Helicity, cohomology, and configuration spaces
Seminar Room 1, Newton Institute
We realize helicity as an integral over the compactified configuration space of 2 points on a domain M in R^3. This space is the appropriate domain for integration, as the traditional helicity integral is improper along the diagonal MxM. Further, this configuration space contains a two-dimensional cohomology class, which we show represents helicity and which immediately shows the invariance of helicity under SDiff actions on M. This topological approach also produces a general formula for how much the helicity of a 2-form changes when the form is pushed forward by a diffeomorphism of the domain. We classify the helicity-preserving diffeomorphisms on a given domain, finding new ones on the two-holed solid torus and proving that there are no new ones on the standard solid torus.
(This is joint work with Jason Cantarella.)