Questions about the landscape in string theory If I understand correctly, the string theory landscape is the totality of possible Calabi-Yau manifolds to make up the compact factor of space in string theory, in which there are of the order of $10^{500}$ elements which are stabilized by fluxes and D-branes, meaning that they are local vacuum energy minima, and essentially determine the physics of the universe. 
My questions are these:


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*Is it assumed that the Calabi-Yau factor of string theory is topologically unique (even though we have no idea which one it is), and is the landscape the totality of complex and metric moduli on it? Or do topologically distinct Calabi-Yau manifolds appear in the same landscape?
Added in edit: More precisely, string theory can be formulated for any topological Calabi-Yau threefold. The action involves different complex and metric structures on them, Yang-Mills fields, etc. When we want to describe the universe, do we have to fix a particular target manifold (up to topology), and then vary the geometrical structures and other fields on it, or should all possible topological types of Calabi-Yau manifolds be allowed to contribute to the same path integral? (Perturbatively it seems obvious that we cannot deal with a change in topology, as this is a discontinous change.)
Classically the distinction seems meaningless, since we will have a single realization anyway (not a superposition), but after quantization we could have contributions to the path integral not only from different metrical and complex structures on a given Calabi-Yau manifold (along with their flux and brance content), but also from topologically non-equivalent ones. 
From what I understand, flux and brane content ensure that critical vacuum energies actually are local minima, so that we don't have massless modes of excitation, but that this construction stabilizes geometrical structures on one and the same topological Calabi-Yau manifold, i.e. without changes in topology. Is that correct?
My question then would be if the landscape is thought of as consisting of these locally flux- and brane-stabilized energy minima as well as all states (i.e. geometrical structures) between them, but all on a fixed topological manifold, or whether different topological manifolds are present in the same landscape?

*Is it possible that the universe at different locations is in different vacuum states? My guess would be that it isn't, since the possible stabilized vacua form a discrete set, and the moduli should vary continuously. If it is, would it even be possible that the Calabi-Yau factor is topologically different at different locations?

*Why is it unavoidable that the moduli of the Calabi-Yau manifold are dynamic? Couldn't it be that the Calabi-Yau manifold with all its structure (metric, complex) is fixed as a model parameter? The only thing that occurs to me now is that in the Polyakov action maybe the complex structure emerges from the embedded world lines. Since the fields determining the embedding are dynamic, all these different complex structures can emerge.


Partial answers are very welcome as well.  
 A: Some informal imprecise answers.

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*There is certainly more than one possible Calabi-Yau allowed in string theory. Indeed, a single Calabi-Yau can correspond to many different vacua, because one must additionally specify flux and brane content.


*Just as a technical point, continuously varying the moduli of one Calabi-Yau can end up with another Calabi-Yau; this is an aspect of mirror symmetry. As for having a truly discontinuous change in the topology of the compact dimensions, this should correspond to some kind of domain wall.
Also, there is the idea that chaotic eternal inflation in string theory should produce a single expanding universe in which diverse vacua are realized in different locations. But this idea has not yet been realized with any kind of rigor. It is extremely difficult to describe the generalized tunneling process responsible for transitions between vacua in such a framework.


*Moduli can be rigid. But it is the exception rather than the rule.
edit: In response to follow-up questions:


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*I meant to ask whether a: we should allow non-homeomorpic CY's to contribute to the same path integral (or the same realization of the
theory), or b: we assume that there is a single CY (up to topological
equivalence), and we only vary the fields (metric, moduli) on it? From
what I understand, flux and brane content ensure that critical vacuum
energies actually are local minima, so that we don't have massless
modes of excitation, but that all this stabilizes structures on a
given topological CY. Is that correct?


One often regards two dimensions as fundamental and ten dimensions as emergent. The two-dimensional picture of perturbative string theory is defined by a choice of 2d superconformal field theory and a sum over 2d manifolds, which in the space-time interpretation are worldsheets of strings in ten dimensions. The 2d theory can have different phases corresponding to different CYs, and even non-geometric phases that have no 10d interpretation.
At a deeper level, one may suppose that the vacua originate in a single nonperturbative string theory, with many local minima around which one may do perturbation theory. These minima would be the vacua of unified string theory. But this unified theory has not been constructed, and in practice even the SCFT for a vacuum is usually not available, and instead one does indeed start with the 10d picture. For more, see Urs Schreiber's remarks.
Also: two instances of the same CY, but with different flux and brane content, correspond to different vacua (assuming that they are both stable). The heuristic number "$10^{500}$" is not counting CYs. It is based on saying that an average CY has several hundred n-cycles in it, and there are multiple possible values of flux along each cycle, so there are about a googol flux vacua for each CY.



*Your technical point cannot be strictly correct: one property of mirror-symmetric CY's is that they have their Hodge numbers inverted,
and since for Kähler manifolds the Hodge numbers are topological
invariants, mirror partners certainly can not be continuously deformed
into each other (unless possibly $h^{1,1}$=$h^{1,2}$).


There are CYs with more than one mirror. It's not that X can be deformed into its mirror Y, but rather that quantum theory on X mirror Y1 can be deformed into quantum theory on X mirror Y2. This is understood by saying that both Y1 and Y2 emerge from different phases of the same worldsheet theory. Brian Greene wrote about this in one of his popular books; the papers in question are Aspinwall Greene Morrison and Witten.



*When you say "moduli can be rigid", do you mean in less common formulations of string theory, or do you mean that it can happen for
specific CY's, e.g. when $h^{1,2}$=0?


For specific CYs.
