Are there geodesics for Calabi–Yau manifold?

In summary: Calabi-Yau spaces.In summary, the conversation discusses the possibility of geodesics on a Calabi-Yau manifold, specifically those that start and end at the same point. It is speculated that a closed string winding around the manifold could be identical to a geodesic, with the next step being a saddlepoint approximation where the string fluctuates around the geodesic. The number of geodesics starting and ending at the same point is discussed, with the idea that some may be relatively short while others are longer. The concept of a homotopy and contractible loops is also mentioned. The conversation also touches on the connection between Cal
  • #1
Spinnor
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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.
 
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  • #2
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.
 
  • #3
tom.stoer said:
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.

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.
 
  • #4
I understand. Unfortunately I never saw anything that relates a closed string to a geodesic. But it sounds rather natural.
 
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  • #5
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?
 
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  • #6
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?
 
  • #7
tom.stoer said:
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.

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!
 
  • #8
Haelfix said:
...
I think instead you are looking for the concept of a homotopy, and more specifically ones that are contractible to a point?

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!
 
  • #9
apeiron said:
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?

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.
 
  • #10
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.
 

1. What is a Calabi-Yau manifold?

A Calabi-Yau manifold is a special type of complex manifold that has important applications in string theory and algebraic geometry. It is a six-dimensional space with special geometric properties that make it a popular subject of study in mathematics and physics.

2. What are geodesics?

Geodesics are the shortest paths between two points on a curved surface. In the context of a Calabi-Yau manifold, they are curves that minimize the length of the path between two points while staying within the manifold.

3. Are there geodesics for all Calabi-Yau manifolds?

No, not all Calabi-Yau manifolds have geodesics. In fact, there are only a few known examples of Calabi-Yau manifolds that have geodesics, and they typically have special symmetries or properties that make them easier to study.

4. Why are geodesics important in the study of Calabi-Yau manifolds?

Geodesics are important because they provide a way to understand the geometry and topology of Calabi-Yau manifolds. They also have applications in string theory, where they are used to describe the behavior of strings moving through these manifolds.

5. How are geodesics on Calabi-Yau manifolds studied?

Geodesics on Calabi-Yau manifolds are studied using techniques from differential geometry and analytic methods. Mathematicians and physicists use a variety of mathematical tools, such as partial differential equations and geometric analysis, to study the properties and behavior of geodesics on these manifolds.

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