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Are there geodesics for Calabi–Yau manifold?

  1. Nov 15, 2009 #1
    Say I sit at some point P in a Calabi-Yau manifold. Are there geodesics which start from P and return to P?

    Are there "geodesics" which start from P and return to P but may make a "side trip first"?

    Is the number of geodesics which start at P and end at P infinite or finite and does that number depend on where you are in the manifold?

    Thanks for any help.
     
    Last edited: Nov 15, 2009
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  3. Nov 16, 2009 #2

    tom.stoer

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    Let me phrase it differently: you want to know if a (classical) closed string winding around a CY can be identical with a geodesic on the CY?

    Let me speculate: the next step would be a saddlepoint approx. where the string fluctuates around the geodesic.
     
  4. Nov 16, 2009 #3
    I was trying to keep it more simple. On the surface of a sphere I can draw geodesics, say lines of longitude on the Earth, of particular interest any geodesic on the surface of a sphere comes back to where it started. Do CY spaces allow one to calculate 1 dimensional geodesics at some given point and of those do some come back to the starting point?

    If so what are some of their properties, a finite or infinite set? For example, it seems (guessing) that some geodesics might be relatively short distances and other geodesics might be considerably longer, just guessing, don't know if one can use the two words together, geodesic and Calabi–Yau manifold %^(

    But then, yes, as you say move not a point but some loop. As different parts of the loop tried to take the shortest path would the loop change shape?

    Thanks for your thoughts.
     
  5. Nov 17, 2009 #4

    tom.stoer

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    I understand. Unfortunately I never saw anything that relates a closed string to a geodesic. But it sounds rather natural.
     
    Last edited: Nov 17, 2009
  6. Nov 17, 2009 #5

    Haelfix

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    Well, a geodesic is the (locally) shortest path between two different points. So for instance, on a plane a geodesic is a line segment.

    On a sphere, their exists antipodal points, so for such a choice of pts A and pts B you would have infinitely many possible geodesics between them. Otoh most other choices on a sphere lead to a single geodesic (some little segment of a great circle, eg an arc). I think its tacitly assumed that we aren't talking about pt A being equal to pt B when we are talking about a geodesic.

    I think instead you are looking for the concept of a homotopy, and more specifically ones that are contractible to a point?
     
    Last edited: Nov 17, 2009
  7. Nov 17, 2009 #6

    apeiron

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    Is this question connected to the old idea that CY space might be spherical, toroid or multi-donut, and the extra windings allowed might explain the mass differences between particle generations - so electron a CY sphere, muon a CY donut, etc?

    Whatever happened to that thought?
     
  8. Nov 17, 2009 #7
    Thanks, that was it exactly!:

    ... you want to know if a (classical) closed string winding around a CY can be identical with a geodesic on the CY?

    I should have read more carefully.

    I was thinking of the "shortest" paths on say a sphere, easy to visualize. But that is it, a string under tension wrapped up in some space might minimize its length by sitting on a geodesic, one which starts and ends at the same point, a loop?

    Except for vibrating about this geodesic does the string move from geodesic to geodesic, different geodesics different stuff? Is it "stuck" or does it move about?

    Thanks for your help!
     
  9. Nov 17, 2009 #8
    We want the loops to get "hung up" in space? If the loops are under tension and they had a way to shrink they would do it, and goodby loop? So our Calabi–Yau manifolds must not allow some closed loops to shrink to a point?

    Thanks for your help!
     
  10. Nov 17, 2009 #9
    From "The Complete Idiot's Guide to String Theory", page 180:

    "The sole 2-D Calabi-Yau space is a torus--a doughnut-shaped space with one hole in the middle."

    A string under tension on such a surface can wrap in 2 interesting ways?

    1. Wrap a string around the small circumference of the doughnut, it can then move around the surface of the doughnut without stretching? A massless mode?

    2. Wrap a string around the large circumference of the doughnut, it then requires stretching to move the string around the surface of the doughnut? A massive mode?

    Any more interesting wrappings?

    Seems it might get more interesting in higher dimensions?

    I see a need for "The Complete Idiot's Guide to Calabi-Yau Spaces"?

    Thanks for your thoughts.
     
  11. Nov 18, 2009 #10

    apeiron

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    Home now so I can indeed check my copy of "The Complete Idiot's Guide to Calabi-Yau Spaces"? (a.k.a: The Elegant Universe, Greene).

    He talks about it on p216. But I misrembered. First generation is analagous to 6D torus, second to a double donut, third to a triple holer.

    Greene appears to be referencing...

    "Vacuum Configurations for Superstrings," P. Candelas, G. Horowitz, A. Strominger, E. Witten, Nucl. Phys. B 258:46-74, 1985.
     
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