All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 11 dimensional manifold in order to model particles. Do they do this because there is no mathematical alternatives or because the nature (mathematical essence) of space at the Planck scale is still not yet discovered?
2 Answers
There are approaches to quantum gravity where spacetime is described as a quantum superposition of labelled piecewiselinear CW complexes or other related combinatorial/algebraic entities. See for example:
John Baez, An introduction to spin foam models of quantum gravity and BF theory, in Geometry and Quantum Physics, eds. Helmut Gausterer and Harald Grosse, Springer, Berlin, 2000, pp. 2593.
John Baez, Higherdimensional algebra and Planckscale physics, in Physics Meets Philosophy at the Planck Length, eds. Craig Callender and Nick Huggett, Cambridge U. Press, Cambridge, 2001, pp. 177195.
Daniele Oriti, Spin Foam Models of Quantum Spacetime, PhD thesis, University of Cambridge, 2003, 337 pp.
However, your question feels more like a physics question than a math question to me.
This paper by Carlip http://arxiv.org/abs/grqc/0108040 is a good, relatively nontechnical explanation of why it's hard to reconcile quantum mechanics (QM) with general relativity (GR).
GR says that spacetime is a real manifold with a semiRiemannian metric. QM says that the possible states of a system form a complex vector space.
If you naively try to combine these two ideas, it's hard to make sense of the result. Given one manifoldwithmetric $M_1$ and another one $M_2$, what would it even mean to talk about the linear combination $c_1M_1+c_2M_2$, where $c_1$ and $c_2$ are complex numbers? The spacetimes $M_1$ and $M_2$ do not have any builtin way of matching up points in one with points in the other. The two spacetimes don't even need to have the same topology. In quantum mechanics, we would also have the Born rule, which says that $c_1^2$ and $c_2^2$ have interpretations as the probabilities of outcomes of measurements. It's not clear what these probabilities would mean in this context.
So should spacetime be described at the Planck scale as a real manifold, or if not, then what? Straightforward application of the fundamental principles of the two theories seems to lead to nonsense answers. We really don't know.

$\begingroup$ That naive attempt at combining the two ideas is way too naive! $\endgroup$ Feb 12, 2015 at 20:22

$\begingroup$ I essentially agree with the general message of this answer, but I disagree with the objection given to the naive proposal. The linear combination $c_1M_1+c_2M_2$ can trivially be taken in the vector space generated by the $M_i$'s and it is usually this kind of thing one has to do in quantum mechanics. For example, in gauge theory, we have classically bundleswithconnections and if $E_1$ and $E_2$ are two such objects then $c_1M_1+c_2M_2$ is a welldefined element in the Hilbert space of the theory. More generally, the "linearity" of quantum mechanics is something which has nothing to do .. $\endgroup$ Feb 15, 2015 at 20:34

$\begingroup$ with the linearity of the classical objects. I agree that the naive proposal does not work but one has to give better reasons. $\endgroup$ Feb 15, 2015 at 20:35