Could quarks be free in higher-dimensional space than 3D? Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space?
To understand why I thought this might be possible, see what we have with electrostatic potential: in 3D it is proportional to $r^{-1}$. This is just what Poisson equation tells us for point charge. If we solve Poisson equation in 2D space, we'll see potential is proportional to $\ln\frac r {r_0}$, and in 1D it's proportional to $r$. We can see that it only allows infinite motion starting form 3D.
Could the same hold for quarks, but with some higher than 3D dimension? Or is their potential of completely different nature with respect to space dimensionality?
 A: If you take the classical analogy of a charge generating field lines then the force at some point can be taken as the density of field lines at that point. In 3D at some distance $r$ the field lines are spread out over a spherical surface of area proportional to $r^2$ so their density and hence force goes as $r^{-2}$ - so far so good.
The trouble with the strong force is that the interactions between gluons cause the field lines to attract each other, so instead of spreading out they group together to form a flux tube or QCD string. In effect all the field lines are compressed into a cylindrical region between the two particles so the field line density, and hence the force, is independant of the separation between the quarks.
This means it doesn't matter what the dimensionality of space is, because the field lines will always organise themselves along the 1D line between the quarks. The quarks woould be confined in any dimension space.
Annoyingly I can't find an authoritative but popular level article on QCD flux tubes, but a Google will find you lots of articles to look through.
