Gauge Theory: Principal G Bundles

In summary: It starts out talking about Chern classes and how they arise in mathematical physics and how they are related to physical quantities. The introduction is worth reading. It starts out talking about Chern classes and how they arise in mathematical physics and how they are related to physical quantities.
  • #1
nateHI
146
4
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?
 
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  • #2
nateHI said:
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?
 
  • #3
nateHI said:
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.
 
  • #4
fresh_42 said:
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.
 
  • #5
nateHI said:
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.
 
  • #6
fresh_42 said:
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 =/
 
  • #7
I don't think Noether's theorem has relevance here.
 
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  • #8
martinbn said:
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):
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.
 
  • #9
I think the OP is talking about characteristic number, but the question doesn't make sense as stated.
 
  • #10
Haelfix said:
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
 
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1. What is a gauge theory?

A gauge theory is a type of physical theory that describes the behavior of fields and particles in terms of mathematical objects known as gauge fields. These fields are defined on a space-time manifold and are used to describe the fundamental forces of nature, such as electromagnetism and the strong and weak nuclear forces.

2. What is a principal G bundle?

A principal G bundle is a mathematical structure that is used to describe the symmetries of a gauge theory. It consists of a space, known as the total space, which is locally similar to a direct product of a space, known as the base space, and a group, known as the structure group. The base space is typically a physical space, such as space-time, while the structure group represents the symmetries of the theory.

3. How is a gauge theory related to principal G bundles?

A gauge theory is described in terms of a principal G bundle, where the gauge fields are associated with the structure group of the bundle. This means that the symmetries of the gauge theory are encoded in the structure of the bundle, and the transformations of the gauge fields are related to the transformations of the bundle.

4. What is the significance of principal G bundles in gauge theory?

Principal G bundles play a crucial role in gauge theory as they provide a mathematical framework for understanding the symmetries and transformations of the theory. They also allow for the description of gauge fields and their interactions with matter fields in a unified and consistent manner.

5. Are there any applications of gauge theory and principal G bundles?

Yes, gauge theory and principal G bundles have many important applications in theoretical physics, particularly in the field of particle physics where they are used to describe the fundamental forces and particles of nature. They also have applications in other areas such as condensed matter physics, cosmology, and string theory.

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