Link Invariants for Flows in Higher Dimensions

TL;DR

This paper extends linking number invariants to higher-dimensional flows, generalizing Arnold's asymptotic Hopf invariant and connecting these concepts to gauge fields, string theory, and topological quantum field theories.

Contribution

It introduces higher-dimensional linking invariants for divergence-free flows, generalizes Arnold's invariant, and explores their relation to gauge fields and string theory.

Findings

Computed linking invariants for n-manifolds with divergence-free flows.

Generalized Arnold's asymptotic Hopf invariant to higher dimensions.

Extended asymptotic Jones-Witten invariants to dimension 2p+1.

Abstract

Linking numbers in higher dimensions and their generalization including gauge fields are studied in the context of BF theories. The linking numbers associated to $n$-manifolds with smooth flows generated by divergence-free p-vector fields, endowed with an invariant flow measure are computed in different cases. They constitute invariants of smooth dynamical systems (for non-singular flows) and generalizes previous results for the 3-dimensional case. In particular, they generalizes to higher dimensions the Arnold's asymptotic Hopf invariant for the three-dimensional case. This invariant is generalized by a twisting with a non-abelian gauge connection. The computation of the asymptotic Jones-Witten invariants for flows is naturally extended to dimension n=2p+1. Finally we give a possible interpretation and implementation of these issues in the context of string theory.