Types of quantum field thories
Urs Schreiber: here is a rough account of my understanding of what the landscape is, to be corrected:
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.
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 with some Calabi-Yau manifold of six real dimensions such that the RR-field in the solution has nontrivial values on . 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.
Here is a review on this topic:
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