A (3+1)-dimensional analog of Witten's (2+1)-dimensional mechanism: unbroken vacuum supersymmetry, a naturally vanishing cosmological constant, and non-degenerate boson and fermion masses

In (2+1)-dimensional supergravity, Witten (1995) showed that unbroken supersymmetry can guarantee a vanishing cosmological constant without forcing boson and fermion masses to be degenerate: the unbroken supercharges correspond to spinors that are covariantly constant at infinity, and any excited state of nonzero energy gravitationally produces an asymptotically conical geometry that admits no such spinors, so the asymptotic supersymmetries act on the vacuum but not on excited states. Whether an analogous separation between the vacuum and the excited states can be realized in a physically realistic (3+1)-dimensional framework remains open. The problem is to construct such a framework, via modified four-dimensional gravity or modified boundary conditions embedded in an ultraviolet-complete theory, in which the vacuum preserves supersymmetry together with the additional structure (such as an R-symmetry) needed for the cosmological constant to vanish naturally, while generic finite-energy excited states admit no asymptotic supercharges and may have non-degenerate spectra; alternatively, to establish a controlled impossibility result. The later literature confirms and elaborates the (2+1) mechanism, but the proposed (3+1) realizations are sector-limited, speculative, or phenomenologically unviable, and no no-go theorem exists.

Background

Four-dimensional physics coupled to gravity faces a notorious dilemma concerning vacuum energy. Supersymmetry (SUSY) is the symmetry that pairs bosons with fermions; its generators (supercharges) enter an algebra in which the anticommutator of a supercharge with its conjugate is proportional to the Hamiltonian, so unbroken SUSY forces the vacuum energy to vanish. The same algebra, however, organizes particles into supermultiplets — groups of bosonic and fermionic states connected by supercharge action — of equal mass, so unbroken SUSY also predicts degenerate boson and fermion masses. In supergravity, supersymmetry made into a local gauge symmetry and therefore necessarily including gravity, Witten's short note 'Is Supersymmetry Really Broken?' sharpened the dichotomy: either SUSY is unbroken and bosons and fermions are degenerate, or SUSY is broken and the vanishing of the cosmological constant — the vacuum energy, which gravitates — loses its symmetry protection. The note adds a caution: even unbroken SUSY forces the cosmological constant to vanish only in combination with an additional structure, such as an R-symmetry (a continuous internal symmetry rotating the phases of the supercharges) or an underlying string theory.

The note then observes that in $(2+1)$ dimensions this dichotomy evaporates. In local supersymmetry the usable, unbroken supercharges are not arbitrary: they are in natural one-to-one correspondence with spinor fields that are covariantly constant at infinity, i.e. spinors $\varepsilon$ with $\nabla\varepsilon=0$ asymptotically, parallel with respect to the Levi-Civita connection of the asymptotic geometry. Now $(2+1)$-dimensional gravity has no propagating gravitational wave quanta, and any state of nonzero energy produces, by gravitational backreaction, a geometry that is asymptotically conical: the spatial metric at infinity has a deficit angle, as for a point mass in $(2+1)$ gravity. A conical asymptotic geometry admits no covariantly constant spinors and hence no unbroken supersymmetries. Consequently, in $(2+1)$-dimensional supergravity, supersymmetry can hold in the vacuum — ensuring vanishing vacuum energy — while failing, for structural rather than dynamical reasons, on every excited state. Witten closed by remarking that this detailed mechanism is special to $(2+1)$ dimensions, but that it at least makes one wonder whether, in a suitable framework, there can be an analogous phenomenon in $(3+1)$ dimensions.

The $(2+1)$ mechanism was confirmed and made concrete soon after. Becker, Becker and Strominger realized it in the supersymmetric Abelian Higgs model (a supersymmetric scalar electrodynamics whose broken phase supports flux-carrying vortex solitons) coupled to $(2+1)$-dimensional supergravity: the vortex solitons are annihilated by half of the supersymmetries, through a surprising cancellation between the Aharonov-Bohm phase (the quantum phase picked up by a charged field transported around the vortex flux) and the holonomy of the spin connection (the parallel transport of spinors around the deficit), while the remaining half, whose action would ordinarily generate the soliton supermultiplet, are simply not well defined; soliton Bose-Fermi degeneracy is therefore absent. Cornalba and Wilczek recast the effect as a general phenomenon of 'symmetry obstruction': in $(2+1)$ dimensions the long-range fields of extended objects obstruct the realization of the ground state's symmetries on the spectrum, and in particular supersymmetry is obstructed by the gravitational effect of any mass. Three-dimensional vacua with a single real supercharge (called $\mathcal{N}=1$ in three dimensions), for which no nontrivial multiplet degeneracy structure exists at all, also arise naturally, for example from M-theory (the eleven-dimensional extension of string theory) compactified on eight-manifolds of Spin(7) holonomy (a special Riemannian holonomy group); this line has been developed recently by Cvetič, Heckman, Torres and Zoccarato.

In $(3+1)$ dimensions the straightforward analog fails for a structural reason: the gravitational field of a finite-energy localized source falls off with distance, so the excited states and the vacuum share the same asymptotically flat structure and hence the same asymptotic covariantly constant spinors; the supercharges defined at infinity act on all sectors and force mass degeneracy whenever the vacuum is supersymmetric. Several attempts to break this structural coincidence have been made. Nishino constructed three- and four-dimensional supergravity models with conical singularities aimed at supersymmetry breaking while maintaining zero cosmological constant. Dvali exhibited four-dimensional theories with unbroken vacuum supersymmetry and R-symmetry in which global monopoles — topological defects of a broken global symmetry, with order-parameter fields winding a sphere at spatial infinity — produce a solid angle deficit that eliminates asymptotically covariantly constant spinors, so that no conserved supercharges exist in that sector; he also sketched infinite-volume extra-dimensional scenarios with a monopole-like brane, while stating explicitly that it is not clear how such scenarios relate to the observable universe. Kaloper later showed that global monopoles in quantum gravity carry negative mass and can catalyze catastrophic instabilities, undermining this building block. Mansouri suggested a 'multilayered universe' in which exact supersymmetry coexists with absent superpartners hidden in other layers, exhibiting a concrete model only in $(2+1)$ dimensions. Most recently, Heckman, Joyce, Sakstein and Trodden explored supersymmetry algebras in Kleinian $(2+2)$ signature — two spatial and two temporal directions — as a speculative route toward the cosmological constant problem. None of these constitutes a physically realistic, ultraviolet-complete (valid to arbitrarily high energies, e.g. as a string theory) $(3+1)$-dimensional realization, and no theorem rules one out; the question Witten left open in his note is still open.

Problem Statement

Witten's note 'Is Supersymmetry Really Broken?' demonstrates that in $(2+1)$-dimensional supergravity supersymmetry can guarantee a vanishing cosmological constant without Bose-Fermi mass degeneracy — because unbroken supercharges correspond to spinors covariantly constant at infinity, while any nonzero-energy state produces an asymptotically conical geometry admitting no such spinors — and leaves open whether an analogous phenomenon can occur in $(3+1)$ dimensions. The problem is to determine whether there exists a physically consistent and realistic $(3+1)$-dimensional framework — obtained by modifying four-dimensional gravity itself, the boundary or asymptotic conditions defining isolated systems, or otherwise — in which:

(1) the vacuum preserves supersymmetry, together with whatever additional structure (such as an R-symmetry or an underlying string-theoretic structure) is needed so that the cosmological constant vanishes naturally, without fine-tuning;

(2) the generic finite-energy excited states belong to asymptotic classes that do not admit asymptotically covariantly constant spinors, so that no conserved supercharge acts on them and their bosonic and fermionic masses need not be degenerate; and

(3) the framework yields well-defined observable physics (no ghosts, instabilities, or pathologies of the kind afflicting known defect-based constructions), is compatible with observed gravitational, particle-physics, and cosmological phenomenology, and can be embedded in an ultraviolet-complete theory of quantum gravity without reintroducing vacuum-energy tuning.

The question is resolved either by an explicit construction establishing all three properties, or by a controlled impossibility result: a proof, under precisely stated and physically motivated assumptions, that no framework of the kind described above can exist, where those assumptions must cover frameworks obtained by modifying four-dimensional gravity or the boundary or asymptotic conditions defining isolated systems, so that a result valid only in standard, unmodified four-dimensional gravity with asymptotically flat boundary conditions does not resolve the question.

References

1. Edward Witten, 'Is Supersymmetry Really Broken?', International Journal of Modern Physics A 10 (1995) 1247-1248, DOI: 10.1142/S0217751X95000590, arXiv:hep-th/9409111, https://arxiv.org/abs/hep-th/9409111
2. 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
3. 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
4. Hitoshi Nishino, 'Conical singularities in three or four dimensions and supersymmetry breaking', Physics Letters B 370 (1996) 65-75, DOI: 10.1016/0370-2693(95)01549-3, arXiv:hep-th/9510153, https://arxiv.org/abs/hep-th/9510153
5. Gia Dvali, 'Cosmological constant and Fermi-Bose degeneracy' (2000), arXiv:hep-th/0004057, https://arxiv.org/abs/hep-th/0004057
6. Freydoon Mansouri, 'Exact local supersymmetry, absence of superpartners, and noncommutative geometries' (1997), arXiv:hep-th/9704187, https://arxiv.org/abs/hep-th/9704187
7. Nemanja Kaloper, 'Troubles with global monopoles in quantum gravity', International Journal of Modern Physics D 31 (2022) 2250035, DOI: 10.1142/S0218271822500353, arXiv:2012.09192, https://arxiv.org/abs/2012.09192
8. 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
9. Jonathan J. Heckman, Austin Joyce, Jeremy Sakstein, Mark Trodden, 'Exploring 2+2 answers to 3+1 questions', International Journal of Modern Physics A 37 (2022) 2250201, DOI: 10.1142/S0217751X22502013, arXiv:2208.02267, https://arxiv.org/abs/2208.02267
10. Stefan Nobbenhuis, 'Categorizing Different Approaches to the Cosmological Constant Problem', Foundations of Physics 36 (2006) 613-680, DOI: 10.1007/s10701-005-9042-8, arXiv:gr-qc/0411093, https://arxiv.org/abs/gr-qc/0411093