How many Calabi–Yau manifolds are possible in ST?

It's unknown how many topologically distinct 6 dimensional Calabi-Yau manifolds there are. There's infinitely many in the sense of how there's infinitely many circles when you count different radii but if you don't count just squashing and scaling (known as the moduli space of a manifold) you're just counting different topologies. It gets even worse when you allow for spaces which aren't manifolds, since being a manifold isn't actually a necessary condition on space-time in string theory, it just makes standard methods possible. Hence why string theory often comes down to pushing mathematical boundaries.

I think you just made strong argument why the Anthropic Principle should be a key component for determining the exact vacua representing our universe. I remember reading Hawking strongly supporting the use of the Anthropic Principle to narrow down the possible solutions. It would be interesting to get an opinion from you. Thanks for the content of your post.

In fact the $$10^{500}$$ count sort of includes that already. The reason why Calabi-Yau manifolds are used is that they break the supersymmetry of string theory down in a nice way.

Type II string theories have N=2 supersymmetry in 9+1 large dimensions. When you role up 6 of the 9 into say 6 circles (so a 6 dimensional torus) you get N=8 supersymmetry in 3+1 dimensions (3+1 being what we see normally). If you role the 6 dimensions into a Calabi-Yau then you get N=2 because Calabi-Yaus break 3/4 of the supersymmetry (proof omitted!). You could then do an orientifold projection to break this down to N=1. Or you could role the 6 into some horrible unsymmetric shape and you break it down to N=0. When you consider ALL of the possible 6 dimensional spaces you could make then you really do have a problem of infinite proportions.

So how do you break infinitely many possibilities down to $$10^{500}$$? You use the anthropic principle. We know that Nature includes chiral phenomena, the electroweak sector doesn't treat left and right handed things the same. The problem is that anything with N>1 supersymmetry is too symmetric and doesn't allow this. Conversely we want N>0 in order to use all the wonderful nice supersymmetry mathematical structure and then we can just softly break it in the same way the Higgs mechanism breaks electroweak symmetry. So that gives us N=1, which you get by doing something to N=2 theories, which are themselves much more mathematically interesting and powerful than N=1 ones. So that's the main motivation, we look at Calabi-Yaus because in Type II theories they give us N=2 supersymmetry, which we can do lots of awesomely complicated fascinating stuff in and which we can easily then break to N=1 to get the most phenomenological supersymmetric model we can make, which is then broken to the vacuum we see normally, all motivated by "It's how we see Nature.".

Ideally, if string theory is true and fundamental, at some point our grasp of compact manifolds of dimensions 0 to 10 would become sufficiently advanced that we find that there's some natural mechanism in supersymmetric string theory which causes 6 and only 6 of the spatial dimensions of space-time to curl up into a space with N=2 structure (the N=1 projection involves orientifolds, which are sort of like branes except they carry negative tension. They are not part of space-time, they are in it like particles are).

This notion has been considered in the literature. I once saw a talk by an horrifically smart guy who made precisely that point, that we presently don't know why its 6, it could in principle be anything. As such he and his collaborators considered if it were 10 because in the early universe they were all small but only 6 remained small. What if they all did? It's too long ago for me to remember the details but I think 10 dimensional compact physics is somewhat trivial in string theory because you don't have a 'big space' to move about in.

Believe, as I clarified in a previous post, Hawking and/or string theorists aren't saying they believe $$10^{500}$$ dimensions/universes exist but rather finding the right solution in a huge list of possible solutions is difficult. If you're referring to people researching somewhat 'out there' ideas like M theory they have been lead there by formal analysis of the implications of pretty straight forward principles. All of standard quantum field theory, the most accurate and tested physical model in history, is based on the quantisation of fields into zero dimensional oscillations (point particles). String theory just says "What if the oscillations aren't 0 dimensional but 1 dimensional?". That then leads to oscillations in any number of dimensions (branes), consistent descriptions of quantised gravity (quantum gravity and gravitons) and then fundamental objects of 2 and 5 dimensions (the M2 and M5 branes) namely M theory. Each step provides us with insight into both physical models we know about (gravity/gauge dualities provide insight into QCD via Type II string theory) and thus far unknown areas like quantum gravity.

Compare all of that to believing in a god. There's no evidence at all so just like Bigfoot the rational conclusion is disbelief, saying "I do not believe the claims about the existence of a deity or deities". Please note that's different from saying "I believe god doesn't exist". The former is a rejection of a claim so has no burden of proof while the latter is a claim and so has a burden of proof. Until

*any* rational, logical reason or evidence can be provided there is no reason to have any belief in a god or gods. Besides, Santa told me not to.