Gauge Theory: Principal G Bundles

[nateHI]

I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold.
My question is for the Physicists in the room, why do you want to know the number of principal G bundles of a manifold?

 

[nateHI]

I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold.
My question is for the Physicists in the room, why do you want to know the number of principal G bundles of a manifold?

I don't think I articulated my question very well. Let me try again: The number of principal G bundles of a manifold is a topological invariant. What I would like to know is, does that invariant correspond to any physical quantity?

 

[fresh_42]

I don't think I articulated my question very well. Let me try again: The number of principal G bundles of a manifold is a topological invariant. What I would like to know is, does that invariant correspond to any physical quantity?

I'm not expert enough to answer this question, but my first thought was the theorem of Noether which connects the mathematical G-operation with physical conversation laws of systems described by differential equations. At least this would be the point where I would start to look for an answer.

In any case I'm as curious, and I think it is a very good question. Maybe @lavinia can shed some light on it.

 

[nateHI]

my first thought was the theorem of Noether which connects the mathematical G-operation with physical conversation laws of systems described by differential equations.

I like where your intuition is going with that. Unfortunately I would need to study the theorem of Noether you mention more extensively before even beginning to consider what you suggest. I do understand enough to see why you might suggest that though.

 

[fresh_42]

I like where your intuition is going with that. Unfortunately I would need to study the theorem of Noether you mention more extensively before even beginning to consider what you suggest. I do understand enough to see why you might suggest that though.

I have a bit of hope that you will share, what you find out. I've taken a look at Noether's original paper, actually it was two (available online), which are written in terms of variation calculus, as well as a modern version in a book about differential geometry, which is quite a bit different. But I don't know enough about the the connection between the conversation of Euler-Lagrange equations and physical conversation laws. Probably not too hard of a question for physicists though.

 

[nateHI]

I have a bit of hope that you will share, what you find out.

Definitely! Don't hold your breath though. I have a heavy course load starting in the fall =/

 

[martinbn]

I don't think Noether's theorem has relevance here.

 

[fresh_42]

I don't think Noether's theorem has relevance here.

You're probably right, although it was tempting: why not connect topological invariants with the origin of why Lie groups are considered at all?

Here's a list of what a quick search on Wikipedia pages about the physical relevance of Chern classes gave (although not always named as such and allegedly by identifying curvature and field strength):

• Aharonov–Bohm effect
• Meissner effect
  
• quantum Hall effect
  
• topological insulator
  
• Yang–Mills theory

And google.com suggested for Chern-Simons Theory: ... string theory, ... lecture notes, ... condensed matter, or ... super gravity.
I also found a ppp about Chern-Simons forms in physics.

 

[Haelfix]

I think the OP is talking about characteristic number, but the question doesn't make sense as stated.

 

[nateHI]

I think the OP is talking about characteristic number, but the question doesn't make sense as stated.

That's quite possible. So then, what are characteristic numbers, how are characteristic numbers related to Dijkgraaf-Witten theory and what physical quantity (if any) do they correspond to in the real world?

The only type of characteristic numbers I'm aware of come from representation theory which has a lot of use in Dijkgraaf-Witten Theory and they play a role in the calculation of the number of principal G bundles of a manifold so maybe you're on to something.

Here is what I'm currently reading if the context helps
http://wwwmath.uni-muenster.de/reine/u/ulrich.pennig/slides/2D-TQFT.pdf

 

[fresh_42]

I have found this paper on a Dutch university website:
http://www.math.ru.nl/~mueger/TQFT/At.pdf
(original: http://www.numdam.org/item?id=PMIHES_1988__68__175_0 ; Grenoble, FR)

I think the introduction is worth reading.