Multiple "parallel" D-branes seen as leaves of foliations and Duminy's theorem

TL;DR

This paper explores the relationship between the Godbillon-Vey class, holonomy, and topological entropy of foliations, and discusses potential implications for string theory, particularly relating the B-field curvature to foliation invariants.

Contribution

It provides a qualitative analysis linking foliation invariants to string theory concepts and suggests a novel connection between the B-field curvature and the Godbillon-Vey class.

Findings

Relation between Godbillon-Vey class and foliation entropy

Potential link between B-field curvature and foliation invariants

Implications for noncommutative geometry in string theory

Abstract

We try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy (which still remains unpublished), relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible consequences on string theory. In particular we shall present a conceptual argument according to which the curvature of the B-field (rank two antisymmetric tensor) of open strings might be related to the Godbillon-Vey class using a suitable generalisation of ``Non-Abelian Geometry'' which has just appeared in physics literature. Our starting point again is the Connes-Douglas-Schwarz article on compactifications of matrix models to noncommutative tori.