Witten (1995) proposed that the vanishing of the cosmological constant and the absence of a massless dilaton in our four-dimensional (4D) world might be explained by a duality between a supersymmetric string vacuum in three dimensions (2+1) and a non-supersymmetric vacuum in four dimensions: a suitable 3D supersymmetric vacuum, taken to infinite coupling λ\lambda — equivalent to decompactifying the circle in the 4D description on R3×S1\mathbb{R}^3\times S^1 — would become a 4D Poincaré-invariant, non-supersymmetric theory whose vacuum energy vanishes by the underlying 3D supersymmetry. Every load-bearing step remains open three decades later: no explicit, stable 3D string vacuum with the required discrete symmetry and supersymmetry unbroken for all λ\lambda is known; the existence of the λ→∞\lambda\to\infty limit as a 4D theory is unverified for any 3D vacuum; the survival of nonzero Bose-Fermi mass splittings under decompactification is unestablished; dilaton decoupling in the uplift is model-dependent and unresolved; and no dual weakly coupled description enabling quantitative 4D predictions exists. The problem asks for an explicit realization of the scenario — a 3D vacuum with unbroken N=1N=1 supersymmetry and the protective discrete CP-type symmetry, stable for all λ\lambda, whose λ→∞\lambda\to\infty limit is a 4D non-supersymmetric Poincaré-invariant theory with exactly vanishing cosmological constant, nonzero Bose-Fermi splittings, and no light dilaton-like scalar, together with calculational control over the resulting strongly coupled 4D theory — or a no-go theorem excluding it.

Background

The cosmological constant problem asks why the vacuum energy density of our four-dimensional (4D) universe — observable as the cosmological constant, which accelerates cosmic expansion — is more than a hundred orders of magnitude below the natural quantum-field-theory estimate built from the zero-point energies of all fields. Supersymmetry (SUSY), a symmetry of spacetime and fields that pairs every boson with a fermion of equal mass, forces the vacuum energy to vanish exactly; but in four dimensions an unbroken supersymmetric vacuum with vanishing cosmological constant also has exactly degenerate boson and fermion spectra, and no superpartner of any known particle has been observed. So in 4D, supersymmetry could explain the vanishing cosmological constant only at the price of an already-falsified prediction.

In 1995, in a companion paper, Witten pointed out a loophole in three spacetime dimensions (two space and one time, abbreviated 3D). In 2+1 dimensions, unbroken supersymmetry can force the cosmological constant to vanish without forcing Bose-Fermi mass degeneracy, because any nonzero mass in three dimensions creates a conical structure at spatial infinity that prevents defining the global conserved supercharges which would pair bosons and fermions; Becker-Becker-Strominger verified this mechanism explicitly in a three-dimensional abelian Higgs model coupled to supergravity, where the topological central charge of the supersymmetry algebra is carried by BPS-saturated vortex solitons (stable topological lumps whose mass saturates the theoretical minimum at fixed charge). A three-dimensional theory can be regarded as a four-dimensional theory compactified on a circle S1S^1 of small radius (Kaluza-Klein compactification: physics on R3×S1\mathbb{R}^3\times S^1, at distances large compared to the circle, looks three-dimensional). Witten's scenario runs this in reverse: take a 3D string vacuum with string coupling λ\lambda (the exponential of the dilaton field), let λ→∞\lambda\to\infty, which corresponds to the radius of the S1S^1 growing to infinity (decompactification), and ask what four-dimensional theory emerges. The conjecture is that a suitable 3D supersymmetric string vacuum at λ=∞\lambda=\infty becomes a 4D Poincaré-invariant, non-supersymmetric string vacuum: the 3D SUSY explains the vanishing 4D cosmological constant, while the absence of Bose-Fermi degeneracy in 3D explains why the 4D world looks non-supersymmetric. Consistency of the picture requires a discrete symmetry: every 4D theory has CPT (the combined charge-conjugation, parity, and time-reversal symmetry), which from the 3D viewpoint acts as a discrete CP-like symmetry, and for the desired vacua all massless bosons must be even and the superpotential odd under it, so that unbroken supersymmetry remains stable. The scenario also promises to solve the dilaton problem: string theory generically contains a massless scalar, the dilaton, whose gravitational-strength couplings would mediate unobserved long-range forces; in the 3D-to-4D uplift the 3D dilaton disappears at large λ\lambda, and if no other field with dilaton-like couplings takes its place, the 4D theory has no light dilaton.

Witten's paper works at this level of abstraction: it posits a three-dimensional string vacuum with unbroken N=1N=1 supersymmetry in 2+1 dimensions — the minimal case, with two real supercharges — together with the discrete symmetry, and constructs no explicit example. The natural candidate class identified in the follow-up literature is M-theory (the strong-coupling limit of type IIA string theory, an eleven-dimensional theory) compactified on an eight-manifold of Spin(7) holonomy — a Ricci-flat compactification space whose single covariantly constant spinor leaves exactly N=1N=1 supersymmetry in 2+1 dimensions. The paper is explicit, however, that the scenario rests on unverified assumptions: (i) the existence of a 3D vacuum with the required discrete symmetry and unbroken supersymmetry stable for all λ\lambda (no explicit example is constructed); (ii) the existence of the λ→∞\lambda\to\infty limit as a Poincaré-invariant 4D theory — in known analogues such strong-coupling limits exist in some cases (e.g. the strong-coupling limit of 5D heterotic string theory is 6D type IIB string theory) while in others existence is unknown; (iii) survival of the nonzero Bose-Fermi splittings in the limit, despite the light states that appear on R3×S1\mathbb{R}^3\times S^1 as the circle decompactifies; (iv) decoupling of the 3D dilaton without a replacement scalar; and (v) the availability of a dual description of the resulting 4D theory, needed for quantitative predictions since the 4D theory is strongly coupled. In the natural 3D mechanisms the mass splittings come out suppressed by powers of Newton's constant, and Witten describes the whole proposal as an optimistic extrapolation, unlikely to be precisely correct in its present form; the paper also records Vafa's suggestion that the mechanism may be connected to holography, the idea that a gravitational theory in a given region can be fully encoded by a theory in fewer dimensions on its boundary.

Subsequent work mapped the terrain without closing it. Sethi-Vafa-Witten derived tadpole-cancellation constraints — the condition that the total charge sourced by branes in the compactification must cancel for the internal equations to be solvable — that render some supersymmetric compactifications below four dimensions impossible, restricting the class of candidate 3D vacua. Curio-Lüst constructed candidate 3D N=1N=1 vacua from U-duality orbits (U-duality being the full non-perturbative duality group of a string compactification, combining perturbative dualities with strong-weak coupling duality) without analyzing the discrete symmetry or the large-λ\lambda limit. The most developed framework, M-theory on Spin(7) manifolds with background flux (field strengths with nonzero values threading the compact space), yields the general 3D N=1N=1 supergravity action but stabilizes all moduli (the scalar fields parametrizing the compactification geometry) except the radial modulus, whose runaway potential — a potential with no minimum, driving the modulus toward infinite value — remains unsolved. In the 2010s, Bonetti-Grimm-Pugh related 3D N=1N=1 M-theory on Spin(7) manifolds to a non-supersymmetric 4D F-theory background (F-theory: a geometric formulation of type IIB string theory in which the string coupling varies over auxiliary two-tori) reduced on a finite interval — a different compactification from the λ→∞\lambda\to\infty decompactification of the original scenario — and Heckman and collaborators revived the program in F-theory on a curved 4D spacetime, obtaining a fractional 'N=1/2N=1/2 supersymmetry' with zero-point-energy cancellation in a related but distinct setting. None of these lines constructs a vacuum satisfying the original assumptions, and no no-go theorem excluding them is known.

Problem Statement

This is the open problem left by E. Witten's 1995 proposal: determine whether the proposed three-to-four-dimensional strong-coupling explanation of the vanishing cosmological constant can be realized. Specifically: does there exist a three-dimensional string vacuum — explicitly constructed, with unbroken N=1N=1 supersymmetry in 2+1 dimensions and a discrete CP-type symmetry (the 3D remnant of 4D CPT) under which all massless bosons are even and the superpotential odd — such that:

(1) supersymmetry remains unbroken and the vacuum stable for every value of the coupling λ\lambda;

(2) the λ→∞\lambda\to\infty limit exists and is a Poincaré-invariant four-dimensional theory (the limit is the decompactification of the S1S^1 of the four-dimensional description on R3×S1\mathbb{R}^3\times S^1);

(3) the Bose-Fermi mass splittings, nonzero at finite λ\lambda, remain nonzero at λ=∞\lambda=\infty, so that the four-dimensional spectrum is genuinely non-supersymmetric;

(4) the three-dimensional dilaton decouples in the uplift and no other light scalar with dilaton-like, gravitational-strength couplings appears in the four-dimensional theory;

so that the λ=∞\lambda=\infty theory is a four-dimensional string vacuum with an exactly vanishing cosmological constant (inherited from the 3D supersymmetry) and no light dilaton? And, as the source additionally requires for the scenario to be predictive: does that four-dimensional theory admit a dual or weakly coupled description — or some other calculational control over its strong coupling — enabling quantitative predictions for particle masses and interactions?

The scenario is realized by an explicit construction within string/M-/F-theory establishing all of (1)-(4) together with a vanishing four-dimensional cosmological constant, and it is ruled out by a no-go theorem showing, under precisely stated assumptions that cover this scenario, that no such three-dimensional vacuum exists or that its λ→∞\lambda\to\infty limit cannot yield the required four-dimensional theory.

References

  1. Edward Witten, "Strong Coupling and the Cosmological Constant", Modern Physics Letters A 10 (1995) 2153-2156. DOI: 10.1142/S0217732395002301; arXiv:hep-th/9506101; https://arxiv.org/abs/hep-th/9506101
  2. Edward Witten, "Is Supersymmetry Really Broken?", International Journal of Modern Physics A 10 (1995) 1247. DOI: 10.1142/S0217751X95000590; arXiv:hep-th/9409111; https://arxiv.org/abs/hep-th/9409111
  3. Katrin Becker, Melanie Becker, Andrew Strominger, "Three-Dimensional Supergravity and the Cosmological Constant", Physical Review D 51 (1995) R6603-R6607. DOI: 10.1103/PhysRevD.51.R6603; arXiv:hep-th/9502107; https://arxiv.org/abs/hep-th/9502107
  4. Lorenzo Cornalba, Frank Wilczek, "Mass Splittings From Symmetry Obstruction", Physics Letters B 411 (1997) 112-116. DOI: 10.1016/S0370-2693(97)00997-0; arXiv:hep-th/9706014; https://arxiv.org/abs/hep-th/9706014
  5. S. Sethi, C. Vafa, E. Witten, "Constraints on Low-Dimensional String Compactifications", Nuclear Physics B 480 (1996) 213-224. DOI: 10.1016/S0550-3213(96)00483-X; arXiv:hep-th/9606122; https://arxiv.org/abs/hep-th/9606122
  6. G. Curio, D. Lüst, "New N=1 Supersymmetric 3-dimensional Superstring Vacua from U-manifolds", Physics Letters B 428 (1998) 95-104. DOI: 10.1016/S0370-2693(98)00391-8; arXiv:hep-th/9802193; https://arxiv.org/abs/hep-th/9802193
  7. M. J. Duff, H. Lü, C. N. Pope, "Supersymmetry without Supersymmetry", Physics Letters B 409 (1997) 136-144. DOI: 10.1016/S0370-2693(97)00687-4; arXiv:hep-th/9704186; https://arxiv.org/abs/hep-th/9704186
  8. Michael Dine, Yuri Shirman, "Truly Strong Coupling and Large Radius in String Theory", Physics Letters B 377 (1996) 36-42. DOI: 10.1016/0370-2693(96)00316-4; arXiv:hep-th/9601175; https://arxiv.org/abs/hep-th/9601175
  9. Michael Dine, "Seeking the Ground State of String Theory", Progress of Theoretical Physics Supplement 134 (1999) 1-17. DOI: 10.1143/PTPS.134.1; arXiv:hep-th/9903212; https://arxiv.org/abs/hep-th/9903212
  10. Melanie Becker, Dragos Constantin, "A Note on Flux Induced Superpotentials in String Theory", Journal of High Energy Physics 08 (2003) 015. DOI: 10.1088/1126-6708/2003/08/015; arXiv:hep-th/0210131; https://arxiv.org/abs/hep-th/0210131
  11. Melanie Becker, Dragos Constantin, S. James Gates Jr., William D. Linch III, Willie Merrell, J. Phillips, "M-theory on Spin(7) Manifolds, Fluxes and 3D, N=1 Supergravity", Nuclear Physics B 683 (2004) 67-104. DOI: 10.1016/j.nuclphysb.2004.01.013; arXiv:hep-th/0312040; https://arxiv.org/abs/hep-th/0312040
  12. Federico Bonetti, Thomas W. Grimm, Tom G. Pugh, "Non-Supersymmetric F-Theory Compactifications on Spin(7) Manifolds", Journal of High Energy Physics 01 (2014) 112. DOI: 10.1007/JHEP01(2014)112; arXiv:1307.5858; https://arxiv.org/abs/1307.5858
  13. Jonathan J. Heckman, Craig Lawrie, Ling Lin, Gianluca Zoccarato, "F-theory and Dark Energy", Fortschritte der Physik 67 (2019) 1900057. DOI: 10.1002/prop.201900057; arXiv:1811.01959; https://arxiv.org/abs/1811.01959
  14. Jonathan J. Heckman, Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, "Pixelated Dark Energy", Fortschritte der Physik 67 (2019) 1900071. DOI: 10.1002/prop.201900071; arXiv:1901.10489; https://arxiv.org/abs/1901.10489
  15. Mirjam Cvetič, Jonathan J. Heckman, Ethan Torres, Gianluca Zoccarato, "Reflections on the Matter of 3d N=1 Vacua and Local Spin(7) Compactifications", Physical Review D 105 (2022) 026008. DOI: 10.1103/PhysRevD.105.026008; arXiv:2107.00025; https://arxiv.org/abs/2107.00025
  16. Jonathan J. Heckman, Austin Joyce, Jeremy Sakstein, Mark Trodden, "Exploring 2+2 Answers to 3+1 Questions" (2022). arXiv:2208.02267; https://arxiv.org/abs/2208.02267