A new cohomological formula for helicity in $\R^{2k+1}$ reveals the effect of a diffeomorphism on helicity

TL;DR

This paper introduces a new cohomological formula for helicity in odd-dimensional spaces, providing computational tools and analyzing how diffeomorphisms affect helicity, with applications to classifying helicity-preserving transformations.

Contribution

It presents a novel cohomological construction of helicity in higher dimensions, a new integral formula, and a classification of diffeomorphisms that preserve helicity.

Findings

New cohomological formula for helicity in ^{k+1}

Explicit formulas for how diffeomorphisms change helicity

Classification of helicity-preserving diffeomorphisms on specific domains

Abstract

The helicity of a vector field is a measure of the average linking of pairs of integral curves of the field. Computed by a six-dimensional integral, it is widely useful in the physics of fluids. For a divergence-free field tangent to the boundary of a domain in 3-space, helicity is known to be invariant under volume-preserving diffeomorphisms of the domain that are homotopic to the identity. We give a new construction of helicity for closed $(k+1)$-forms on a domain in $(2k+1)$-space that vanish when pulled back to the boundary of the domain. Our construction expresses helicity in terms of a cohomology class represented by the form when pulled back to the compactified configuration space of pairs of points in the domain. We show that our definition is equivalent to the standard one. We use our construction to give a new formula for computing helicity by a four-dimensional integral. We provide a Biot-Savart operator that computes a primitive for such forms; utilizing it, we obtain another formula for helicity. As a main result, we find a general formula for how much the value of helicity changes when the form is pushed forward by a diffeomorphism of the domain; it relies upon understanding the effect of the diffeomorphism on the homology of the domain and the de Rham cohomology class represented by the form. Our formula allows us to classify the helicity-preserving diffeomorphisms on a given domain, finding new helicity-preserving diffeomorphisms on the two-holed solid torus, and proving that there are no new helicity-preserving diffeomorphisms on the standard solid torus. We conclude by defining helicities for forms on submanifolds of Euclidean space. In addition, we provide a detailed exposition of some standard `folk' theorems about the cohomology of the boundary of domains in $\R^{2k+1}$.