Exploring the Significance of Calabi-Yau Spaces in String Theory

[Chaos' lil bro Order]

I was wondering what Calabi-Yau spaces were and their significance to string theory. Any help is appreciated, ty.

 

[Wiemster]

From popular science books I know that they are supposed to be the 'shapes' (manifolds) in which the extra dimensions predicted by string theory are 'curled up' (compactified). Try google!

 

[neutrino]

For a non-technical intro, you may want to read Brian Greene's first pop. book,The Elegant Universe. I think he covers them extensively, at that level.

 

[DaveC426913]

Wiki seems to give a pretty concise description.

 

[AlphaNumeric]

They are Ricci flat complex manifolds which admit a closed Kahler form. The ones in string theory are of 3 complex dimensions.

I have a lot of time on my hands while waiting for stupidly slow mathematica programs I've written to compute and since I do a bit of work on Calabi Yau manifolds I typed up http://www.hep.phys.soton.ac.uk/~g.j.weatherill/papers/Articles/Complex%20Manifolds.pdf over the last few days. Goes through the various mathematical structures which relate to a CY manifold so as to put them in context of manifolds, Kahler geometry and Riemannian geometry.

To anyone who actually knows their stuff, if you spot a huge mistake or whatnot, please say. I often lack the ability to explain myself properly. I used Nakahara as a vague template for that and I was half teaching myself diff geom as I went through (typing up stuff helps me learn it) so comments more than welcome.

 

[Chaos' lil bro Order]

Fantastic replies, ty.

Alpha, I'll try to read your paper even thought I'm a math novice. ty

 

[quasar987]

To anyone who actually knows their stuff, if you spot a huge mistake or whatnot, please say.

Alpha, I don't know any stuff and it's not a huge mistake; it barely qualifies as a whatnot, but it's interesting at least: Equation (1) is not, as written, a proper homeomorphism btw $\mathbb{R}^2$ (nor even $\mathbb{R}^2 \setminus \{0\}$ for that matter) and its image. It is something I became aware of only recently because in every physics books, the polar coordinates are introduced precisely as your equation (1) and I never looked further into it. But if you do for one second, it becomes apparent that (1) does not describe an injection. See http://en.wikipedia.org/wiki/Polar_coordinates#Converting_between_polar_and_Cartesian_coordinates for the "real" polar coordinates.

 

[AlphaNumeric]

Thanks Quasar. Yeah, the coordinate change has a problem at r=0 (the Jacobian which validates the smoothness of the change is 1/r). I've seen it mentioned before in a GR textbook so I should have remembered.

It's more mentioned as a familiar example of different coordinates on the same space for people who are not sure what I'm talking about. I think it'd be simpler to put a qualifier "Warning, not a precise homeomorphism, for illustrative purposes only" than to do what the Wikipedia link does which it go a bit deeper than a passing mention.

Or I might just change it so that instead of cartestian -> polar, I just do a rotation in cartesians since that's well defined everywhere.

 

[Sojourner01]

Where do you people pick up your mathematical vocabulary? Pretty much all mathematical explanations here go straight over my head - am I really a minority? That is, would you experts expect undergrad students in general to comprehend what you've said based entirely on what you'd assume was their current education level?

 

[AlphaNumeric]

I'm in my 5th year of uni having spent the first 4 doing maths, that's what I got mine ;)

The problem with Calabi Yau spaces is that they are pretty deep into maths, there's no real way to define their properties other than vigorous arm waving if you are talking to someone who hadn't done at least some level of vector calculus and geometry.

You aren't in a minority, but I imagine that there's a higher proportion of people on these forums who do maths or physics at at least undergraduate level so it might feel at times like you're way behind. Go on other forums and you'll feel way ahead.

I don't expect many people who've done just calculus and basic vectors to follow what my pdf said, but people in their second year of uni are getting close, particularly if they like to read around. It's hard to judge how fast people learn and how much grounding/examples they need, particularly if you're trying to make such an advanced topic vaguely accessible. If I'd typed much more elaboration it would have ballooned the pdf from a "Quick guide to..." to "A course in..." but if I cut any more out it'd be unattainable to anyone who doesn't already know what Ricci flat or Kahler means.

The best end result would be for someone to get half way, find themselves stuck and to go away to research what some of the things I mentioned were, so they could read further. That way it stimulates learning on a wider topic.

I wouldn't worry too much about it, maths can seem like that. One week it's all gibberish, the next you can't believe once you couldn't understand it. Moments were the penny drops make it worthwhile :)

 

[Sojourner01]

Heh, well. Your article wasn't too bad, I suppose I understood what it was trying to tell me, just didn't catch any of the significance.

I suppose I particularly suffer because most of my lecturers are extremely sloppy with notation and terminology. This is not to say that their teaching suffers for it, merely that it is of limited use in understanding anything else outside their courses.