Shiing-Shen Chern, one of the great geometers of the twentieth century, died last Friday at Nankai University. He was 93 years old. An article about his life is posted on the web-site of MSRI, the mathematics institute in Berkeley of which he was the founding director.

A lot of what I know about geometry was learned from his beautiful short book entitled “Complex Manifolds Without Potential Theory”, published by Springer in 1979. Some of his most important work concerned the topology and geometry of fiber bundles, and its significance can be seen in the number of crucial ideas of this field that carry his name, for instance: Chern classes, the Chern character, Chern-Weil theory, the Chern-Simons secondary characteristic class.

Update: The New York Times has an obituary.

OK, much better,

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

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

Thanks, much clearer.

-drl

Maybe the confusion here is over the term “monopole”. For lay physicists, it implies some sort of source.

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).

Peter,

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

Chern classes are purely topological. All they do is look at the field strength on a sphere, and use it to tell you the number of monopoles inside the sphere (and that this is an integer).

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.

Peter,

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

Chern classes and Chern-Simons theory are discussed in Baez and Muniain, Gauge Fields, Knots and Gravity.

Peter,

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 .

Actually the syllabus that’s up for that course is for the entire year. In the first semester we just did general manifold and bundle theory, will start next semester with Riemannian geometry, hope to get to the index theorem at the end.

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)

I just looked at the table of contents and excerpts of Morita’s book on amazon and it looks like it is clearly written and even I should be able to grasp its contents!

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).

Since my last class for the semester was yesterday, right now I’m just enjoying not having to think about geometry for a while, and being able to get to work on other projects. I don’t intend to write up notes for the geometry course this year (it will continue next semester) for several reasons:

1. I’m way too busy.

2. This is the first time I’ve taught this, and whenever you teach something for the first time, you only realize after you’ve started explaining some topics that there is a better way of approaching things.

3. While there’s no book out there I’m completely happy with, there are quite a few good ones of various kinds, and pretty much everything I’ve been talking about is covered reasonably in one or another. One book I’ve grown to like as I have looked at it over the semester is “Geometry of Differential Forms” by Shigeumi Morita. In the case of the representation theory course, the standard books didn’t cover much of what I wanted to explain (especially connections to physics).

If I teach the course again (probably not for a few years), then maybe I will try and write up some formal notes for it.

Danny,

For better or worse, mathematicians have a whole apparatus for dealing with vector potentials and field strengths that avoids having to introduce singular vector potentials when the field strength is not singular. From this point of view Dirac singularities come about because you are trying to write down a trivialization of a topologically non-trivial bundle. When you do have a topologically non-trivial bundle, like in the case of a monopole, basically you can’t choose the same gauge globally, unless you are willing to introduce singularities. By thinking of a choice of gauge as something you can only do locally, you avoid the whole problem of these singularities.

Peter

Actually, it would not be a bad idea if Peter (who may have a lot of time on his hands :)) could write up some notes on the Modern Geometry course he is teaching for the current (and next?) semester . Just like he did for the Lie group and representation course.

If I understand your previous reply, Chern classes are a formalization of what Dirac did when he concocted his “singular potential” by imagining the A field associated with an infinitely thin, infinitely long solenoid stretching off to infinity. Now why couldn’t one just write

E = -curl V

B = -grad v – dV/dt

and pretend charges are “Dirac electropoles”, thus replacing the nice potential A with a singular one V? Is it because you could not smoothly pack these electropoles into a volume and come up with an average density (you’d have “singular potential hair” – the electric solenoid strings) uniformly stretching out to infinity)? So is Chern theory must basically be a theory about the nature of sources…

-drl

I’ve put off writing about the geometric Langlands conjecture for a little while, partly because I wanted to take some time to see if I could understand it a bit better. But also I couldn’t help myself from writing about the latest on the Landscape.

Chern’s title is a somewhat strange one. I think he was trying to indicate that he concentrates not on the theory of holomorphic functions, but on the complex geometry in the sense that a complex manifold is one where you can consistently locally choose complex coordinates. Of course once you do this, you can’t really avoid thinking about holomorphic functions. But a lot of what Chern studies involves complex vector bundles over complex manifolds, but not necessarily imposing the condition that these are holomorphic vector bundles (i.e. the transition functions you use to glue together the bundles may not be holomorphic).

BTW I meant to ask the meaning of the title “Complex analysis without potential theory”. Doesn’t analyticity imply potential theory?

-drl

Actually I was just about to write something about the Langlands program and physics. Either later today or tomorrow.

I hadn’t heard about the hole in Dean’s proof, will add an update to that weblog entry

Hi Peter,

I second Levi and drl; I enjoy your mathematical posts, especially when I can understand a significant portion ðŸ™‚ I think it is especially useful for physicists who would otherwise think that there is not much in mathematics outside current fashions in string theory-related mathematics (typically those results derivale using analytic methods).

Someday, maybe take up what was it about Grothendieck’s work (schemes, Grothendieck group, etc.) that makes it so powerful even now (like Voevodsky’s Field medal work). Or Langland’s conjectures and their impact on mathematics, so we physicists can appreciate more the remarkable work being done in other areas of mathematics. In such instances, your dual background in math and physics is very helpful.

PS: Sorry to hear about a loophole in Carolyn’s proof on the Jacobian conjecture; hope she fixes it.

I love the mathematical asides – otherwise I’d probably never know what was happening out there.

-drl

Thanks for the encouraging comments about the more mathematical postings. I certainly intend to continue to write more things of that kind.

I wholeheartedly endorse Levi’s comment; the postings that generate the least polemic and diatribe are almost invariably the most interesting

Peter,

In the comments to the post “String Theory at 20 Explains It All-Not” you note that there aren’t many comments when you post on the subject of mathematics and the mathematics of quantum field theory.

I hope you realize that a lack of comments doesn’t necessarily indicate a lack of interest. It often just indicates a lack of controversy. I wanted to mention this since your (too infrequent) posts on mathematics are my favorite posts on this blog.

Hi Danny,

Too bad you’re not near here, I just spent the last week or two lecturing about Chern classes in my graduate geometry class.

A good reference for physicists is

Eguchi, Gilkey and Hanson Physics Reports Vol. 66, number 6, 1980

There are also various books about geometry and topology for physicists that should discuss this.

To understand what is going on, first think about U(1) gauge theory. The field strength is a closed 2-form, so represents a cohomology class of the manifold. Physically, if you consider a sphere S^2 in R^3, integrating the field strength 2-form over the sphere counts the number of monopoles inside the sphere. So this cohomology class (the first Chern class) detects the non-trivial topology of a U(1) bundle on a sphere surrounding a monopole.

For U(1) you just get the first Chern class, for U(n) you get n different Chern classes, basically by taking powers of the field strength (=curvature) and then taking a trace. The algebra gets a little intricate, but it’s basically the algebra of symmetric polynomials.

Peter

Does anyone have a good reference for the appearence of Chern classes in physics? I’m extremely foggy on this stuff (mainly because I can’t understand modern math langauge and have no access to the oral tradition required to translate it).

-drl

This is easily proved:

http://www.genealogy.ams.org/html/id.phtml?id=6424

Note how many mathematical grandchildren he has!

Sad news indeed. He seems to have been generous and an excellent mentor as well. Among his many outstanding students was the Fields Medallist Shing-Tung Yau of Calabi-Yau fame.

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