I mean, I also just didn't want to read a huge wall of text. *shrug*
In any case, it's pretty generic boilerplate stuff, and doesn't describe the diagram at all!
Maybe you could clarify what your specific question is?
I can't read French well enough to say for sure what this is supposed to be. However:
The Higgs vev is the distance between pairs of parallel branes. This causes the ##W_\pm## strings to have to stretch between the separated branes, giving them a nonzero tension, and thus a mass.
The tricky...
I don't think any of your interpretations are correct. I wouldn't use "rescale" to describe any of these, especially not for ##\tau## and ##\varphi##. The new coordinates are a mixture of all of the old ones, not merely "rescaled".
To give the best interpretion of these, I would draw some...
It looks like the only difference between the barred and unbarred sigma matrices is that ##\sigma_0 = -1## while ##\bar \sigma_0 = 1##. Since 1 is just the identity matrix, I don't see how you could have gotten different results for the barred vs. unbarred case.
Are you actually writing the LaTeX, or is Scientific Workplace generating it? Because some of your LaTeX is simply incorrect. \textperiodcentered should not appear in mathmode. The correct command is much simpler: \cdot. Also, I'm not sure where \small\fbox{2074} came from. From the image...
Mathematicians call this "neutral signature" and you can find loads of math papers on the subject. You are not going to find much physics, though, as physics doesn't make much sense with more than one time direction.
A holonomic basis is a basis where all of the basis vectors commute. Given a holonomic basis, it is possible to choose coordinates ##x^\mu## such that the basis vectors are the set of partial derivative operators ##\partial/\partial x^\mu##.
Classically, it is true that there is no reason for the gauge group to be ##U(1)## rather than ##\mathbb{R}## (which is simply the universal cover of ##U(1)##).
For non-Abelian groups, the kinetic term ##F^2## has its gauge indices contracted via the group's Killing form, and then questions of...
The piecewise-linear idea works just as well. You'll have to do some linear algebra to find the boundaries between the pieces, but I'm sure it isn't hard.
What did you intend this to be an example of, exactly? As you point out, ##\mathbb{R}## has only one differential structure. If I am understanding your notation correctly, I think the point is that ##x^3## only gives you a differential structure on ##\mathbb{R}^+##, not on ##\mathbb{R}##...
Say I give you a basis of vectors at a point. Forget about differential geometry for a moment, and let's just think about a vector space. So, you have an n-dimensional vector space, and I give you a set of n linearly-independent little arrows at the origin.
The little arrows don't have to be...
This statement is your error. The correct statement is
$$D = d + \rho(A) \wedge$$
where ##\rho : G \to GL(n)## is the homomorphism which defines the representation of the gauge group you need. If you have two 1-forms ##\alpha## and ##\beta##, each in the fundamental representation, then the...
The Christoffel connection tells you which vectors in two nearby tangent spaces are "the same". A spinning particle is a gyroscope and wants to keep its angular momentum vector pointing in the same direction as it moves from point P to a nearby point. So the Christoffel connection is needed in...