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This paper establishes that the symmetry superalgebra generated by Killing vectors, Killing spinors, and infinitesimal R-symmetry gauge transformations on a spin-R manifold is always a filtered subdeformation of the R-symmetry-extended Poin

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2026-08-03 23:04 UTC pith:OPOBL4S6

load-bearing objection Real content for spin-R Killing superalgebras, but geometric realisability is hand-imposed and the reconstruction is only a partial converse. the 3 major comments →

arxiv 2511.07246 v3 pith:OPOBL4S6 submitted 2025-11-10 math.DG hep-th

Killing (super)algebras for generalised spin manifolds

Andrew D.K. Beckett This is my paper
classification math.DG hep-th MSC 17B7017B8153C2753C3053C50
keywords Killing superalgebrasspin-R structuresgeneralised spin structuresR-symmetryfiltered subdeformationsSpencer cohomologyKilling spinorshomogeneous supersymmetric backgrounds
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a Killing (super)algebra for a connection on a spinor bundle over a spin-R manifold, where the usual spin structure is twisted by an internal R-symmetry group so that fermions can exist on spaces that are not spin. To make the algebra close, the even part must include not only Killing vectors but also infinitesimal gauge transformations of the R-bundle, and the spinor-spinor bracket needs a map pairing spinors into such gauge transformations alongside the Dirac current. The central structural claim is that every such Killing superalgebra is a filtered subdeformation of the r-extended flat model superalgebra, meaning it is obtained from the R-symmetry-extended Poincaré superalgebra by controlled bracket deformations. Using Spencer cohomology, the paper identifies, in Lorentzian signature with more than half the spinor module parallel, which deformations are geometrically realisable, and proves a converse reconstruction theorem from such deformations to homogeneous spin-R backgrounds. If correct, this yields a precise algebraic characterisation of symmetry superalgebras for supersymmetric backgrounds with gauged R-symmetry.

Core claim

The core claim is Theorem 20: for any admissible pair (D, ρ) on a spin-R manifold, the Killing (super)algebra bK(D,ρ) = bV(D,ρ) ⊕ R(D,ρ) ⊕ bS_D is a filtered subdeformation of the r-extended flat model superalgebra associated to (R^{s,t}, S, κ). Theorem 17 gives the exact pointwise conditions under which this object is a genuine Lie superalgebra, and Theorem 50 shows that a geometrically realisable deformation reconstructs a homogeneous spin-R background whose Killing superalgebra contains the deformation. The net geometric-algebraic correspondence is that the symmetry superalgebras of spin-R backgrounds are exactly the realisable filtered subdeformations of the r-extended Poincaré superalge

What carries the argument

The central object is the Killing (super)algebra of an admissible pair (D, ρ), which pairs a connection D on the twisted spinor bundle with a map ρ that sends pairs of spinors into infinitesimal R-symmetry gauge transformations. Its flat template is the r-extended flat model superalgebra bs = V ⊕ S ⊕ (so(V)⊕r), the Poincaré superalgebra extended by the R-symmetry algebra. The proof machinery is Spencer (2,2)-cohomology of the graded subalgebra a = V′ ⊕ S′ ⊕ (h⊕r′): a₀-invariant cohomology classes control the infinitesimal deformations, and the Homogeneity Theorem makes κ surjective onto V in the highly supersymmetric Lorentzian case, reducing homological conditions to simple algebraic ones.

Load-bearing premise

The load-bearing premise is geometric realisability (Definition 44): the R-symmetry gauge-sector deformation maps λ₂ and θ̃₂ are set to zero by hand, so the correspondence excludes the very gauged backgrounds (F ≠ 0) that motivate the paper, and the classification also depends on the Lorentzian Homogeneity Theorem with more than half the spinor module Killing.

What would settle it

A single concrete highly supersymmetric Lorentzian spin-R background with non-zero R-symmetry field strength F (for example, a gauged supergravity solution with charged gravitini) whose Killing superalgebra has localised deformation data with θ̃₂ ≠ 0 would falsify the claim that realisable deformations (λ₂ = θ̃₂ = 0) exhaust the symmetry superalgebras of spin-R backgrounds; equivalently, an admissible integrable Spencer cocycle with λ₂ ≠ 0 that integrates to a filtered deformation satisfying all Jacobi identities, yet is not geometrically realisable, would disprove completeness of the correspo

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Any Killing superalgebra on a spin-R manifold is always a filtered subdeformation of the r-extended Poincaré superalgebra, so structural questions about such symmetry algebras reduce to Spencer cohomology of the flat model.
  • Theorem 17 gives a concrete pointwise criterion for when a connection and a spinor-pairing map define a genuine Lie superalgebra, making admissibility checkable in examples.
  • In highly supersymmetric Lorentzian signature, only admissible, integrable Spencer cohomology classes can arise as Killing superalgebras, yielding a classification scheme for such symmetries.
  • Every geometrically realisable deformation is realised as a subalgebra of the Killing superalgebra of a homogeneous spin-R background, so the algebraic classification produces actual geometries.
  • The reconstruction applies only when the R-symmetry field strength F vanishes on contraction with the restricted Killing vectors; genuinely gauged backgrounds with F ≠ 0 lie outside the current framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The paper's own realisability definition (λ₂ = θ̃₂ = 0) is acknowledged to be unnatural, and since θ̃₂ is naturally interpreted as the R-symmetry field strength, I infer that a future relaxation of the admissibility conditions would let the correspondence extend to the gauged F ≠ 0 case that motivates the whole construction.
  • Because the Homogeneity Theorem is the keystone, I infer that the classification is genuinely tied to Lorentzian signature, causal Dirac currents, and the > ½ dim S threshold; other signatures or smaller Killing-spinor spaces may admit Killing superalgebras not captured by the Spencer-based list.
  • The reconstruction theorem produces homogeneous backgrounds, so I infer that a local version obtained by patching homogeneous models could characterise all highly supersymmetric spin-R backgrounds, not only those with a transitive isometry group.
  • Since the δ-cocycle structure is complicated by the r-extension and the normalised cochains are not a₀-invariant without a section of the Dirac current, I infer that a practical classification scheme will require additional data beyond the Lie-pair data used in the unextended case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper defines Killing (super)algebras associated to connections on spinor bundles over pseudo-Riemannian manifolds equipped with a spin-R structure, i.e. a generalised spin structure twisted by an R-symmetry group. The even part consists of restricted Killing vector fields together with infinitesimal R-gauge transformations, and the odd part consists of parallel spinors for an admissible pair (D,rho). The main structural result, Theorem 20, shows that such a Killing (super)algebra is a filtered subdeformation of the r-extended flat model superalgebra. The algebraic part of the paper uses Spencer cohomology to study filtered subdeformations of this extended flat model, obtaining an integration theorem (Theorem 42) for admissible, integrable cocycles, and introduces the class of geometrically realisable deformations (Definition 44). The final reconstruction theorem (Theorem 50) asserts that, with some caveats, a geometrically realisable highly supersymmetric deformation arises from a homogeneous spin-R background and embeds into its Killing superalgebra.

Significance. If the main theorems are correct, the paper provides a substantial generalisation of earlier work by de Medeiros–Figueroa-O’Farrill–Santi and by the author: it places R-twisted Killing superalgebras in a global geometric setting, introduces a covariant Cartan calculus for spin-R structures, and connects the geometry to Spencer-cohomology classification of filtered subdeformations. The paper is also commendably honest: it repeatedly flags its own caveats, notably in Remarks 19 and 43, and it does not claim more than the proofs support in the final theorem. The derivation is parameter-free and largely self-contained, and the writing is careful. The main caveat is that the core geometric-algebraic correspondence is ultimately conditional: the class of geometrically realisable deformations is defined by hand-imposed vanishing conditions, and the reconstruction theorem leaves admissibility of the constructed pair open. These issues are acknowledged by the author, but they currently prevent the advertised 'correspondence' from being a fully established theorem in the gauged R-symmetry case.

major comments (3)
  1. [Definition 44, Remarks 19 and 43, Theorem 50] Geometric realisability is imposed by hand: a representative cocycle is required to satisfy λ2 = 0 and θ̃2 = 0. Remark 43 identifies θ̃2 with the R-symmetry field strength F, and Remark 19 notes that local transitivity forces F = 0. Since Theorem 50 reconstructs backgrounds only from geometrically realisable deformations, and its proof uses θ̃2 = 0 to conclude F = 0, the advertised correspondence excludes exactly the gauged F ≠ 0 sector that motivates the paper. The author explicitly says Definition 44 is 'not entirely satisfactory', but as it stands this is a load-bearing restriction rather than a derived condition.
  2. [Theorem 50, Definition 18, equations (96)–(99)] The proof of Theorem 50 constructs a connection D and a bundle map ρ, but it does not verify the four admissibility conditions of Definition 18 for the pair (D,ρ). The closing sentence says the embedding holds 'if (D,ρ) is an admissible pair, in particular if Ψ(g1) = bS_D', but neither hypothesis is established. Thus the unconditional part of the theorem is only the existence of the homogeneous structure and a linear map Ψ; the stronger statement that the deformation embeds into the Killing superalgebra is conditional. If Ψ(g1) is strictly smaller than bS_D, admissibility of the full pair is not automatic, and the reconstruction claim in the abstract is therefore overstated.
  3. [§4.3.1, Theorem 27] The algebraic classification and reconstruction are confined to Lorentzian signature, causal Dirac currents, and dim S′ > ½ dim S. This restriction is quoted from [11,48] and is clearly stated, so it is not internally inconsistent. However, it means the central correspondence is not a general statement for all generalised spin manifolds, despite the title's generality. The paper's own abstract and introduction are careful about this, but the reader should be aware that the 'highly supersymmetric' analysis is a special case rather than the full setting.
minor comments (3)
  1. [Theorem 42 statement] In the hypothesis of Theorem 42, the expression 'i∗(α+β+γ+δ)' should read 'i∗(α+β+γ+ρ)'; there is no δ in the Spencer cocycle at this point.
  2. [Theorem 42, bracket formulas] In the bracket list (225), the term '[a, v] = [a, λ2(v)]' is labelled with underbrace 'h', but the preceding discussion and equation (204) show it should take values in r′, not h. This is presumably a typo, but it is in a central theorem.
  3. [Throughout] Several minor typographical issues: Definition 2 contains 'let Let κ'; Definition 18 has 'the the Killing spinor equation'; and in the discussion after equation (24) the notation R is sometimes used where R = R/Z₂ is meant. These do not affect the mathematics.

Circularity Check

1 steps flagged

Realisability (Def. 44) is defined by setting λ2=0 and θ̃2=0, so the reconstruction direction of Thm. 50 is partly secured by definition; the forward structure theorems are independent.

specific steps
  1. self definitional [Definition 44; Remark 43; proof of Theorem 50 (Eq. (248))]
    "A highly supersymmetric filtered subdeformation of bs is geometrically realisable if ... a representative cocycle can be chosen such that λ2 = 0 and θ̃2 = 0. This definition is not entirely satisfactory since it is rather unnatural to set some of the deformation maps to zero “by hand” rather than via some homological condition; we choose this definition now for compatibility with Definition 18 but leave open the possibility for both to be modified in future work."

    Definition 44 selects the realisable subclass by killing exactly the two deformation maps λ2 and θ̃2 that Remark 43 interprets as the R-symmetry gauge-field strength and as the [a,v] bracket. Theorem 50 then reconstructs a background from a geometrically realisable deformation; its proof uses the definition to conclude θ̃2=0, whence F=0 by Eq. (248). This reproduces the flat-R / no-gauging structure already built into Definition 18 through ı_X F=0 and cL_X a=0. Thus the ‘vice versa’ direction is enforced by the definition of realisability, not derived from a homological criterion. The paper is transparent about this, but it is a reduction by construction.

full rationale

The forward mathematical chain is genuine and parameter-free: Theorem 17 derives necessary-and-sufficient Jacobi conditions for bK(D,ρ), Theorem 20 proves via Killing transport that bK(D,ρ) is a filtered subdeformation of the r-extended flat model, and Theorem 42 integrates admissible Spencer cocycles into filtered deformations. None of these steps reduces to its input. Self-citations, including the author’s [10] and the Homogeneity Theorem [11,48], are load-bearing but are external, stated with fixed assumptions and not assuming the present conclusion, so they are not circular in the sense of the rubric. The one by-construction element is the reverse/reconstruction direction: geometric realisability is defined by imposing λ2=0 and θ̃2=0 explicitly ‘for compatibility with Definition 18’, and Theorem 50 uses θ̃2=0 to set F=0. Moreover, Theorem 50's embedding into bK(D,ρ) is conditional on admissibility (‘if (D,ρ) is an admissible pair, in particular if Ψ(g1)=bSD’); the proof checks the bracket relations but does not verify conditions (96)–(99). These are acknowledged caveats rather than disguised fits, so the overall circularity is partial and limited to the realisability subclass.

Axiom & Free-Parameter Ledger

1 free parameters · 6 axioms · 1 invented entities

No fitted numerical parameters; the paper is a pure derivation. Its dependence is structural: (i) the Homogeneity Theorem (quoted from the author's research lineage) restricts the classification to Lorentzian signature, causal Dirac currents and dim S′ > ½ dim S; (ii) geometric realisability (λ₂=0, θ̃₂=0) is imposed by hand so the algebraic class matches the geometric definition; (iii) time-orientability is required for a global Dirac current; (iv) compact R-symmetry is used to obtain a faithful action. The ρ map is a new structural ingredient with no independent evidence.

free parameters (1)
  • Geometric realisability constraints (λ₂ = 0, θ̃₂ = 0) = 0 (by hand)
    Definition 44 imposes λ₂=0 and θ̃₂=0 on the deformation data so that the classified deformations coincide with the geometric Definition 18. The author flags it as 'rather unnatural'; it is a hand-chosen constraint, not derived.
axioms (6)
  • domain assumption Homogeneity Theorem: for Lorentzian V with dim>2 and causal symmetric Dirac current κ, any S′⊆S with dim S′ > ½ dim S has κ|J²S′ surjective onto V
    Quoted from [11,48]; it is the engine behind the 'highly supersymmetric' classification (§4.3.1) and underlies Lemma 31, Lemma 33 and Theorem 42. The paper restricts to causal currents and Lorentzian signature to invoke it.
  • domain assumption Time-orientability (in indefinite signature) so a global bundle Dirac current exists
    Lemmas 7-8: without a holonomy-invariant reduction of the structure group, the bundle Dirac current κ does not exist globally, and the odd-odd bracket of the Killing superalgebra cannot be defined.
  • standard math Cheng–Kac Spencer-cohomology framework: filtered deformations of Z-graded Lie superalgebras are governed by degree-2 Spencer cohomology with defining sequences (µ, θ, 0, …)
    External machinery [47], recapitulated in [10]; used throughout §4 to phrase cocycle conditions, integrability and the classification of deformations.
  • domain assumption Compact R-symmetry group in the Lorentzian case, so the kernel of the action on S′ splits off
    Lemmas 29-30: needed for faithfulness/transitivity so the homological machinery (Lemma 31, Lemma 25) applies. The paper notes a compact R-symmetry is always available for a suitable causal Dirac current.
  • ad hoc to paper Geometric realisability (Def. 44): λ₂=0, θ̃₂=0
    Imposed by hand 'for compatibility with Definition 18'; the paper itself calls it 'not entirely satisfactory'. Necessary for Theorem 50 and it excludes F≠0 deformations (see Remark 43, where θ̃₂ is interpreted as the would-be field strength).
  • domain assumption Closedness of the subgroups K ⊆ G₀ and R′ ⊆ R in Lemma 48 / Theorem 50
    Needed to lift the isotropy representation to Spin_R(V) and produce a homogeneous spin-R structure. The paper notes the classical spin-structure analogue already fails without such hypotheses.
invented entities (1)
  • ρ map (spinor-pairing into infinitesimal R-gauge transformations) no independent evidence
    purpose: Serves as the odd-odd bracket component ρ(ϵ,ζ) ∈ Γ(ad Q) so the Killing superalgebra can have a non-trivial gauge sector; without it the construction collapses to the untwisted case of [10].
    Introduced in Theorem 17 / Definition 18 as an undetermined bundle section whose existence is constrained by the Jacobi identities (eqs. 85-99). Purely mathematical; no falsifiable handle outside the framework. The paper argues it makes the algebra 'much richer' but gives no worked example.

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read the original abstract

We define the notion of a Killing (super)algebra for a connection on a spinor bundle associated to a generalised spin structure on a pseudo-Riemannian manifold of any signature. We are led naturally to include in the even subspace not only Killing vectors but also certain infinitesimal gauge transformations, and we show that the definition of the (super)algebra requires, in addition to the spinor connection and a Dirac current, a map to pair spinor fields into infinitesimal gauge transformations. We show that these (super)algebras are filtered subdeformations of (an analogue of) the Poincar\'e superalgebra extended by the R-symmetry algebra. By employing Spencer cohomology, we study such deformations from a purely algebraic point of view and, at least in the case of Lorentzian signature and high supersymmetry, identify the subclass of deformations to which the Killing superalgebras belong. Finally, we show that, with some caveats, one can reconstruct a supersymmetric background geometry from such a deformation as a homogeneous space on which the deformation is realised as a subalgebra of the Killing superalgebra.

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Reference graph

Works this paper leans on

53 extracted references · 14 canonical work pages

  1. [1]

    C. S. Shahbazi.Differential spinors and Kundt three-manifolds with skew-torsion. 2024. arXiv: 2405.03756 [math.DG](cit. on p. 2)

    Pith/arXiv arXiv 2024

  2. [2]

    C. S. Shahbazi and A. Gil-Garc ´ ıa.Differential spinors forG ∗ 2 and isotropic structures. 2024. arXiv: 2409.08553 [math.DG](cit. on p. 2)

    Pith/arXiv arXiv 2024

  3. [3]

    C. I. Lazaroiu, E.-M. Babalic and I.-A. Coman. ‘Geometric algebra techniques in flux compactific- ations’. In:Adv. High Energy Phys.2016 (2016), p. 7292534.doi:10.1155/2016/7292534. arXiv: 1212.6766 [hep-th](cit. on p. 2)

    Pith/arXiv arXiv 2016

  4. [4]

    Cort´ es, C

    V. Cort´ es, C. Lazaroiu and C. S. Shahbazi. ‘Spinors of real type as polyforms and the generalized Killing equation’. In:Math. Z.299.3-4 (2021), pp. 1351–1419.doi:10.1007/s00209-021-02726-6. arXiv:1911.08658 [math.DG](cit. on p. 2)

    Pith/arXiv arXiv 2021

  5. [5]

    Figueroa-O’Farrill and A

    J. Figueroa-O’Farrill and A. Santi. ‘Spencer cohomology and 11-dimensional supergravity’. In: Commun. Math. Phys.349 (2017), pp. 627–660.doi:10 . 1007 / s00220 - 016 - 2700 - 1. arXiv: 1511.08737 [hep-th](cit. on pp. 2, 4, 6). 53

    Pith/arXiv arXiv 2017

  6. [6]

    Figueroa-O’Farrill and A

    J. Figueroa-O’Farrill and A. Santi. ‘On the algebraic structure of Killing superalgebras’. In:Adv. Theor. Math. Phys.21.5 (2017), pp. 1115–1160.doi:10.4310/ATMP.2017.v21.n5.a1. arXiv: 1608.05915 [hep-th](cit. on pp. 2–4, 6, 23, 24, 27, 45)

    Pith/arXiv arXiv 2017

  7. [7]

    Beckett and J

    A. Beckett and J. Figueroa-O’Farrill. ‘Killing superalgebras for lorentzian five-manifolds’. In:JHEP 07 (2021), p. 209.doi:10.1007/JHEP07(2021)209. arXiv:2105.05775 [hep-th](cit. on p. 2)

    Pith/arXiv arXiv 2021

  8. [8]

    de Medeiros, J

    P. de Medeiros, J. Figueroa-O’Farrill and A. Santi. ‘Killing superalgebras for lorentzian six-manifolds’. In:J. Geom. Phys.132 (2018), pp. 13–44.doi:10.1016/j.geomphys.2018.05.019. arXiv:1804 .00319 [hep-th](cit. on pp. 2–4, 12, 23)

    doi:10.1016/j.geomphys.2018.05.019 2018

  9. [9]

    Figueroa-O’Farrill

    J. Figueroa-O’Farrill. ‘A Geometric Construction of the Exceptional Lie AlgebrasF 4 andE 8’. In:Commun. Math. Phys.283 (2008), pp. 663–674.doi:10.1007/s00220- 008- 0581- 7. arXiv: 0706.2829 [math.DG](cit. on p. 2)

    Pith/arXiv arXiv 2008

  10. [10]

    A. D. K. Beckett. ‘Killing (super)algebras associated to connections on spinors’. In:SIGMA Sym- metry Integrability Geom. Methods Appl.21 (2025), Paper No. 081.doi:10.3842/SIGMA.2025.081. arXiv:2409.11306 [math.DG](cit. on pp. 2–5, 13–15, 21–25, 27–33, 36, 38–40, 42–49, 52)

    arXiv 2025

  11. [11]

    Hustler.Homogeneity in Supergravity

    N. Hustler.Homogeneity in Supergravity. PhD Thesis, U. of Edinburgh. 2016.url:http://hdl.h andle.net/1842/19576(visited on 22/04/2024) (cit. on pp. 3, 36, 49)

    2016

  12. [12]

    S. W. Hawking and C. N. Pope. ‘Generalized Spin Structures in Quantum Gravity’. In:Phys. Lett. B73 (1978), pp. 42–44.doi:10.1016/0370-2693(78)90167-3(cit. on p. 3)

    work page doi:10.1016/0370-2693(78)90167-3(cit 1978

  13. [13]

    A. Back, P. G. O. Freund and M. Forger. ‘New Gravitational Instantons and Universal Spin Struc- tures’. In:Phys. Lett. B77 (1978), pp. 181–184.doi:10.1016/0370-2693(78)90616-0(cit. on p. 3)

    work page doi:10.1016/0370-2693(78)90616-0(cit 1978

  14. [14]

    Forger and H

    M. Forger and H. Hess. ‘Universal Metaplectic Structures and Geometric Quantization’. In:Com- mun. Math. Phys.64 (1979), pp. 269–278.doi:10.1007/BF01221734(cit. on p. 3)

    work page doi:10.1007/bf01221734(cit 1979

  15. [15]

    S. J. Avis and C. J. Isham. ‘Generalized Spin Structures on Four Dimensional Space-Times’. In: Commun. Math. Phys.72 (1980), p. 103.doi:10.1007/BF01197630(cit. on p. 3)

    work page doi:10.1007/bf01197630(cit 1980

  16. [16]

    M. F. Atiyah, R. Bott and A. Shapiro. ‘Clifford modules’. In:Topology3.suppl (1964), pp. 3–38. doi:10.1016/0040-9383(64)90003-5(cit. on p. 3)

    work page doi:10.1016/0040-9383(64)90003-5(cit 1964

  17. [17]

    G. S. Whiston. ‘Lorentzian characteristic classes’. In:Gen. Relativity Gravitation6.5 (1975), pp. 463–475.doi:10.1007/bf00762451(cit. on p. 3)

    work page doi:10.1007/bf00762451(cit 1975

  18. [18]

    Atiyah, R

    M. Atiyah, R. Bott and V. K. Patodi. ‘On the Heat equation and the index theorem’. In:Invent. Math.19 (1973), pp. 279–330.doi:10.1007/BF01425417(cit. on p. 3)

    work page doi:10.1007/bf01425417(cit 1973

  19. [19]

    H. B. Lawson Jr. and M.-L. Michelsohn.Spin geometry. Vol. 38. Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1989, pp. xii+427.isbn: 0-691-08542-0 (cit. on p. 3)

    1989

  20. [20]

    M. Nagase. ‘Spin q structures’. In:J. Math. Soc. Japan47.1 (1995), pp. 93–119.doi:10.2969/jm sj/04710093(cit. on pp. 3, 6)

    arXiv 1995

  21. [21]

    H. B. Lawson Jr. ‘Spin h Manifolds’. In:SIGMA19 (2023), p. 012.doi:10.3842/SIGMA.2023.012. arXiv:2301.09683 [math.DG](cit. on p. 3)

    Pith/arXiv arXiv 2023

  22. [22]

    Albanese and A

    M. Albanese and A. Milivojevi´ c. ‘Spin h and further generalisations of spin’. In:J. Geom. Phys. 164 (2021). [Corrigendum: J. Geom. Phys. 184 (2023), 104709], Paper No. 104174, 13.doi:10.10 16/j.geomphys.2021.104174. arXiv:2008.04934 [math.AT](cit. on pp. 3, 8)

    arXiv 2021

  23. [23]

    Espinosa and R

    M. Espinosa and R. Herrera. ‘Spinorially twisted spin structures, I: Curvature identities and eigen- value estimates’. In:Differ. Geom. Appl.46 (2016), pp. 79–107.doi:10.1016/j.difgeo.2016.0 1.008. arXiv:1409.6246 [math.DG](cit. on pp. 3, 8). 54

    Pith/arXiv arXiv 2016

  24. [24]

    Artacho and M.-A

    D. Artacho and M.-A. Lawn. ‘Generalised spin r structures on homogeneous spaces’. In:Differential Geom. Appl.101 (2025), Paper No. 102291.doi:10.1016/j.difgeo.2025.102291. arXiv:2303 .05433 [math.DG](cit. on pp. 3, 8, 46)

    arXiv 2025

  25. [25]

    C. I. Lazaroiu and C. Shahbazi. ‘Real pinor bundles and real Lipschitz structures’. In:Asian J. Math.23 (2019), pp. 749–836.doi:10.4310/AJM.2019.v23.n5.a3. arXiv:1606.07894 [math.DG] (cit. on pp. 3, 6)

    Pith/arXiv arXiv 2019

  26. [26]

    C. I. Lazaroiu and C. S. Shahbazi. ‘On the spin geometry of supergravity and string theory’. In: Geometric methods in physics XXXVI. Trends Math. Birkh¨ auser/Springer, Cham, 2019, pp. 229– 235.isbn: 978-3-030-01155-0.doi:10 . 1007 / 978 - 3 - 030 - 01156 - 7 _ 25. arXiv:1607 . 02103 [hep-th](cit. on p. 3)

    2019

  27. [27]

    Janssens

    B. Janssens. ‘Generalised Spin Structures in General Relativity’. In:Annales Henri Poincare19.5 (2018), pp. 1587–1610.doi:10.1007/s00023-018-0667-5(cit. on p. 3)

    work page doi:10.1007/s00023-018-0667-5(cit 2018

  28. [28]

    A. D. K. Beckett.Notes on generalised spin structures. 2025. arXiv:2511.03627 [math.DG](cit. on pp. 3, 4, 6, 9, 10, 12–16, 46, 47, 53)

    Pith/arXiv arXiv 2025

  29. [29]

    R. Haag, J. T. Lopusza´ nski and M. Sohnius. ‘All Possible Generators of Supersymmetries of theS Matrix’. In:Nucl. Phys. B88 (1975), p. 257.doi:10.1016/0550-3213(75)90279-5(cit. on p. 3)

    work page doi:10.1016/0550-3213(75)90279-5(cit 1975

  30. [30]

    J. A. Strathdee. ‘Extended Poincar´ e Supersymmetry’. In:Int. J. Mod. Phys. A2 (1987). Ed. by A. Salam and E. Sezgin, p. 273.doi:10.1142/S0217751X87000120(cit. on pp. 3, 6)

    work page doi:10.1142/s0217751x87000120(cit 1987

  31. [31]

    Van Proeyen

    A. Van Proeyen. ‘Tools for supersymmetry’. In:Ann. U. Craiova Phys.9.I (1999), pp. 1–48. arXiv: hep-th/9910030(cit. on pp. 3, 6, 37)

    Pith/arXiv arXiv 1999

  32. [32]

    Gall and T

    L. Gall and T. Mohaupt. ‘Supersymmetry algebras in arbitrary signature and theirR-symmetry groups’. In:JHEP10 (2021), p. 203.doi:10 . 1007 / JHEP10(2021 ) 203. arXiv:2108 . 05109 [hep-th](cit. on pp. 3, 5, 6, 37)

    2021

  33. [33]

    D’Auria, S

    R. D’Auria, S. Ferrara and M. A. Lled´ o. ‘On the embedding of space-time symmetries into simple superalgebras’. In:Lett. Math. Phys.57 (2001), pp. 123–133.doi:10.1023/A:1017950711091. arXiv:hep-th/0102060(cit. on p. 3)

    Pith/arXiv arXiv 2001

  34. [34]

    Ceresole and G

    A. Ceresole and G. Dall’Agata. ‘General matter coupledN= 2,D= 5 gauged supergravity’. In:Nucl. Phys. B585 (2000), pp. 143–170.doi:10 . 1016 / S0550 - 3213(00 ) 00339 - 4. arXiv: hep-th/0004111(cit. on p. 4)

    Pith/arXiv arXiv 2000

  35. [35]

    G¨ unaydin and M

    M. G¨ unaydin and M. Zagermann. ‘Gauging the fullR-symmetry group in five-dimensional,N= 2 Yang-Mills Einstein tensor supergravity’. In:Phys. Rev. D63 (2001), p. 064023.doi:10.1103/Ph ysRevD.63.064023. arXiv:hep-th/0004117(cit. on p. 4)

    Pith/arXiv arXiv 2001

  36. [36]

    Bellor ´ ın and T

    J. Bellor ´ ın and T. Ort ´ ın. ‘Characterization of all the supersymmetric solutions of gaugedN= 1, D= 5 supergravity’. In:JHEP08 (2007), p. 096.doi:10.1088/1126-6708/2007/08/096. arXiv: 0705.2567 [hep-th](cit. on p. 4)

    Pith/arXiv arXiv 2007

  37. [37]

    Bellor ´ ın

    J. Bellor ´ ın. ‘Supersymmetric solutions of gauged five-dimensional supergravity with general matter couplings’. In:Class. Quant. Grav.26 (2009), p. 195012.doi:10.1088/0264-9381/26/19/195012. arXiv:0810.0527 [hep-th](cit. on p. 4)

    Pith/arXiv arXiv 2009

  38. [38]

    Beckett.Spencer cohomology, supersymmetry and the structure of Killing superalgebras

    A. Beckett.Spencer cohomology, supersymmetry and the structure of Killing superalgebras. PhD Thesis, U. of Edinburgh. 2024.url:http : / / dx . doi . org / 10 . 7488 / era / 4477(visited on 06/08/2025) (cit. on pp. 4, 53)

    2024

  39. [39]

    D. V. Alekseevsky and V. Cort´ es. ‘Classification ofN-(super)-extended Poincar´ e algebras and bilinear invariants of the spinor representation of Spin(p, q)’. In:Commun. Math. Phys.183 (1997), pp. 477–510.doi:10.1007/s002200050039. arXiv:math/9511215 [math.RT](cit. on pp. 4, 5). 55

    Pith/arXiv arXiv 1997

  40. [40]

    ˇCap and K

    A. ˇCap and K. Neusser. ‘On automorphism groups of some types of generic distributions’. In: Differ. Geom. Appl.27.6 (2009), pp. 769–779.doi:10 . 1016 / j . difgeo . 2009 . 05 . 003. arXiv: 0807.0974 [math.DG](cit. on p. 4)

    Pith/arXiv arXiv 2009

  41. [41]

    Cort´ es, L

    V. Cort´ es, L. Gall and T. Mohaupt. ‘Four-dimensional vector multiplets in arbitrary signature (I)’. In:Int. J. Geom. Methods Mod. Phys.17.10 (2020), p. 2050150.doi:10.1142/S0219887820501509. arXiv:1907.12067 [hep-th](cit. on p. 5)

    Pith/arXiv arXiv 2020

  42. [42]

    D. W. Morris.Introduction to Arithmetic Groups. 2015. arXiv:math/0106063 [math.DG](cit. on p. 6)

    Pith/arXiv arXiv 2015

  43. [43]

    Figueroa-O’Farrill and A

    J. Figueroa-O’Farrill and A. Santi. ‘Eleven-dimensional supergravity from filtered subdeformations of the Poincar´ e superalgebra’. In:J. Phys. A49.29 (2016), p. 295204.doi:10.1088/1751-8113/4 9/29/295204. arXiv:1511.09264 [hep-th](cit. on p. 6)

    Pith/arXiv arXiv 2016

  44. [44]

    Y. Kosmann. ‘D´ eriv´ ees de Lie des spineurs’. In:Ann. Mat. Pura Appl. (4)91 (1972), pp. 317–395. doi:10.1007/BF02428822(cit. on p. 12)

    work page doi:10.1007/bf02428822(cit 1972

  45. [45]

    B. Kostant. ‘Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold’. In: Trans. Amer. Math. Soc.80 (1955), pp. 528–542.doi:10.2307/1993001(cit. on pp. 13, 24)

    work page doi:10.2307/1993001(cit 1955

  46. [46]

    R. Geroch. ‘Limits of spacetimes’. In:Comm. Math. Phys.13 (1969), pp. 180–193 (cit. on p. 24)

    1969

  47. [47]

    Cheng and V

    S.-J. Cheng and V. G. Kac. ‘Generalized Spencer cohomology and filtered deformations ofZ-graded Lie superalgebras’. In:Adv. Theor. Math. Phys.2.5 (1998). [Erratum: Adv.Theor.Math.Phys.8.5, 697–709 (2004)], pp. 1141–1182.doi:10 . 4310 / ATMP . 1998 . v2 . n5 . a7. arXiv:math / 9805039 [math.RT](cit. on pp. 27, 28, 30, 31, 38)

    1998

  48. [48]

    Figueroa-O’Farrill and N

    J. Figueroa-O’Farrill and N. Hustler. ‘The homogeneity theorem for supergravity backgrounds’. In: JHEP10 (2012), p. 014.doi:10.1007/JHEP10(2012)014. arXiv:1208.0553 [hep-th](cit. on pp. 36, 49)

    Pith/arXiv arXiv 2012

  49. [49]

    A. Santi. ‘Remarks on highly supersymmetric backgrounds of 11-dimensional supergravity’. In: Geometry, Lie theory and applications—the Abel Symposium 2019. Vol. 16. Abel Symp. Springer, Cham, 2022, pp. 253–277.isbn: 978-3-030-81295-9.doi:10.1007/978-3-030-81296-6_12. arXiv: 1912.10688 [hep-th](cit. on p. 45)

    Pith/arXiv arXiv 2019

  50. [50]

    C. B¨ ar. ‘The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces’. In:Arch. Math. (Basel)59.1 (1992), pp. 65–79.doi:10.1007/BF01199016(cit. on pp. 46, 47)

    work page doi:10.1007/bf01199016(cit 1992

  51. [51]

    Cahen, S

    M. Cahen, S. Gutt and A. Trautman. ‘Spin structures on real projective quadrics’. In:J. Geom. Phys.10.2 (1993), pp. 127–154.doi:10.1016/0393-0440(93)90025-A(cit. on pp. 46, 47)

    work page doi:10.1016/0393-0440(93)90025-a(cit 1993

  52. [52]

    H.-C. Wang. ‘On invariant connections over a principal fibre bundle’. In:Nagoya Math. J.13 (1958), pp. 1–19 (cit. on p. 48)

    1958

  53. [53]

    Kobayashi and K

    S. Kobayashi and K. Nomizu.Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15. Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969, pp. xv+470.isbn: 978-0-471-15732-8 (cit. on p. 48). 56

    1969