Homology/Cohomology in Theoretical Particle Physics?

["pi"mp]

Hi all, I apologize if this should have been posted in a math section instead, I wasn't sure. But I'm wondering if homology or cohomology ever comes up in theoretical physics? I'm being introduced to it at the same time in an algebra class and a manifolds class. There's a commutative/homological algebra class offered next semester, just wondering if it'd be worth it to sit in!

If it does come up, exactly in what respect?

Thanks!

 

[Bill_K]

I once heard Geoffrey Chew say, "Every physics student should learn homology theory." But that was a long time ago, at a time when he thought that the analyticity properties of the S-matrix were going to be the answer to the world's problems. Turns out he was wrong.

These subjects might have distant uses in physics, depending on the emphasis. I'd suggest asking the instructor if he himself is aware of any physics applications. And if his answer is no, look for another course.

 

[fzero]

(Co)homology is an important ingredient in understanding geometry, so it's worthwhile learning for many reasons. There are areas of physics where topology is directly applied.

For example, half-integer spins can be understood as so-called projective representations of the rotation group. Furthermore, the possible spins are associated to the fundamental group $\pi_1(G)$ of the Lorentz group. In 3+1 dimensions, this group is $\pi_1(SO(3,1)=\mathbb{Z}_2$ leading to the existence of bosons and fermions. In 2+1 dimensions, one considers $\pi_1(SO(2,1)=\mathbb{Z}$, which leads to additional representations that are neither fermions nor bosons, called anyons.

A more obvious example which I won't try to over-elaborate on is electromagnetism and more general gauge theories, where the connection between de Rahm cohomology and things like Stokes' theorem are directly relevant. There are additional deep connections between topology and gauge theories. You might browse through Nakahara, Geometry, Topology and Physics for some examples.

Commutative algebra does crop up in some fairly esoteric physics theories. You'd be better off studying as much representation theory of Lie groups/algebras first, since that has applications to almost everything.

 

[Bill_K]

If you're saying that an understanding of group theory is more important, I agree. If you say that homological algebra will help you understand group theory, I disagree.