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I have heard for quite a few times that there are $10^{500}$ possible compactifications in string theory. And in one of his lectures, physicist Leaonard Susskind explains that, this comes from the fact that, if you have a torus with 500 holes in it, in 10 dimensions, there are $10^{500}$ possibilities.

I don't really understand why he took a torus(10 dimensions is of course clear) and why is it 500 holed?

I suppose the 10-torus is $ (S^{1})^{10}$. How this 10-torus and the Calabi-Yau manifolds are connected? And does it have indeed 500 holes in it?

can any one illuminate on this?

Qmechanic
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Eden Zane
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1 Answers1

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This number - $10^{500}$ is not precise; it's an estimate. We simply don't know the exact number. Susskind's "explanation" is a hand-wavy way of explaining the estimate (note he doesn't explain why there are 500 holes).

Source:

One can roughly estimate the number of choices at each step, and argue that they combine to produce a combinatorially large number of metastable vacua. These arguments are still in their early days and there is as yet no consensus on the number; estimates range from $10^{500}$ which at the time it was made seemed large, to the recent $10^{272,000}$.

Here's the paper estimating the high end number.

Allure
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