The ungauged maximal supergravity in seven spacetime dimensions was constructed in 1982 by Sezgin and Salam, who obtained a locally supersymmetric Lagrangian and transformation rules with global SL(5,R) symmetry, local USp(4) symmetry (USp(4) being the compact symplectic group whose Lie algebra is that of SO(5)), ten vector fields, five antisymmetric two-form potentials, and fourteen scalars parametrizing SL(5)/SO(5). The construction explicitly deferred several elements: the quartic fermion terms of the Lagrangian, the higher-order fermion contributions to the supersymmetry rules, the supercovariantization of the spin connection, field strengths and scalar currents, the Yang-Mills gauging, and the four-dimensional compactification patterns. Subsequent literature resolved the gauging question - the SO(5) gauging via dual three-form potentials subject to an odd-dimensional self-duality condition, noncompact gaugings, and the general classification by the embedding tensor (the object that specifies which subgroup of the global symmetry group is promoted to a local gauge symmetry) - and the compactification question was subsumed by later classifications. However, every existing construction gives the Lagrangian and supersymmetry rules only up to (that is, neglecting) higher-order fermion terms. The surviving open problem is the complete fermionic sector of maximal seven-dimensional supergravity, ungauged and gauged: all quartic fermion terms of the action, the supersymmetry transformation laws to all orders in fermions, the supercovariant curvatures, verification of supersymmetry invariance and closure of the algebra through quartic order, and the determination of whether the gauging modifies the higher-order fermion terms.

Background

Supergravity is the extension of general relativity to local supersymmetry, that is, to symmetry transformations whose anticommuting parameter is a spacetime-dependent spinor; 'maximal' supergravity carries 32 supercharges (independent supersymmetry generators), the largest number compatible with fields of spin at most two. In seven spacetime dimensions (D=7) the maximal theory contains 128 bosonic and 128 fermionic on-shell degrees of freedom (independent propagating field components): the metric, ten vector fields, five antisymmetric two-form tensor potentials (gauge two-forms, invariant under generalized gauge shifts of the potential), and fourteen real scalars parametrizing the symmetric space SL(5,R)/SO(5) as a nonlinear sigma model (the scalars serve as coordinates on this coset space), together with gravitini (the spin-3/2 superpartners of the graviton) and spin-1/2 fermions. The reduced theory exhibits a global SL(5,R) 'hidden' symmetry, not manifest before field dualizations (trading a form potential for its Hodge-dual potential), while the fermions carry indices of USp(4), the group of unitary symplectic 4x4 matrices, whose Lie algebra is that of SO(5) (in the standard physics shorthand, USp(4) ~ SO(5)); it acts as the R-symmetry (the automorphism group of the supersymmetry algebra acting on fermion indices). The theory can be obtained by torus reduction of eleven-dimensional supergravity - the unique theory with 32 supercharges above ten dimensions, whose Lagrangian, including all terms quartic in the gravitino (its only fermion), was constructed by Cremmer, Julia and Scherk in 1978 - followed by dualizations that make SL(5,R) manifest.

Sezgin and Salam constructed the ungauged maximal D=7 supergravity in 1982 by the Noether procedure, the iterative order-by-order construction of interactions for which the supersymmetry algebra closes (order-by-order cancellations governed by Fierz identities, the algebraic identities that reorganize products of spinor bilinears). Their paper identified the field content and coset structure above, established the generalized gauge invariance needed for the antisymmetric tensor fields, and conjectured possible Yang-Mills gaugings, including SO(5)xSO(5) structures in seven dimensions and SO(4)xSO(5)xSO(5) structures in four dimensions with the last SO(5) gauging composite fields. Several elements were explicitly deferred: the quartic fermion terms of the Lagrangian (supersymmetry requires specific four-fermion terms beyond the bilinear kinetic and mass terms), the corresponding higher-order fermion contributions to the supersymmetry transformation rules, the supercovariantization of the spin connection, field strengths and scalar currents (their completion by fermion bilinears, which makes the fermion field equations transform covariantly under local supersymmetry and is required for the algebra to close beyond leading order), the Yang-Mills gaugings, and the compactification patterns of the resulting four-dimensional theories.

The subsequent history resolved the gauging question. The 1982 paper itself noted that its ten vector fields cannot gauge the compact SO(5) subgroup of SL(5) because the two-form potentials transform nontrivially under SL(5). Pernici, Pilch and van Nieuwenhuizen (1984) achieved the SO(5) gauging - motivated by the compactification of eleven-dimensional supergravity on the four-sphere analyzed by Pilch, van Nieuwenhuizen and Townsend - by trading the two-forms for dual three-form potentials whose field strengths obey a first-order duality relation of the form dC proportional to g times the Hodge dual of C ('self-duality in odd dimensions'), and constructed the full nonlinear gauged action and supersymmetry rules with the fermion mass terms ('fermion shifts') induced by the gauging; the three- and four-fermion terms were still omitted. Noncompact gaugings followed (Pernici, Pilch, van Nieuwenhuizen and Warner, 1985), and the general gauging - with gauge groups CSO(p,q,r) with p+q+r=5, the family of groups acting on a five-dimensional real vector space of which SO(5) is the compact case - was classified using the embedding tensor, the object that encodes which subgroup of the global symmetry is promoted to a local Yang-Mills symmetry (Samtleben and Weidner, 2005; Weidner, 2007). The eleven-dimensional origin of the SO(5) theory was established by the nonlinear consistent truncation on AdS7 x S4 (anti-de Sitter space times the four-sphere, the maximally supersymmetric compactification of eleven-dimensional supergravity) in Nastase, Vaman and van Nieuwenhuizen (1999), and a bosonic exceptional field theory (a reformulation on an extended coordinate space that renders the hidden SL(5) symmetry manifest) exists (Musaev, 2016). The compactification question was subsumed by later work: the spontaneous compactifications of the seven-dimensional theories were analyzed by Pernici (1985), and all maximal four-dimensional gaugings were classified by de Wit, Samtleben and Trigiante (2007).

What has never been done in over forty years is the completion of the fermionic sector. Every existing construction - ungauged or gauged, component or embedding-tensor - gives the Lagrangian and supersymmetry rules only up to (that is, neglecting) higher-order fermion terms. Samtleben and Weidner (2005) state this explicitly and add the expectation that the omitted terms receive no corrections of order g in the gauge coupling from the gauging, an expectation they do not prove; Sezgin's 2023 survey of supergravities confirms that for this theory, as for most others, only the bosonic Lagrangian and the supertransformations up to leading terms in fermions are known. Since the eleven-dimensional quartic-fermion terms are known, the ungauged seven-dimensional ones are in principle fixed by torus reduction, but the required dualization into the SL(5)-covariant two-form/self-dual-three-form basis in which the gaugings are formulated has never been carried out in print, and for the gauged theory no derivation of these terms exists at all.

Problem Statement

Construct the complete fermionic sector of maximal supergravity in seven spacetime dimensions for the ungauged theory and for its gaugings, in particular for the compact SO(5) gauging and, in the general embedding-tensor (CSO(p,q,r), p+q+r=5) case, for the associated family of gaugings. Concretely, the required results are: (1) the full locally supersymmetric Lagrangian including all quartic (four-fermion) terms; (2) the supersymmetry transformation laws to all orders in fermions, including the three-fermion (cubic) corrections to the gravitino and spin-1/2 transformations; (3) the supercovariant versions of the spin connection, of the vector and (self-dual) tensor field strengths, and of the scalar coset currents, such that the fermionic field equations transform covariantly under local supersymmetry; (4) a verification that the action is invariant under the full transformation rules and that the supersymmetry algebra closes on all fields through quartic order in fermions (on the fermion field equations at higher orders, as is standard in supergravity); and (5) an explicit determination of whether and how the higher-order fermion terms are modified by the gauging, proving or refuting the recorded expectation that they receive no order-g corrections (g being the Yang-Mills gauge coupling) relative to the ungauged theory. Each of (1)-(5) is mandatory and carries independent standing; no proper subset of them resolves the task. The answer must reproduce, in the appropriate limits, the published bosonic Lagrangian, quadratic fermion terms, fermion shifts and lower-order supersymmetry rules of Pernici-Pilch-van Nieuwenhuizen (1984) and Samtleben-Weidner (2005).

References

  1. E. Sezgin and Abdus Salam, 'Maximal extended supergravity theory in seven dimensions', Physics Letters B 118 (1982) 359-364, DOI 10.1016/0370-2693(82)90204-0, https://doi.org/10.1016/0370-2693(82)90204-0
  2. E. Cremmer, B. Julia and J. Scherk, 'Supergravity in theory in 11 dimensions', Physics Letters B 76 (1978) 409-412, DOI 10.1016/0370-2693(78)90894-8, https://doi.org/10.1016/0370-2693(78)90894-8
  3. K. Pilch, P. van Nieuwenhuizen and P. K. Townsend, 'Compactification of d = 11 supergravity on S4 (or 11 = 7 + 4, too)', Nuclear Physics B 242 (1984) 377-392, DOI 10.1016/0550-3213(84)90400-0, https://doi.org/10.1016/0550-3213(84)90400-0
  4. M. Pernici, K. Pilch and P. van Nieuwenhuizen, 'Gauged maximally extended supergravity in seven dimensions', Physics Letters B 143 (1984) 103-107, DOI 10.1016/0370-2693(84)90813-x, https://doi.org/10.1016/0370-2693(84)90813-x
  5. M. Pernici, K. Pilch, P. van Nieuwenhuizen and N. P. Warner, 'Noncompact gaugings and critical points of maximal supergravity in seven dimensions', Nuclear Physics B 249 (1985) 381-395, DOI 10.1016/0550-3213(85)90046-x, https://doi.org/10.1016/0550-3213(85)90046-x
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  7. H. Nastase, D. Vaman and P. van Nieuwenhuizen, 'Consistent nonlinear KK reduction of 11d supergravity on AdS7 x S4 and self-duality in odd dimensions', Physics Letters B 469 (1999) 96-102, arXiv:hep-th/9905075, DOI 10.1016/S0370-2693(99)01266-6, https://arxiv.org/abs/hep-th/9905075
  8. H. Samtleben and M. Weidner, 'The maximal D = 7 supergravities', Nuclear Physics B 725 (2005) 383-419, arXiv:hep-th/0506237, DOI 10.1016/j.nuclphysb.2005.07.028, https://arxiv.org/abs/hep-th/0506237
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  10. B. de Wit, H. Samtleben and M. Trigiante, 'The maximal D = 4 supergravities', Journal of High Energy Physics 06 (2007) 049, arXiv:0705.2101, DOI 10.1088/1126-6708/2007/06/049, https://arxiv.org/abs/0705.2101
  11. E. T. Musaev, 'Exceptional field theory: SL(5)', Journal of High Energy Physics 02 (2016) 012, arXiv:1512.02163, DOI 10.1007/JHEP02(2016)012, https://arxiv.org/abs/1512.02163
  12. E. Sezgin, 'Survey of supergravities', arXiv:2312.06754 (2023), chapter for the Handbook of Quantum Gravity, https://arxiv.org/abs/2312.06754
  13. H. Samtleben, '11D Supergravity and Hidden Symmetries', arXiv:2303.12682 (2023), chapter for the Handbook of Quantum Gravity, https://arxiv.org/abs/2303.12682