There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejudice. You may find this related blog discussion here useful: Local and global supersymmetry

There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejudice. You may find this related blog discussion here useful: Local and global supersymmetry

There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejedudice. You may find this related blog discussion here useful: Local and global supersymmetry

There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejudice. You may find this related blog discussion here useful: Local and global supersymmetry