[Diff Forms & Skyrmions] What is this called?

[vsv86]

Hello

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

 

[jedishrfu]

Can you provide a book context here? Some authors use alternative notation and its good to know who they are and what book they wrote.

 

[vsv86]

That's the thing. I don't have a book context, and I would like to have one, but from a book on maths if possible. What I have is a suggestion from a colleague that a phenomenon I am looking at may be linked to magnetic Skyrmions by analogy (he cannot give any more information on this). I found some articles that talk physics of Skyrmions, i.e. these excitations occur in some solids etc. What it boils down to, it seems, is analysis of expressions such as the one I gave above. For example, the Wikipedia article on magnetic Skyrmions (https://en.wikipedia.org/wiki/Magnetic_skyrmion) gives an expression that is very similar to mine, they suggest to find the topological index by integrating the form

$\vec{M}.\left(\partial_x \vec{M} \times \partial_y \vec{M} \right) dx \wedge dy$

where $\vec{M}$ is the 3d magnetization. So they have a 3-1 form based on 3d vectors (I used 2-1 form based on 2d vectors as an example). If I follow the wikipedia link, I get to Nature Physics paper, which sends me to a PRL paper on Berry phase, and I am sure the list will continue.

What I did is to write down the form

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

and look at what it gives me. If I put in a radial field ($\vec{a}$) and integrate the form on the surface of a circle, I get $2\pi \times 1$, if I put azimuthal field I get $2\pi \times 1$, if I put in field constant field in one direction (e.g. $\vec{a}=\hat{\vec{x}}$) I get zero. So I understand that form $\omega$ has something to do with how the vector field ($\vec{a}$) behaves on the surface of the unit circle. My hunch is that this type of form did not come from nowhere and has some motivation behind it, and that motivation is mathematical.

Can you suggest a book where this motivation would be explained?

 

[protonsarecool]

The 2-form you have written down is just the Berry curvature, which happens in this context to be also the first Chern form. Forms like this do pop up a lot in the context of topological insulators, and i guess skyrmions as well.
The 1-form you are writing would look like a Berry connection (of which the Berry curvature is the exterior derivative), but the Levi-Civita symbol confuses me.

A book which i like that touches on these Chern classes and also connections on fibre bundles (and particularly the Berry connection) is "Geometry, Topology and Physics" by Mikio Nakahara, but it doesn't really talk about the physical applications.