In other words, the monopole is in the *pure gauge theory without matter* (like instantons in Y-M).

Thanks, much clearer.

-drl

However, in this case, it is the *gauge field* A_{\mu}(x) that can have “monopole-like” properties, i.e., the field strength corresponding to the gauge configuration falls off like a monopole; it is as if there was a monopole at the origin causing this field.

Chern classes (and characters) are a way of systematically classifying the possible types of field configurations: trivial (0), monopole (1), etc…

An example is from the Fubini-Study metric (U(1) over S^2):
F=(2i rdr d\theta)/(1+r^2)^2. Integral of (iF/2\pi) is -1; there is no “charge”!

In other words, the monopole is in the *pure gauge theory without matter* (like instantons in Y-M).

Below, you used the example of the Dirac monopole to illustrate a Chern class. What I was saying is – apparently you can use this classification to distinguish between the end of a long string of end-to-end magnetic dipoles, and the (on the face of it) equivalent distribution of acutal poles. This isn’t obvious and depends on the duality invariance of the energy tensor.

What you just described was simply Gauss’ theorem, other than the integer part.

(I just used Maxwell as an example because it has a potential theory.)

-drl

So this has nothing to do with the Maxwell equations, Chern’s formalism doesn’t care at all whether the field strength satisfies any equations at all. Neither does it have anything to do with the sources that occur in the Maxwell equations.

I don’t think I was clear. Let’s just say we replaced every pole by a charge and so, in the magnetodymanic world we’d have

E = -D x A
B = -Da – dt A

and of course Maxwell is

div E = 0
div B = m

curl B – dt E = 0
-curl E – dt B = M

with D^2 A = M etc.

In this world electric charges are associated with a singular magnetic vector potential. Because the energy tensor is duality invariant, nothing is any different as long as you stick to countably many discrete charges.

The point is, Maxwell-Lorentz really needs a density as a source, not a countable collection of poles each carrying a singular potential.

So the Chern must say something about the sources, in terms of how they can be locally “smeared out”.

-drl

Many thanks for the pointers on Index Theorem; Freed’s notes look very nice!

DRL-

Here is a rough way to understand the Chern class.

Consider D=det(I+(i/2pi)F), where F is the curvature 2-form transforming as UFU^{-1} under gauge transformation U. Note that D is invariant under gauge transformations.

But D can also be expanded as a polynomial in F, in terms of homogeneous polynomial, P_j(F), of elements of F of order j. The j-th Chern class is the cohomology class determined by P_j(F), which is clearly independent of gauge (connection) choice .

There are quite a few good books about the index theorem from various perspectives. For the point of view I’ll be taking in my course (if I get to this…), a good reference is Dan Freed’s notes that are on the web, see his webpage for a whole course he taught on the index theorem:

http://www.ma.utexas.edu/users/dafr/Index/index.html

For the more abstract K-theory point of view, it’s hard to beat some of the expository things Atiyah wrote about this. There are several such articles in Volume 3 of his collected works, one (Classical groups and classical differential operators on manifolds) in Volume 4 (this last one is highly recommended)

But, unfortunately, no discussion of the Atiyah-Singer Index theorem, which you covered in the first part of your course.Do you know a good reference for that (that does more than just quote the theorem “analytical index=topological index”)?

Perhaps there is a book that gives a glimpse of whatever has been happening since then in modern geometry (heard Gromov has done a fair bit, of which I know little).