1
$\begingroup$

I have seen many places claiming that the given a collection of topological defects on a 2-dimensional surface, the sum of the topological charges is $2\pi\chi$ (where $\chi$ is the Euler characteristic).

What is the proof of this statement? Can you give me any references for the general case (not in a special case, like liquid crystals)? If not, what is the proof for ordinary 2 dimensional crystals?

Thanks.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

You don't explain what you mean by "topological charges" but I expect you mean the sum of the Hopf indices of the zeros of a tangent-vector field on a manifold. The resulting Poincare-Hopf theorem says that the sum of these numbers is indeed $\chi$.

A discussion and sketch of a proof can be found starting on on page 547 of my book with Paul Goldbart Mathematics for Physics, an online draft version of which can be found here.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.