De Rham Theory
Now we can marry the two lines of argument and arrive at one of the most beautiful and useful advanced mathematical area, de Rham cohomology. I cannot cover this in only one post, and there will be some back to physics posts (one or more guest posts) to prevent the math topics to become too dry. The end goal of the math series is to be able to talk about Yang-Mills theory and the Standard Model, so there is light (physics) at the end of the tunnel.
Let us start gentle into the topic and consider the real 3 dimensional space. Let us talk about differential forms (the stuff that you pull back). We have 0-forms, 1-forms, 2-forms, and 3-forms. On R3 there are no 4 or higher forms? Why? Read on…
What 0-forms might be? They are simply the usual functions. Let’s call them f.
Take the differential of a function:
df =∂f/∂x dx + ∂f/∂y dy + ∂f/∂z dz
and you get the gradient which is a 1-form. 1-forms are isomorphic with vector fields X:
f1 dx + f2 dy + f3 dz ~ X = (f1, f2 , f3)
Take the differential of 1-form and you get 2-forms (the curl):
d ( f1 dx + f2 dy + f3 dz ) =
+(∂f3/∂y - ∂f2/∂z) dy dz
– (∂f1/∂z - ∂f3/∂x) dx dz
+ (∂f2/∂x - ∂f1/∂y) dx dy
Then take the differential of 2-forms and you get a 3-form (the divergence)
d ( f1 dy dz - f2 dx dz + f3 dx dy ) = (∂f1/∂x + ∂f2/∂y + ∂f3/∂z) dx dy dz
The gradient, curl, and divergence are the bread and butter of Maxwell’s equations in college.
When working with differentials remember 2 rules:
- (dx dx) = 0
- dx dy = - dydx
Now the key idea is the d d = 0 and we are getting somewhere interesting. We will be able to study topology by investigating partial differentials equations. Mathematically this is very surprising because it reveals a bridge between two very different domains. In topology you deal with accumulation points, closed and open sets, while in differential equations you encounter say Schrodinger equation. What can Schrodinger equation tell you about open sets?
Physically on the other hand it is not surprising at all!!! You know that a violin sounds like a violin, and a drum like a drum. This is because the solution to the wave equation in a cavity depends on the shape of the cavity. By the way, this is why I like physics: it provides an easy context for intractable mathematical abstractions.
So from above we have:
- d(0-forms) = gradient
- d(1-forms) = curl
- d(2-forms) = divergence
What is the curl of a gradient? What is the divergence of a curl? They are zero because dd = 0 (recall how long it took to prove those things in college?)
Now suppose we have a vector field v
and we ask if it is a gradient of some potential:
v= grad (A)
We know that curl (grad (A)) = 0 and locally this is always obeyed, but not globally. If we have a loop (recall homotopy) the integral of v along the loop depends only on the homotopy class and the following vector space:
v for which curl (v) = 0 BUT v is not grad (A) for some A.
is the same as the homotopy class!
In general we can define the following p-th de Rham cohomology (this is a vector space):
α | d α = 0
HpdR = --------------------------------
α | α = dβ
where α is a p-form and β is a (p-1)-form.
In other words: closed forms which are not exact. This is the same Ker/Image pattern we encountered in homology. To be continued…