliations, demonstrating that the classical quasi-isomorphism between singular cochains and de Rham forms lifts to an $A_{\infty}$ quasi-isomorphism in the foliated setting. Furthermore, we investigate the Riemann-Hilbert correspondence for foliations, building upon the established higher Riemann-Hilbert correspondence for manifolds. By constructing an integration functor, we prove a higher Riemann-Hilbert correspondence for foliations, revealing an equivalence between $\infty$-representations of $L_{\infty}$-algebroids and $\infty$-representations of Lie $\infty$-groupoids within the context of foliations. This work generalizes the classical Riemann-Hilbert correspondence to foliations, providing a deeper understanding of the relationship between representations of Lie algebroids and Lie groupoids in this framework.

Higher Riemann-Hilbert correspondence for foliations

TL;DR

This work generalizes classical de Rham and Riemann–Hilbert correspondences to foliations by developing an -enhanced framework. Using Chen's iterated integrals and Igusa-style cube-to-simplex maps, it constructs an -quasi-isomorphism between foliated forms and foliated cochains . It then builds an integration-based -functor RH from cohesive modules over the foliated de Rham dga to -local systems, proving an -quasi-equivalence with representations of the monodromy -groupoid and connecting representations of -algebroids to Lie -groupoids. Together, these results extend the RH correspondence to foliations, providing a higher-categorical bridge between foliated -algebroid representations and leafwise -groupoid representations, with potential extensions to singular foliations via the monodromy and integration framework.

Abstract

This paper explores foliated differential graded algebras (dga) and their role in extending fundamental theorems of differential geometry to foliations. We establish an de Rham theorem for foliations, demonstrating that the classical quasi-isomorphism between singular cochains and de Rham forms lifts to an quasi-isomorphism in the foliated setting. Furthermore, we investigate the Riemann-Hilbert correspondence for foliations, building upon the established higher Riemann-Hilbert correspondence for manifolds. By constructing an integration functor, we prove a higher Riemann-Hilbert correspondence for foliations, revealing an equivalence between -representations of -algebroids and -representations of Lie -groupoids within the context of foliations. This work generalizes the classical Riemann-Hilbert correspondence to foliations, providing a deeper understanding of the relationship between representations of Lie algebroids and Lie groupoids in this framework.

Paper Structure

This paper contains 11 sections, 26 theorems, 97 equations.

Table of Contents

  1. $A_{\infty}$ de Rham theorem for foliations
  2. de Rham theorem for foliations
  3. Riemann-Hilbert correspondence
  4. Chen's iterated integral
  5. Cube's to simplices
  6. $A_{\infty}$ de Rham theorem for foliation
  7. Riemann-Hilbert correspondence for $\infty$-foliated local systems
  8. Iterated integrals on vector bundles
  9. $\infty$-holonomy of ${\mathbb{Z}}$-connection over ${\mathcal{F}}$
  10. Riemann-Hilbert correspondence
  11. Integrate $\infty$-representations of $L_{\infty}$-algebroids

Key Result

Theorem 1.1

Given a foliation $(M, {\mathcal{F}})$, there exists a isomorphism

Theorems & Definitions (57)

  • Theorem 1.1: de Rham theorem for foliations
  • Lemma 1.2: Poincare lemma for foliations
  • proof
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • proof : Proof of de Rham theorem
  • Theorem 1.6: Riemann-Hilbert correspondance for foliation
  • Definition 1.7
  • ...and 47 more