- #1

vsv86

- 25

- 13

I was playing with maths for magnetic Skyrmions. There is very prominent mathematical construct in there that I would like to understand, but I do not know where to look.

It is easiest to state it for simple 2d space. We can define a 1-form:

##\omega=\sqrt{\left| g \right|} \epsilon_{\alpha\beta} a^{\alpha} \nabla_\nu a^\beta dx^\nu##

where ##g=det g_{\alpha\beta}## is the determinant of the metric, ##\epsilon_{\alpha\beta}## is the Levi-Civita (relative) tensor ##\vec{a}=a^\alpha\partial_\alpha## is a vector field, ##\nabla_\nu## is the covariant derivative, and ##dx^\nu## is the basis of the vector space of 1-forms (i.e. ##dx^\mu (\partial_\nu)=\delta^\mu_\nu##). It is easy to check that this quantity does transform as a 1-form and is independent of coordinate choice.

What is this structure called? It would seem that for Skyrmions what they do is define the equivalent of ##\omega##, i.e. an n-1 form in n-dimensional space and then integrate ##\omega## on the boundary of some volume, an n-dimensional sphere for example, to extract topological charge. What are the general properties of this thing? Are there any? or it all depends on ##a^\alpha##

Thank you