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The string theory landscape comprises $10^{500}$ (give or take some) different Universes. What feature(s) of string theory you have to vary to "go" from one Universe to the next?

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$10^{500}$ is an approximate number of possible $6D$ Calabi-Yau compactifications of String Theory, which is purely geometrical fact (look at this book on Mirror Symmetry for a very detailed and thorough discussion of the subject). For lower dimensional compactifications the situation is much simpler -- just $S^1 \times S^1$ for $2D$ case, and $S^1 \times S^1 \times S^1 \times S^1$ or so-called K3 manifold for $4D$ case.

Calabi-Yau compactifications are relevant, if one restricts the VeV's of background fields to be zero. In principle, it doesn't have to be the case, and there are many very interesting and important solutions of String Theory with non-zero background fields. For example, Type IIB String Theory on $AdS_5 \times S^5$ background with non-zero flux of 5-strength of 4-form field through $S^5$, which appeared to be dual to ${\cal N}=4$ Super Yang-Mills theory in $4D$. There are also many possible flux compactifications in the form $\mathbb{R}^4 \times M_6$, where $M_6$ is a compact manifold with non-zero fluxes of certain String Theory fields through it (see, for example, this review).

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    $\begingroup$ Compactifications with nonzero fluxes are also interesting in mathematics, in the form of e.g. generalized (Kähler) geometry, introduced by Hitchin and his students. $\endgroup$ – Danu Feb 9 '17 at 9:15
  • $\begingroup$ @AndreyFeldman-Are all the possible compactifications "eigenstates" of some (probability) function? $\endgroup$ – descheleschilder Feb 9 '17 at 19:25
  • $\begingroup$ @descheleschilder Sorry, I totally can't understand what do you mean. $\endgroup$ – Andrey Feldman Feb 10 '17 at 13:23
  • $\begingroup$ @AndreyFeldman-What I mean to ask if that if a certain compactification becomes real, is that because it is projected out of a superposition of all the possible configurations, like the spin of a particle gets projected on the z-axis when a measurement is made. To put it differently, what determines which compactification becomes real, for example, the one that gives rise to our Universe? Or do they all exist at once? $\endgroup$ – descheleschilder Feb 10 '17 at 13:40
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    $\begingroup$ @descheleschilder Unfortunately, full non-perturbative formulation of String Theory is non known yet, so these results are classical -- all the geometries are solutions of certain equations of motion for background fields, which arose from the requirement of the worldsheet conformal invariance, and have the form of Einstein equations or their generalizations. Only in the case of Topological Strings there is a full quantum mechanical description of the theory, where one sums over many backgrounds. See this review by Vafa and Neitzke for the details. $\endgroup$ – Andrey Feldman Feb 10 '17 at 13:54

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