moduli stabilization



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In physics, moduli stabilization refers to the problem of rendering Kaluza-Klein compactifications stable.

A Kaluza-Klein compactification is a model of gravity where spacetime is assumed to be a higher dimensional fiber bundle, with compact fibers of tiny extension, such that the resulting physics looks effectively lower-dimensional, but inheriting extra fields. Namely the size and shape of the compactified extra dimension is encoded in the Riemannian metric, hence in the field of gravity, hence are themselves dynamical fields. Since these fields parameterize the moduli space of the KK-compactification, they are called moduli fields.

The problem of moduli stabilization is the problem of identifying mechanisms or conditions that ensure that as these fields dynamically evolve, the compact spatial dimensions remain stably so, neither opening up nor collapsing. For phenomenoloigcally realistic KK-compactifications the compact volume has to stably be a tiny but finite value (“volume stabilization”).

Equivalently, since fast varying moduli appear as light or massless particles in the low-dimensional effective field theory which would show up in accelerator experiments (such as the LHC) but don’t, the problem is to identify mechanisms or conditions that would render these moduli fields massive.

In classical field theory of gravity then KK-compactifications are argued (Penrose 03, section 10.3) to generically be unstable by the Penrose-Hawking singularity theorem.

In string theory it is argued that string theoretic effects may stabilize the compact dimensions, namely a combination of flux compactification and non-perturbative brane effects (Acharya 02, KKLT 03). However, these arguments typically focus on fluctuations that preserve given special holonomy (Calabi-Yau 3-folds in type II or G2-manifolds in M-theory). There is also a more generic argument for volume compactification by string winding modes (“Brandenberger-Vafa mechanismBrandenberger-Vafa 89, Watson-Brandenberger 03) and the claim (Kim-Nishimura-Tsuchiya 12) that in the non-perturbative IKKT model computer simulations show a spontaneous stable compactification to 3+1 dimensions.

In string theory

The issue of stabilization of compact dimensions arises notably in string theory Kaluza-Klein compactifications.

In the context of type II string theory one way to design the model such that the moduli fields are massive is to consider the case where higher background gauge fields vacuum expectation values (VEVs) F pF_p are present on the compactification space. Since these fields are characterized by their higher field strength/curvature forms which are referred to as “flux” terms in physics, these models are called flux compactification models (KKLT 03).

Because the standard kinetic action term

S kinF p gF p S_{kin} \propto \int F_p \wedge \star_g F_p

couples the flux VEV to the metric gg (via the Hodge star operator) and hence to the moduli, it generically induces an effective potential energy for these, which may stabilize them (when including non-perturbative effects).

Similarly in M-theory on G2-manifolds the 4-form flux of the supergravity C-field leads to potentials for the moduli, which is argued to generically stabilize them (Acharya 02).

Since for these flux compactifications only the periods of the form fields on the compact space matter, under a bunch of further assumptions on the nature of the compactification, one can reduce the number of possible such compactifications to a combinatorial problem. The resulting space of possibilities is also known as the landscape of string theory vacua.

The moduli stabilization in (KKLT 03) was demonstrated in two steps. First, all moduli were stabilized at a fixed minimum with a negative cosmological constant. This was achieved by combining fluxes with non-perturbative effects. Second, the minimum was lifted to a metastable vacuum with a positive cosmological constant. This was accomplished by adding anti D-branes and using previous results, obtained in (Kachru-Pearson-Verlinde 01), that the flux-anti D-brane system can form a metastable bound state with positive energy. In (KKLT 03) it was also shown that one can fine tune various parameters to make the value of the cosmological constant consistent with the observed amount of dark energy.


In field theory

The problem of generic in-stability of moduli of gravity KK-compactifications is highlighted in

  • Roger Penrose, section 10.3 in On the stability of extra space dimensions in Gibbons, Shellard, Rankin (eds.) The Future of Theoretical Physics and Cosmology, Cambridge (2003) (spire)

In string theory

In type II string theory

A generic argument for stabilization of compact dimensions in type II string theory via string winding modes at the self-T-duality radius is the Brandenberger-Vafa mechanism, see e.g.

Discussion of moduli stabilization via flux compactification of and non-perturbative effects in type II string theory/F-theory originates with the influential article (“KKLT”)

which led to a little burst of discussion of the landscape of string theory vacua. The analysis there relies on

Further developments include

A variant via Kähler uplifting is

Review includes

Analogous discussion in type IIA string theory includes (Acharya 02) and

Discussion of volume stabilization of compact dimensions in the context of cosmic inflation is in


  • S.-W. Kim, J. Nishimura, and A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions, Phys. Rev. Lett. 108, 011601 (2012), (arXiv:1108.1540).

  • S.-W. Kim, J. Nishimura, and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, JHEP 10, 147 (2012), (arXiv:1208.0711).

it is claimed that computer simulation shows that the IKKT matrix model description of, supposedly, non-perturbative type II string theory exhibits spontanous decompactification of 3+1 large dimensions, with the other 6 remaining tiny.

In M-theory

Discussion of moduli stabilization in M-theory on G2-manifolds for stabilization via “flux” (non-vanishing bosonic field strength of the supergravity C-field) is in

and moduli stabilization for fluxless compactifications via nonperturbative effects, claimed to be sufficient and necessary to solve the hierarchy problem, is discussed in

and specifically for the G2-MSSM in

In heterotic string theory

Discussion of moduli stabilization in heterotic string theory includes

Revised on September 18, 2016 13:55:06 by Urs Schreiber (