Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Gauge Theory: Principal G Bundles

  1. Jul 28, 2017 #1
    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?
     
  2. jcsd
  3. Jul 31, 2017 #2
    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?
     
  4. Jul 31, 2017 #3

    fresh_42

    Staff: Mentor

    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.
     
  5. Aug 1, 2017 #4
    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.
     
  6. Aug 1, 2017 #5

    fresh_42

    Staff: Mentor

    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.
     
  7. Aug 1, 2017 #6
    Definitely! Don't hold your breath though. I have a heavy course load starting in the fall =/
     
  8. Aug 2, 2017 #7

    martinbn

    User Avatar
    Science Advisor

    I don't think Noether's theorem has relevance here.
     
  9. Aug 2, 2017 #8

    fresh_42

    Staff: Mentor

    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.
     
  10. Aug 2, 2017 #9

    Haelfix

    User Avatar
    Science Advisor

    I think the OP is talking about characteristic number, but the question doesn't make sense as stated.
     
  11. Aug 2, 2017 #10
    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
     
    Last edited: Aug 2, 2017
  12. Aug 2, 2017 #11

    fresh_42

    Staff: Mentor

    Last edited: Aug 2, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Gauge Theory: Principal G Bundles
  1. Gauge theories (Replies: 10)

  2. What is a gauge theory? (Replies: 15)

Loading...