Gauge theory and foliations I; germ cords versus quantum cords

TL;DR

This paper uses gauge theory to analyze the space of smooth codimension-k foliations, extending holonomy concepts and relating moduli spaces to classifying spaces, revealing new topological insights.

Contribution

It introduces a gauge-theoretic framework for studying foliations, connecting moduli spaces of Maurer-Cartan elements to classifying spaces and comparing classical and quantum foliation theories.

Findings

Moduli space contains foliation space as a subspace

Holonomy extends naturally to the moduli space

Bott cohomology coincides with cohomology on non-singular foliations

Abstract

We apply gauge theory to study the space $F_k(M)$ of smooth codimension-$k$ framed foliations on a smooth manifold $M$. The quotient of Maurer-Cartan elements by the action of an infinite dimensional non-abelian gauge groupoid forms a moduli space, which contains $F_k(M)$ as a subspace. The notion of holonomy is naturally extended to this moduli space and the cohomology theory associated with points of this moduli space which correspond to non-singular foliations coincides with Bott cohomology. The quotient of the moduli space under concordance is identified as the space of homotopy classes of maps to the classifying spaces $B\Gamma^g_k$ and $B\Gamma^q_k$. While $B\Gamma^g$ is a classic and has been studied since Haefliger, $B\Gamma^q$ (which is a quotient of $B\Gamma^g$) carries a simpler topology and offers a rival theory.