In general, if you construct a manifold out of a combinatorial graph, or out of patches, then you're not finding anything "more fundamental". You're just describing the most straightforward and most superficial definition of the concept of topology.

I realize that this very modest sentence contradicts the whole philosophy of Stephen Wolfram's book, and the reason behind this contradiction is that Stephen Wolfram's book is misguided in this very basic respect.

If you want to find more fundamental laws of physics that produce topology, you always deal with phases of a physical system that have to be found by solving some equations; and/or with a geometric interpretation of individual terms in an expansion of a physical observable that doesn't have a topological interpretation.

To see the latter example, check e.g. the quantum foam paper

http://arxiv.org/abs/hep-th/0309208

by Okounkov (a Fields medalist), Reshetikhin, and (most importantly) Vafa. A partition sum of a topological string theory may be obtained as a sum over manifolds of many different topologies - but it may also be viewed as an expansion of a function that describes the propagation of (topological) strings in a flat background.

Both ways of writing the partition sum are also equivalent to the partition sum of a melting crystal. Their paper is a particular, quantitative realization of John Wheeler's concept of a "quantum foam".

In other contexts of physics, different topologies are simply allowed and they have to be summed over. That's the case of perturbative string theory where the scattering amplitudes are calculated as sums over world sheets of all topologies (Riemann surfaces).

This sum may be obtained in various other formalisms where it looks "derived". In the non-perturbative light cone gauge description of perturbative string theory, the so-called matrix string theory,

http://arxiv.org/abs/hep-th/9701025

http://arxiv.org/abs/hep-th/9703030

the very states containing $K$ strings are obtained from the same Hilbert space, by identifying various holonomies of the gauge field around a circular dimension of a Yang-Mills theory. The possible holonomies for which low-energy states exist may be labeled by permutations of $N$ eigenvalues, and the cycles from which these permutations are composed may literally be interpreted as strings of various lengths.

Now, 1 string is topologically different from $K$ strings for $K\neq 1$ but in matrix string theory, all these states are configurations of a single field theory in different limits. Analogously, the sum over histories will contain world sheets of all topologies which are generated as histories of switching in between the phases of the Yang-Mills theory.

In string theory, you get generically dual descriptions of a compactification in which the topology of the space is completely different. For example, mirror symmetry relates two very different topologies of a six-dimensional Calabi-Yau manifold. M-theory or string theory compactified on a four-dimensional K3 manifold is equivalent to heterotic strings on tori; the individual excitations of the heterotic string are mapped to nontrivial submanifolds of the K3 manifold, and so on.

So the summary is that there are many exciting ways in which topology of space may turn out to be "emergent", or at least "as fundamental as other, non-topological descriptions", but none of them is similar to your "discretized" template.

This post imported from StackExchange Physics at 2014-04-01 16:39 (UCT), posted by SE-user Luboš Motl