Quantum and structural viability of scalar-induced spacetime compactification (the Gell-Mann-Zwiebach sigma-model mechanism)

Gell-Mann and Zwiebach (1984) showed that a nonlinear sigma model coupled to (4+E)-dimensional Einstein gravity can induce spontaneous compactification: when the scalar target manifold carries an Einstein metric with $R_{ij}=\alpha^2 h_{ij}$ and the sigma-model coupling is tuned to $\lambda^2=2/\alpha^2$, the ansatz $\phi^i=x^i$ (the scalars equal the internal coordinates) makes spacetime factorize as flat four-dimensional Minkowski space times an internal space isomorphic to the target manifold, with an identically vanishing classical four-dimensional cosmological constant and an unfixed internal size. The source paper left the mechanism's physical viability open. Parts of the original bundle were subsequently settled by the authors' own companion papers (fluctuation spectra for compact symmetric spaces; a working noncompact $SU(1,1)/U(1)$ example) and by later work (half-supersymmetric realizations in $D=7,8,9$ supergravities; constructions retaining a massless photon). The surviving open core is threefold: (i) the quantum-corrected four-dimensional cosmological constant and the radiative fate of the size modulus - only a single one-loop effective-potential check for $M^4\times S^3$ exists; (ii) the complete stability classification of Einstein target spaces beyond the symmetric, group, and coset cases analyzed; (iii) realization with controlled scalar potentials in a supersymmetric or string-level theory retaining massless gauge bosons. These components jointly decide whether a purely geometric, classically flat, scalar-induced compactification can survive as a consistent quantum compactification.

Background

Kaluza-Klein theory unifies gauge fields with gravity by adding extra spatial dimensions to spacetime; the extra dimensions must close up (compactify) into a small internal manifold, and four-dimensional gauge fields then appear as components of the higher-dimensional metric along the internal directions (Kaluza-Klein vectors). For the picture to be predictive the internal geometry should follow from the higher-dimensional field equations rather than being imposed by hand - the program of spontaneous compactification. The benchmark mechanism of Freund and Rubin (1980) compactifies supergravity (the extension of Einstein gravity by supersymmetry, a symmetry relating bosons and fermions) using the flux of an antisymmetric tensor field, but generically leaves an unacceptably large cosmological constant - the vacuum energy density of the observed spacetime - in the noncompact dimensions.

Gell-Mann and Zwiebach (1984) proposed a purely geometric alternative. A nonlinear sigma model is a scalar field theory in which the fields $\phi^i$ are coordinates on a Riemannian manifold $(M_E,h_{ij})$, the target space; such scalar sectors are generic in higher-dimensional supergravity, where the target spaces are usually coset spaces $G/H$ (the space of left cosets of a subgroup $H$ in a Lie group $G$) and are typically noncompact and negatively curved. Coupling $E$ such scalars to Einstein gravity in $D+E$ dimensions with no bare cosmological constant and no scalar potential, they used the ansatz $\phi^i(x)=x^i$: the scalar fields are literally the extra-dimensional coordinates. If the target metric is an Einstein metric, $R_{ij}(h)=\alpha^2 h_{ij}$ (the Ricci tensor, which measures the curvature of the metric, is proportional to the metric itself), and the sigma-model coupling is tuned to $\lambda^2=2/\alpha^2$, then taking the internal metric proportional to the target metric, $\overline{g}_{ij}=-a^2 h_{ij}$ in the source's signature convention with $a$ the internal size, solves all field equations, and spacetime factorizes as flat $D$-dimensional Minkowski space times $M_E$. The Ricci tensor of the noncompact directions vanishes identically, $R_{\alpha\beta}=0$: the four-dimensional cosmological term is exactly zero at the classical level, with no tuning of a vacuum energy against gravity.

Four structural features define the problem. First, the internal size $a$ is not fixed by the classical equations - it is a modulus (a flat direction of the classical solution space, a breathing degree of freedom). Second, every Kaluza-Klein vector acquires the same mass, of order $(a^2\lambda^2)^{-1/2}$, so the target-space isometry group (the group of distance-preserving transformations of the target metric) survives only as a global symmetry at low energies and no massless gauge boson remains. Third, the strict classical solution requires the scalar potential to vanish, $V(\phi)=0$. Fourth, a higher-dimensional cosmological constant $\Lambda$ extends the mechanism to (anti-)de Sitter noncompact factors (de Sitter and anti-de Sitter space are the maximally symmetric vacuum solutions of Einstein's equations with, respectively, a positive and a negative cosmological constant) through $a^2\Lambda=\pm\alpha^2-2/\lambda^2$; negatively curved target spaces then induce noncompact internal spaces, which the authors suggested might still be acceptable if the spectrum of Kaluza-Klein masses exhibits a mass gap separating the massless sector from the continuum of massive states - a suggestion reinforced by the fact that supergravity scalar manifolds are typically noncompact, such as the $SO(n,1)/SO(n)$ scalar cosets of $d=9$ supergravity constructed by Gates, Nishino and Sezgin, or the gauged supergravities (supergravity theories in which a global symmetry of the scalar manifold is promoted to a local gauge symmetry, which generically generates a scalar potential; for example the $SU(2)$-gauged $N=2$, $d=8$ theory of Salam and Sezgin).

The authors listed what would have to hold for the mechanism to be physically viable and left all of it open: quantum corrections to the vanishing cosmological term were not computed; the applicability to supergravity and superstring theories was unclear; the equally massive Kaluza-Klein vectors could not be identified with observed gauge bosons; the stability of the background against fluctuations was deferred to a companion study; and the admissibility of scalar potentials was unresolved. Classic results in Kaluza-Klein theory show that one-loop vacuum energy on product compactifications can both shift the effective cosmological constant and stabilize a free internal size, which makes the quantum fate of this classically flat background a sharp, well-posed question.

Problem Statement

Consider Einstein gravity in $D+E$ dimensions (the physical case $D=4$) coupled to $E$ scalar fields forming a nonlinear sigma model with target metric $h_{ij}(\phi)$ and coupling $\lambda$, with vanishing higher-dimensional cosmological constant and vanishing scalar potential. The Gell-Mann-Zwiebach background is

$\phi^i(x)=x^i,\qquad \overline{g}_{ij}=-a^2h_{ij},\qquad R_{ij}(h)=\alpha^2 h_{ij},\qquad \lambda^2=\frac{2}{\alpha^2},$

with field equations $R_{\mu\nu}=(2/\lambda^2)h_{ij}\,\partial_\mu\phi^i\partial_\nu\phi^j$; it solves them with the noncompact factor Ricci-flat, $R_{\alpha\beta}=0$, so the four-dimensional cosmological term vanishes identically at the classical level, while the internal size $a$ remains a free modulus. A higher-dimensional cosmological constant generalizes the solution to (anti-)de Sitter noncompact factors with $a^2\Lambda=\pm\alpha^2-2/\lambda^2$.

Determine whether this mechanism yields a physically viable compactification, in three jointly defining senses:

(1) Quantum stability of the vanishing cosmological term. Compute the quantum-corrected effective four-dimensional cosmological constant for scalar-induced compactifications - at minimum at one loop - including the radiatively generated effective potential for the size modulus $a$, and establish whether the classically exact $R_{\alpha\beta}=0$ survives quantization or quantum effects reintroduce a nonzero cosmological constant and destabilize or freeze the modulus. (The only existing computation is the one-loop effective potential for the special case $M^4\times S^3$, which addressed stability of the scalar sector but not the general cosmological term.)

(2) Complete stability classification. Classify the Einstein target spaces as stable or unstable under the full coupled graviton-scalar fluctuation spectrum about the background - beyond the compact symmetric spaces $G/H$ (homogeneous spaces whose geometry is invariant under a reflection about every point), shown free of tachyonic modes (fluctuations of negative mass-squared) by Gell-Mann and Zwiebach's companion analysis, and beyond the group-manifold and coset cases later examined - giving in each case which fluctuating fields remain massless and which acquire masses.

(3) Controlled potentials and supersymmetric or string-level realization. Determine for which nonzero scalar potentials $V(\phi)$ scalar-induced compactification with an exactly flat four-dimensional spacetime still exists, and exhibit such realizations in a supersymmetric higher-dimensional theory, ultimately in string theory, retaining at least one massless gauge boson. (The half-supersymmetric realizations in $D=7,8,9$ supergravities and the massless-photon constructions already in the literature use the vanishing classical potential $V(\phi)=0$ and therefore do not discharge this sub-question; the new ingredient required is a controlled nonzero $V(\phi)$ giving an exactly flat four-dimensional spacetime, ultimately at string level.)

These three components jointly decide the same question: whether a purely geometric, classically flat, sigma-model-induced compactification can survive as a consistent quantum compactification.

References

1. Murray Gell-Mann and Barton Zwiebach, Spacetime compactification induced by scalars, Physics Letters B 141 (1984) 333-336. DOI: 10.1016/0370-2693(84)90256-9, https://doi.org/10.1016/0370-2693(84)90256-9
2. Murray Gell-Mann and Barton Zwiebach, Curling up two spatial dimensions with SU(1,1)/U(1), Physics Letters B 147 (1984) 111-114. DOI: 10.1016/0370-2693(84)90603-8, https://doi.org/10.1016/0370-2693(84)90603-8
3. Murray Gell-Mann and Barton Zwiebach, Dimensional reduction of space-time induced by nonlinear scalar dynamics and noncompact extra dimensions, Nuclear Physics B 260 (1985) 569-592. DOI: 10.1016/0550-3213(85)90051-3, https://doi.org/10.1016/0550-3213(85)90051-3
4. Peter G. O. Freund and Mark A. Rubin, Dynamics of dimensional reduction, Physics Letters B 97 (1980) 233-235. DOI: 10.1016/0370-2693(80)90590-0, https://doi.org/10.1016/0370-2693(80)90590-0
5. R. Parthasarathy, On space-time compactification induced by a general non-linear sigma model, Physics Letters B 181 (1986) 91-97. DOI: 10.1016/0370-2693(86)91261-X, https://doi.org/10.1016/0370-2693(86)91261-X
6. S. J. Gates, H. Nishino and E. Sezgin, Supergravity in d=9 and its coupling to the non-compact sigma model, Classical and Quantum Gravity 3 (1986) 21-28. DOI: 10.1088/0264-9381/3/1/005, https://doi.org/10.1088/0264-9381/3/1/005
7. C. M. Hull and N. P. Warner, Non-compact gaugings from higher dimensions, Classical and Quantum Gravity 5 (1988) 1517-1530. DOI: 10.1088/0264-9381/5/12/005, https://doi.org/10.1088/0264-9381/5/12/005
8. B. Chakraborty and R. Parthasarathy, Quantum effects in Kaluza-Klein theory with spontaneous compactification induced by a non-linear sigma model, Classical and Quantum Gravity 6 (1989) 1455-1462. DOI: 10.1088/0264-9381/6/10/014, https://doi.org/10.1088/0264-9381/6/10/014
9. B. Chakraborty and R. Parthasarathy, Dynamical compactification in Kaluza-Klein cosmology, Physics Letters A 142 (1989) 75-78. DOI: 10.1016/0375-9601(89)90163-1, https://doi.org/10.1016/0375-9601(89)90163-1
10. Nicolaos A. Batakis and Alexandros A. Kehagias, Electroweak boson masses from gravity-coupled sigmas, Physics Letters B 238 (1990) 45-49. DOI: 10.1016/0370-2693(90)92098-4, https://doi.org/10.1016/0370-2693(90)92098-4
11. N. A. Batakis and A. A. Kehagias, Generic SU(2) multidimensional sigma-model vacuum, Classical and Quantum Gravity 7 (1990) L113-L117. DOI: 10.1088/0264-9381/7/6/002, https://doi.org/10.1088/0264-9381/7/6/002
12. Nicolaos A. Batakis and Alexandros A. Kehagias, Higgs mass determined in the gravity-coupled sigma model, Physics Letters B 253 (1991) 149-153. DOI: 10.1016/0370-2693(91)91376-7, https://doi.org/10.1016/0370-2693(91)91376-7
13. E. I. Guendelman, The no-scale non-linear sigma model, magnetic charge, the cosmological constant, compactification and symmetry breaking, Classical and Quantum Gravity 12 (1995) 1893-1911. DOI: 10.1088/0264-9381/12/8/007, https://doi.org/10.1088/0264-9381/12/8/007
14. M. Vasilić, Consistency analysis of Kaluza-Klein geometric sigma models, General Relativity and Gravitation 33 (2001) 1783-1798. DOI: 10.1023/A:1013079318311, https://doi.org/10.1023/A:1013079318311
15. M. Vasilić, Vacuum stability in Kaluza-Klein geometric sigma models, General Relativity and Gravitation 34 (2002) 1383-1391. DOI: 10.1023/A:1020074701286, https://doi.org/10.1023/A:1020074701286
16. Alex Kehagias and Constantina Mattheopoulou, Scalar-induced compactifications in higher dimensional supergravities, Journal of High Energy Physics 2005 (2005) 106. DOI: 10.1088/1126-6708/2005/08/106 (preprint arXiv:hep-th/0507010), https://arxiv.org/abs/hep-th/0507010
17. Alex Kehagias and Constantina Mattheopoulou, Flat-brane compactifications in supergravity induced by scalars, Nuclear Physics B 797 (2008) 117-136. DOI: 10.1016/j.nuclphysb.2007.12.027, https://doi.org/10.1016/j.nuclphysb.2007.12.027
18. Durmus A. Demir and Beyhan Puliçe, Non-gravitating scalars and spacetime compactification, Physics Letters B 638 (2006) 1-7. DOI: 10.1016/j.physletb.2006.05.015, https://doi.org/10.1016/j.physletb.2006.05.015
19. Nima Arkani-Hamed, Savas Dimopoulos, Nemanja Kaloper and Raman Sundrum, A small cosmological constant from a large extra dimension, Physics Letters B 480 (2000) 193-199. DOI: 10.1016/S0370-2693(00)00359-2 (preprint arXiv:hep-th/0001197), https://doi.org/10.1016/S0370-2693(00)00359-2
20. Fotis Farakos, Alex Kehagias and Emmanuel N. Saridakis, Vanishing cosmological constant by gravitino-dressed compactification of 11D supergravity, Journal of High Energy Physics 2011 (2011) 107. DOI: 10.1007/JHEP12(2011)107 (preprint arXiv:1111.1577), https://arxiv.org/abs/1111.1577
21. Abdus Salam and E. Sezgin, d=8 supergravity, Nuclear Physics B 258 (1985) 284-304. DOI: 10.1016/0550-3213(85)90613-3, https://doi.org/10.1016/0550-3213(85)90613-3
22. Milovan Vasilić, Geometric sigma model of the Universe, Chinese Physics C 41 (2017) 055102. arXiv:1507.03093, https://arxiv.org/abs/1507.03093
23. Beyhan Pulice and Sukru Hanif Tanyildizi, Self-Compactifying Gravity (2010). arXiv:1009.4952, https://arxiv.org/abs/1009.4952