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The undertaking called string theory started out as perturbative string theory where the idea was to encode spacetime physics in perturbation theory by an S-matrix that is obtained by a sum of the integrals of the correlators of a fixed 2-dimensional conformal field theory over the moduli spaces of conformal structures on surfaces of all possible genera – thought of as the second quantization of a string sigma-model.
The S-matrix elements obtained this way from the string perturbation series could be seen to be approximated by an ordinary effective QFT (some flavor of supergravity coupled to gauge theory and fermions) on target space.
Moreover, the precise choice of 2d CFT also encodes a classical solution of the effective background QFT, hence a vacuum of that theory.
The first superstring revolution was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of quantum anomalies.
The second superstring revolution was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbatve definitions.
Aspects of this bigger non-perturbative context are known as M-theory. While one couldn’t figure out what that actually is, the circumstancial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time 11-dimensional supergravity.
In a different but similar manner, other background field theories were found whose classical solutions are thought to encode “stable solutions” (“vacuum solutions”) of whatever physical theory this non-perturbative definition of string theory is.
Here, when talking about a “stable solution” one thinks of solutions of these theories of gravity with plenty of extra fields that look like Minkowski space times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what “constant in time” means), hence sitting at the bottom of their corresponding effective potentials.
Solutions with this property, in particular for all the scalar fields that appear, are said to have stabilized moduli : the scalar fields that encode various properties of the geometry of the solution are constant in time.
Since these geometric properties determine, in the fashion of Kaluza-Klein theory, the effective physics in the remaining Minkowski space factor, it is these “moduli-stabilized” solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world.
At some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term landscape for the collection of all such solutions was invented.
So, to summarize in a few words, the landscape of string theory vacua is…
One widely studied class of modli-stabilized solutions to the string-theory background equations is that of flux compactifications.
These are classical solutions to the corresponding supergravity theory that are of the form $M^4 \times CY$ with $CY$ some Calabi-Yau manifold of six real dimensions such that the RR-field in the solution has nontrivial values on $CY$. Its components are called the fluxes .
The presence of this RR-field in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the RR-field suitably one can find classical solutions in which all these moduli have values that are constant in time.
A review of flux compactifications is for instance in (Graña 05)
at least one thing missing in the discussion here is the subtlety explained out by Jacques Distler in blog dicussion here
Surveys of the general story of flux compactification in F-theory includes
Some general thoughts on what a moduli space of 2d CFTs should be are in
The compactness results mentioned there are discussed in
based on conjectures in
Discussion of aspects of effective field theories which might rule them out as having a UV-completion by a string theory vacuum has been initiated in
Early and technical articles that amplified the existence of a finite but very large number of string theory compactifications are
which says on p. 2
Although the consistenty requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. [] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below.
and
which says in conclusion on page 45-46
Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible.
A popular account of these observations was given in
a commented translation of which later appeared as
Similarly
These articles, and this speech, did not cause much of excitement then. Also they did not discuss moduli stabilization, which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory:
That there are $10^{hundreds}$ different flux compactifications was maybe first said explicitly in
The idea became popular with the articles
Leonard Susskind, The Anthropic Landscape of String Theory (arXiv:hep-th/0302219)
Shamit Kachru, Renata Kallosh, Andrei Linde, Sandip Trivedi, de Sitter Vacua in String Theory, Phys. Rev. D68:046005, 2003 (arXiv:hep-th/0301240)
Joseph Polchinski, The Cosmological Constant and the String Landscape (arXiv:hep-th/0603249)
The specific (but arbitrary) value “$10^{500}$” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in
Michael Douglas, Basic results in vacuum statistics, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 (arXiv:hep-th/0409207)
Ralph Blumenhagen, Florian Gmeiner, Gabriele Honecker, Dieter Lüst, Timo Weigand, p.3 of The Statistics of Supersymmetric D-brane Models, Nucl.Phys.B713:83-135, 2005 (arXiv:hep-th/0411173)
Previously
had considered $10^{120}$ and earlier Lerche-Lüst-Schellekens 87 had $10^{1500}$.
A review of the issue of flux compactifications is in
General considerations on this state of affairs are in
The fact that in principle all the parameters of the “landscape” of string theory vacua are dynamical (are moduli fields) and the idea that an eternal cosmic inflation might be something like an ergodic process in this landscape has led to ideas to connect this to phenomenology and the standard model of cosmology/standard model of particle physics by way of statistical mechanics.
Summaries of this line of thinking include
For more on this see the references at multiverse and eternal inflation.