landscape of string theory vacua



recalling the context

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 2d superconformal 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.)

Hence it is the chocie of worldsheet 2d SCFT which in perturbative string theory translates products of “field insertions” into scattering amplitudes. In perturbative AQFT it is the choice of vacuum state which does this, and therefore 2d SCFTs are the perturbative string theory vacua.

narrowing in on the issue

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.

the landscape

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…

Flux compactifications

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×CYM^4 \times CY with CYCY some Calabi-Yau manifold of six real dimensions such that the RR-field in the solution has nontrivial values on CYCY. 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


F-theory flux compactification

Surveys of the general story of flux compactification in F-theory includes

Moduli space of 2d CFTs

Scan of the moduli space of semi-realistic type IIB CFTs compactfied on orbifolds of Gepner models is in

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

See also

Phenomenological speculation

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 consistency 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.


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

  • Bert Schellekens, Naar een waardig slot, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4

a commented translation of which later appeared as


The articles Lerche-Lüst-Schellekens 86, Lerche-Lüst-Schellekens 87, and the speech Schellekens 98, 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 hundreds10^{hundreds} different flux compactifications was maybe first said explicitly in

The idea became popular in discussion of the cosmological constant with the articles

The specific (but arbitrary) value “10 50010^{500}” for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in


had considered 10 12010^{120} and earlier Lerche-Lüst-Schellekens 87 had 10 150010^{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

  • Raphael Bousso, The State of the Multiverse: The String Landscape, the Cosmological Constant, and the Arrow of Time, 2011 (pdf)

For more on this see the references at multiverse and eternal inflation.

Revised on February 27, 2018 04:16:03 by Urs Schreiber (