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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 phenomenologically 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 pure classical gravity KK-compactifications are argued (Penrose 03, section 10.3) to generically be unstable by the Penrose-Hawking singularity theorem.
But if extra gauge fields or higher gauge field beyond pure gravity are admitted in the higher dimensions, then stable compactifications may exist if there is “magnetic flux” in the compact fiber spaces. These are called Freund-Rubin compactifications, or flux compactifications.
A well-studied example is 6-dimensional Einstein-Maxwell theory with magnetic flux on a 2-dimensional fiber spaces over a 4-dimensional base space (Freund-Rubin 80, RDSS 83).
(On the other hand, Freund-Rubin compactifications usually have fibers the site of the curvature radius of the base, and hence not “small”.)
Similarly, in string theory it is argued that the extra fields and further 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 mechanism” Brandenberger-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.
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) $F_p$ 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
couples the flux VEV to the metric $g$ (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.
The problem of generic in-stability of moduli of pure gravity KK-compactifications is highlighted in
Freund-Rubin flux compactifications are due to
A class of stable compactifications of 6d Einstein-Maxwell theory down to four dimensions is due to
and the special case of compactifications of 6d Einstein-Maxwell theory to 4d is in
Further discussion of these models as toy models for flux compactifications in string theory is in
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.
Robert Brandenberger, Cumrun Vafa, Superstrings In The Early Universe, Nucl. Phys. B 316, 391 (1989) (spire)
Scott Watson, Robert Brandenberger, Stabilization of Extra Dimensions at Tree Level, JCAP 0311 (2003) 008 (arXiv:hep-th/0307044)
Brian Greene, Daniel Kabat, Stefanos Marnerides, On three dimensions as the preferred dimensionality of space via the Brandenberger-Vafa mechanism, 10.1103/PhysRevD.88.043527 (arXiv:1212.2115)
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
Frederik Denef, Michael Douglas, Bogdan Florea, Antonella Grassi, Shamit Kachru, Fixing All Moduli in a Simple F-Theory Compactification, Adv.Theor.Math.Phys.9:861-929, 2005 (arXiv:hep-th/0503124)
Vijay Balasubramanian, Per Berglund, Joseph Conlon, Fernando Quevedo, Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications, JHEP 0503:007,2005 (arXiv:hep-th/0502058)
A variant via Kähler uplifting is
Review includes
Renata Kallosh, Stabilization of moduli in string theory, lectures 2005 (part I pdf, part II pdf)
Joseph Conlon, Moduli Stabilisation and Applications in IIB String Theory, Fortsch.Phys.55:287-422,2007 (arXiv:hep-th/0611039)
Sibasish Banerjee, Calabi-Yau compactification of type II string theories (arXiv:1609.04454)
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
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.
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
Bobby Acharya, Konstantin Bobkov, Gordon Kane, Piyush Kumar, Diana Vaman, An M theory Solution to the Hierarchy Problem, Phys.Rev.Lett.97:191601,2006 (arXiv:hep-th/0606262)
Bobby Acharya, Konstantin Bobkov, Gordon Kane, Piyush Kumar, Jing Shao, Explaining the Electroweak Scale and Stabilizing Moduli in M Theory, Phys.Rev.D76:126010,2007 (arXiv:hep-th/0701034)
Bobby Acharya, Piyush Kumar, Konstantin Bobkov, Gordon Kane, Jing Shao, Scott Watson, Non-thermal Dark Matter and the Moduli Problem in String Frameworks,JHEP 0806:064,2008 (arXiv:0804.0863)
and specifically for the G2-MSSM in
Discussion of moduli stabilization in heterotic string theory includes
Evgeny I. Buchbinder, Burt Ovrut, Vacuum Stability in Heterotic M-Theory, Phys.Rev. D69 (2004) 086010 (arXiv:hep-th/0310112)
Sergei Gukov, Shamit Kachru, Xiao Liu, Liam McAllister, Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants, Phys.Rev.D69:086008,2004 (arXiv:hep-th/0310159)