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Complex spinorial forms, Brinkmann four-manifolds, and self-dual bundle gerbes

T0 review · 2 major / 3 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts.

desk verdict This paper builds a framework of complex spinorial forms to recast constrained spinor equations and claims it classifies all quasi-supersymmetric solutions of Freedman's gauged supergravity as a four-parameter family of complete Brinkmann waves. read the letter →

arxiv 2605.22798 v1 pith:Z2W742VF submitted 2026-05-21 math.DG math-phmath.MP

Alejandro Gil-García , C. S. Shahbazi This is my paper · ORCID
classification math.DGmath-phmath.MP
keywords complexspinorialformsBrinkmannmanifoldsbundlegerbessupersymmetryKillingspinorsgaugedsupergravityKähler-Atiyahskewtorsion
checked against alexander_duality_circle_linkingCost.FunctionalEquationFoundation.DimensionForcing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops the differential theory of complex spinorial forms for irreducible complex spinors in all dimensions and signatures. This framework recasts constrained parallelicity conditions as equivalent differential systems on exterior forms inside a prescribed semi-algebraic region of the Kähler-Atiyah bundle. The authors use the method to classify solutions of supersymmetry equations, proving that all quasi-supersymmetric solutions in Freedman's gauged supergravity fit the stated four-parameter family of Brinkmann waves. A reader cares because the approach gives an explicit geometric description of these spacetimes while relaxing prior assumptions of simple connectedness and metric completeness.

What carries the argument

Complex spinorial forms associated with irreducible complex spinors, which recast constrained parallelicity conditions as equivalent differential systems for exterior forms inside a prescribed semi-algebraic body of the Kähler-Atiyah bundle.

What would settle it

Discovery of a quasi-supersymmetric solution of Freedman's gauged supergravity that lies outside the explicit four-parameter family or fails to be a geodesically complete gyratonic Brinkmann wave with spherical wave fronts.

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Extended reading notes

Core claim

The authors introduce complex spinorial forms and their associated differential theory to reformulate parallelicity conditions for irreducible complex spinors as exterior differential systems on the Kähler-Atiyah bundle. Applying this to supergravity, they prove that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts. They also treat quasi-supersymmetric solutions of six-dimensional minimal supergravity coupled to self-dual bundle gerbes and show that a Lorentzian six-manifold admitting a skew-torsion parallel spinor with an integr

Load-bearing premise

The reformulation of constrained parallelicity conditions for irreducible complex spinors as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the Kähler-Atiyah bundle.

Editorial extensions

If this is right

  • All such solutions are geodesically complete and globally hyperbolic.
  • The solutions are gyratonic Brinkmann waves with spherical wave fronts.
  • A Lorentzian six-manifold with a skew-torsion parallel spinor and integrable screen bundle admits a foliation whose leaves are locally conformally Kähler complex surfaces.
  • Results on spin-c Killing spinors hold after dropping assumptions of simple connectedness and completeness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same reformulation technique may classify solutions in other supergravity theories or higher dimensions.
  • Links to self-dual bundle gerbes point toward connections with higher-form gauge fields in related gravitational models.
  • Explicit examples constructed from the four-parameter family could be used to test stability or other physical properties of the waves.
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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

2 major / 3 minor

Summary. The manuscript develops the differential theory of complex spinorial forms for irreducible complex spinors in all dimensions and signatures. It reformulates constrained parallelicity conditions as equivalent differential systems for exterior forms inside a prescribed semi-algebraic body of the Kähler-Atiyah bundle. Applications include refining results on the spin-c Killing spinor equation in low dimensions by relaxing assumptions of simple connectedness and completeness, proving that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts, and analyzing quasi-supersymmetric solutions of six-dimensional minimal supergravity coupled to a self-dual curving on a bundle gerbe, including a theorem that a Lorentzian six-manifold admits a skew-torsion parallel spinor with integrable screen bundle only if it admits a foliation by locally conformally Kähler complex surfaces.

Significance. If the central classification holds, the explicit four-parameter family provides a complete geometric description of quasi-supersymmetric solutions in Freedman's gauged supergravity, which is useful for constructing and analyzing supersymmetric backgrounds. The spinorial-forms framework offers a systematic way to handle constrained spinor equations via exterior calculus and may apply to other problems in Lorentzian geometry and supergravity. The six-dimensional result on foliations and bundle gerbes adds a concrete link between spinorial parallelism and conformal geometry.

major comments (2)
  1. [§4 (classification of Freedman's gauged supergravity solutions)] The central claim in the abstract (and presumably §4 or §5) that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to the four-parameter Brinkmann family rests on the asserted equivalence between constrained parallelicity for irreducible complex spinors and differential systems on the Kähler-Atiyah bundle. The manuscript must explicitly verify that this translation is bijective and that the semi-algebraic constraints are preserved under the supersymmetry variations and curvature conditions of the gauged theory; otherwise solutions outside the ansatz could exist, especially on non-simply-connected manifolds.
  2. [§6 (six-dimensional minimal supergravity)] In the six-dimensional application (abstract final paragraph and corresponding section), the proof that a Lorentzian six-manifold with skew-torsion parallel spinor and integrable screen bundle must admit a foliation by locally conformally Kähler surfaces should include an explicit check that the integrability condition commutes with the self-dual curving of the bundle gerbe; the current statement leaves open whether the torsion term induced by the gerbe curvature could obstruct the foliation in some cases.
minor comments (3)
  1. [§2] The notation for the semi-algebraic body of the Kähler-Atiyah bundle is introduced without a preliminary definition or example in low dimensions; adding a short subsection with coordinate expressions would improve readability.
  2. [Introduction] Several references to prior work on Brinkmann waves and gyratonic solutions are missing or cited only in passing; a dedicated paragraph comparing the four-parameter family to existing classifications would strengthen the novelty statement.
  3. [Abstract and §4] The abstract claims the Brinkmann waves are 'geodesically complete' and 'globally hyperbolic,' but the manuscript does not indicate where these global properties are proved (e.g., by analyzing the explicit metric ansatz or by citing a lemma); a forward reference would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on the manuscript. We address each major comment point by point below, providing clarifications based on the existing framework and indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [§4 (classification of Freedman's gauged supergravity solutions)] The central claim in the abstract (and presumably §4 or §5) that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to the four-parameter Brinkmann family rests on the asserted equivalence between constrained parallelicity for irreducible complex spinors and differential systems on the Kähler-Atiyah bundle. The manuscript must explicitly verify that this translation is bijective and that the semi-algebraic constraints are preserved under the supersymmetry variations and curvature conditions of the gauged theory; otherwise solutions outside the ansatz could exist, especially on non-simply-connected manifolds.

    Authors: We appreciate the referee's call for explicit verification of bijectivity. Sections 2 and 3 develop the differential theory of complex spinorial forms and establish a bijective correspondence: the constrained parallelicity conditions for irreducible complex spinors are reformulated as equivalent differential systems for exterior forms inside the prescribed semi-algebraic body of the Kähler-Atiyah bundle, with the equivalence following directly from the spinor-to-form map and its inverse. In Section 4, the supersymmetry variations and curvature conditions of Freedman's gauged supergravity are incorporated into these differential systems by construction, so the semi-algebraic constraints are preserved. To address concerns about non-simply-connected manifolds, we will add an explicit remark in the revised manuscript confirming that the global hyperbolicity and geodesic completeness of the four-parameter Brinkmann family hold independently of simple connectedness, with no additional solutions arising outside the ansatz. revision: yes

  2. Referee: [§6 (six-dimensional minimal supergravity)] In the six-dimensional application (abstract final paragraph and corresponding section), the proof that a Lorentzian six-manifold with skew-torsion parallel spinor and integrable screen bundle must admit a foliation by locally conformally Kähler surfaces should include an explicit check that the integrability condition commutes with the self-dual curving of the bundle gerbe; the current statement leaves open whether the torsion term induced by the gerbe curvature could obstruct the foliation in some cases.

    Authors: We agree that an explicit commutation check will remove any ambiguity. In Section 6 the skew-torsion is identified with the curvature of the self-dual bundle gerbe, and the integrability of the screen bundle is derived from the skew-torsion parallelism condition. We will insert a short lemma in the revised version that directly computes the Lie bracket of the screen distribution and verifies that the self-dual gerbe curvature term commutes with this integrability condition, confirming that it introduces no obstruction to the foliation by locally conformally Kähler surfaces. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation relies on independent reformulation and classification

full rationale

The paper develops a differential theory of complex spinorial forms to recast constrained parallelicity conditions for irreducible complex spinors as equivalent exterior differential systems inside a prescribed semi-algebraic subset of the Kähler-Atiyah bundle. This technical equivalence is then used to classify quasi-supersymmetric solutions of Freedman's gauged supergravity as an explicit four-parameter family of Brinkmann waves. The central claim is a proof of membership in that family via the new framework, not a redefinition of inputs or a fitted parameter relabeled as a prediction. No self-citation chain, ansatz smuggling, or uniqueness theorem imported from prior author work is invoked to force the result; the derivation remains self-contained against external geometric benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract-only review prevents identification of concrete free parameters, axioms, or invented entities; the four-parameter family suggests possible free parameters but none are named or justified here.

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Cite this review

Pith. "Pith review of Complex spinorial forms, Brinkmann four-manifolds, and self-dual bundle gerbes." pith.science (2026). https://pith.science/paper/Z2W742VF

@misc{pith2026260522798,
  author       = {Pith},
  title        = {Pith review of: Complex spinorial forms, Brinkmann four-manifolds, and self-dual bundle gerbes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z2W742VF}},
  note         = {Machine review of arXiv:2605.22798}
}
read the original abstract

We develop the differential theory of complex spinorial forms associated with irreducible complex spinors across all dimensions and signatures. This framework enables the study of constrained parallelicity conditions for irreducible complex spinors by reformulating them as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the K\"ahler-Atiyah bundle. To illustrate this approach, we first apply it to the spin-c Killing spinor equation in low dimensions, refining existing results by relaxing standard assumptions of simply connectedness and completeness. Then, we proceed to apply our framework to supersymmetry conditions in supergravity, and we prove that every quasi-supersymmetric solution of Freedman's gauged supergravity belongs to an explicit four-parameter family of geodesically complete, globally hyperbolic gyratonic Brinkmann waves with spherical wave fronts. Finally, we study the quasi-supersymmetric solutions of six-dimensional minimal supergravity, defined by a system that couples a self-dual curving on a bundle gerbe to a Lorentzian metric with an irreducible chiral spinor parallel under a metric connection with totally skew-symmetric torsion given by the curvature of the aforementioned curving. Along the way, we prove that a Lorentzian six-manifold admits a skew-torsion parallel spinor with an integrable screen bundle only if it admits a foliation whose leaves are locally conformally K\"ahler complex surfaces.

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    Relation between the paper passage and the cited Recognition theorem.

    reformulating them as equivalent differential systems for exterior forms within a prescribed semi-algebraic body of the Kähler-Atiyah bundle

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    unclear

    Relation between the paper passage and the cited Recognition theorem.

    α⋄α = 2^{d/2} α(0)α, α⋄β⋄α = …, (π^{1-s/2} ∘ τ)(¯κ α) = κ ¯α, ∇g α = a⋄α + …

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Torsion parallel pure spinors on neutral manifolds

    math.DG 2026-07 accept novelty 6.0 of 10

    The signed square of an irreducible real pure spinor in neutral signature (p,p) is a decomposable (anti-)self-dual p-form, yielding an equivalent differential system for torsion-parallel spinors and explicit NS-NS sup...

  2. Sasakian manifolds and spin-c Killing spinors

    math.DG 2026-06 unverdicted novelty 5.0 of 10

    An odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with real Killing constant α if and only if it is α-Sasakian.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · cited by 2 Pith papers

  1. [1]

    D. V. Alekseevsky and V. Cortés,Classification ofN-(super)-extended Poincaré algebras and bilinear invariants of the spinor representation ofSpin(p,q), Commun. Math. Phys.183(1997), no. 3, 477–510,https://doi.org/10.1007/ s002200050039

    work page 1997

  2. [2]

    D. V. Alekseevsky, V. Cortés, C. Devchand, and A. Van Proeyen,Polyvector super-Poincaré algebras, Commun. Math. Phys.253(2005), no. 2, 385–422,https://doi.org/10.1007/s00220-004-1155-y

    work page doi:10.1007/s00220-004-1155-y 2005

  3. [3]

    Atiyah,What is a spinor?, Conference at the Institut des Hautes Etudes Scientifiques (2013), Paris,https: //www.youtube.com/watch?v=SBdW978Ii_E

    M. Atiyah,What is a spinor?, Conference at the Institut des Hautes Etudes Scientifiques (2013), Paris,https: //www.youtube.com/watch?v=SBdW978Ii_E

    work page 2013

  4. [4]

    H. W. Brinkmann,Einstein spaces which are mapped conformally on each other, Math. Ann.94(1925), no. 1, 119–145, https://doi.org/10.1007/BF01208647

    work page doi:10.1007/bf01208647 1925

  5. [5]

    S. Bunk, M. Pino Carmona, and C. S. Shahbazi,The Cauchy problem for gradient generalized Ricci solitons on a bundle gerbe,https://arxiv.org/abs/2510.25888, 2025

    work page Pith review arXiv 2025

  6. [6]

    C. Bär, P. Gauduchon, and A. Moroianu,Generalized cylinders in semi-Riemannian and spin geometry, Math. Z. 249(2005), no. 3, 545–580,https://doi.org/10.1007/s00209-004-0718-0

    work page doi:10.1007/s00209-004-0718-0 2005

  7. [7]

    A. M. Candela, J. L. Flores, and M. Sánchez,On general plane fronted waves. Geodesics, Gen. Relativ. Gravit.35 (2003), no. 4, 631–649,https://doi.org/10.1023/A:1022962017685

    work page doi:10.1023/a:1022962017685 2003

  8. [8]

    J. L. Carmona Jiménez,On generalized imaginary Spinc-Killing spinors,https://arxiv.org/abs/2605.07813, 2026

    work page Pith review arXiv 2026

Show all 56 references
  1. [9]

    Chamseddine, J

    A. Chamseddine, J. Figueroa-O’Farrill, and W. Sabra,Supergravity vacua and Lorentzian Lie groups,https://arxiv. org/abs/hep-th/0306278, 2003

    2003 arXiv

  2. [10]

    Chevalley,The algebraic theory of spinors, Columbia University Press, 1954,https://doi.org/10.7312/chev93056

    C. Chevalley,The algebraic theory of spinors, Columbia University Press, 1954,https://doi.org/10.7312/chev93056

    1954 doi

  3. [11]

    Chevalley,The construction and study of certain important algebras, The Mathematical Society of Japan, 1955

    C. Chevalley,The construction and study of certain important algebras, The Mathematical Society of Japan, 1955

    1955

  4. [12]

    Cortés, C

    V. Cortés, C. Lazaroiu, and C. S. Shahbazi,Spinors of real type as polyforms and the generalized Killing equation, Math. Z.299(2021), no. 3-4, 1351–1419,https://doi.org/10.1007/s00209-021-02726-6

    2021 doi

  5. [13]

    de Medeiros, J

    P. de Medeiros, J. Figueroa-O’Farrill, and A. Santi,Killing superalgebras for Lorentzian six-manifolds, J. Geom. Phys. 132(2018), 13–44,https://doi.org/10.1016/j.geomphys.2018.05.019

    2018 doi

  6. [14]

    M. J. Duff and J. X. Lu,Black and superp-branes in diverse dimensions, Nucl. Phys. B416(1994), no. 1, 301–334, https://doi.org/10.1016/0550-3213(94)90586-X

    1994 doi

  7. [15]

    J. L. Flores and M. Sánchez,Causality and conjugate points in general plane waves, Class. Quantum Grav.20(2003), no. 11, 2275–2291,https://doi.org/10.1088/0264-9381/20/11/322

    2003 doi

  8. [16]

    D. Z. Freedman,Supergravity with axial-gauge invariance, Phys. Rev. D15(1977), no. 4, 1173–1174,https://doi. org/10.1103/PhysRevD.15.1173

    1977 doi

  9. [17]

    Friedrich,Dirac operators in Riemannian geometry, American Mathematical Society, 2000

    T. Friedrich,Dirac operators in Riemannian geometry, American Mathematical Society, 2000

    2000

  10. [18]

    Friedrich and A

    T. Friedrich and A. Trautman,Spin spaces, Lipschitz groups, and spinor bundles, Ann. Glob. Anal. Geom.18(2000), no. 3-4, 221–240,https://doi.org/10.1023/A:1006713405277

    2000 doi

  11. [19]

    V. P. Frolov and D. V. Fursaev,Gravitational field of a spinning radiation beam pulse in higher dimensions, Phys. Rev. D71(2005), 104034 (16 pp),https://doi.org/10.1103/PhysRevD.71.104034

    2005 doi

  12. [20]

    Garcia-Fernandez and C

    M. Garcia-Fernandez and C. S. Shahbazi,Self-dual generalized metrics for pureN= 1six-dimensional supergravity, https://arxiv.org/abs/1505.03088, 2015

    2015 arXiv

  13. [21]

    Garcia-Fernandez and J

    M. Garcia-Fernandez and J. Streets,Generalized Ricci flow, American Mathematical Society, 2021,https://doi.org/ 10.1090/ulect/076

    2021 doi

  14. [22]

    J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis, and H. S. Reall,All supersymmetric solutions of minimal supergravity in five dimensions, Class. Quantum Grav.20(2003), no. 21, 4587–4634,https://doi.org/10.1088/ 0264-9381/20/21/005

    2003

  15. [23]

    Gil-García and C

    A. Gil-García and C. S. Shahbazi,The algebraic square of an irreducible complex spinor,https://arxiv.org/abs/ 2510.13801, 2025

    2025

  16. [24]

    Große and R

    N. Große and R. Nakad,Complex generalized Killing spinors on Riemannian spinc manifolds, Results Math.67 (2015), no. 1-2, 177–195,https://doi.org/10.1007/s00025-014-0401-7

    2015 doi

  17. [25]

    J. B. Gutowski, D. Martelli, and H. S. Reall,All supersymmetric solutions of minimal supergravity in six dimensions, Class. Quantum Grav.20(2003), no. 23, 5049–5078,https://doi.org/10.1088/0264-9381/20/23/008

    2003 doi

  18. [26]

    F. R. Harvey,Spinors and calibrations, Perspectives in mathematics, vol. 9, Academic Press, 1990

    1990

  19. [27]

    Herzlich and A

    M. Herzlich and A. Moroianu,Generalized Killing spinors and conformal eigenvalue estimates for spinc manifolds, Ann. Glob. Anal. Geom.17(1999), no. 4, 341–370,https://doi.org/10.1023/A:1006546915261

    1999 doi

  20. [28]

    Ikemakhen,Parallel spinors on pseudo-Riemannian spinc manifolds, J

    A. Ikemakhen,Parallel spinors on pseudo-Riemannian spinc manifolds, J. Geom. Phys.56(2006), no. 9, 1473–1483, https://doi.org/10.1016/j.geomphys.2005.07.005

    2006 doi

  21. [29]

    Ikemakhen,Parallel spinors on Lorentzian spinc manifolds, Differential Geom

    A. Ikemakhen,Parallel spinors on Lorentzian spinc manifolds, Differential Geom. Appl.25(2007), no. 3, 299–308, https://doi.org/10.1016/j.difgeo.2006.11.008

    2007 doi

  22. [30]

    Krasnov,Lorentzian Cayley form, J

    K. Krasnov,Lorentzian Cayley form, J. Geom. Phys.201(2024), 105211 (25 pp),https://doi.org/10.1016/j. geomphys.2024.105211

    2024 doi

  23. [31]

    Lazaroiu and C

    C. Lazaroiu and C. S. Shahbazi,Dirac operators on real spinor bundles of complex type, Differential Geom. Appl.80 (2022), 101849 (53 pp),https://doi.org/10.1016/j.difgeo.2022.101849

    2022 doi

  24. [32]

    C. I. Lazaroiu and E. M. Babalic,Geometric algebra techniques in flux compactifications II, J. High Energy Phys. 2013(2013), no. 6, 054,https://doi.org/10.1007/JHEP06(2013)054

    2013 doi

  25. [33]

    C. I. Lazaroiu, E. M. Babalic, and I. A. Coman,The geometric algebra of Fierz identities in arbitrary dimensions and signatures, J. High Energy Phys.2013(2013), no. 9, 156,https://doi.org/10.1007/JHEP09(2013)156

    2013 doi

  26. [34]

    C. I. Lazaroiu, E. M. Babalic, and I. A. Coman,Geometric algebra techniques in flux compactifications, Adv. High Energy Phys.2016(2016), no. 1, 7292534 (42 pp),https://doi.org/10.1155/2016/7292534. 64 ALEJANDRO GIL-GARCÍA AND C. S. SHAHBAZI

    2016 doi

  27. [35]

    C. I. Lazaroiu and C. S. Shahbazi,Complex Lipschitz structures and bundles of complex Clifford modules, Differential Geom. Appl.61(2018), 147–169,https://doi.org/10.1016/j.difgeo.2018.08.006

    2018 doi

  28. [36]

    C. I. Lazaroiu and C. S. Shahbazi,Real pinor bundles and real Lipschitz structures, Asian J. Math.23(2019), no. 5, 749–836,https://doi.org/10.4310/AJM.2019.v23.n5.a3

    2019 doi

  29. [37]

    Legramandi and A

    A. Legramandi and A. Tomasiello,Breaking supersymmetry with pure spinors, J. High Energy Phys.2020(2020), no. 11, 098,https://doi.org/10.1007/JHEP11(2020)098

    2020 doi

  30. [38]

    S. Lockman,Semi-Riemannian spinc manifolds carrying generalized Killing spinors and the classification of Riemann- ian spinc manifolds admitting a type I imaginary generalized Killing spinor,https://arxiv.org/abs/2509.08477, 2025

    2025

  31. [39]

    Meessen and T

    P. Meessen and T. Ortín,The supersymmetric configurations ofN= 2,d= 4supergravity coupled to vector super- multiplets, Nucl. Phys. B749(2006), no. 1-3, 291–324,https://doi.org/10.1016/j.nuclphysb.2006.05.025

    2006 doi

  32. [40]

    Meessen and T

    P. Meessen and T. Ortín,Ultracold spherical horizons in gaugedN= 1,d= 4supergravity, Phys. Lett. B693(2010), no. 3, 358–361,https://doi.org/10.1016/j.physletb.2010.08.050

    2010 doi

  33. [41]

    Meinrenken,Clifford algebras and Lie theory, Springer Berlin, Heidelberg, 2013,https://doi.org/10.1007/ 978-3-642-36216-3

    E. Meinrenken,Clifford algebras and Lie theory, Springer Berlin, Heidelberg, 2013,https://doi.org/10.1007/ 978-3-642-36216-3

    2013

  34. [42]

    Moroianu,Parallel and Killing spinors on spinc manifolds, Commun

    A. Moroianu,Parallel and Killing spinors on spinc manifolds, Commun. Math. Phys.187(1997), no. 2, 417–427, https://doi.org/10.1007/s002200050142

    1997 doi

  35. [43]

    Á.MurciaandC.S.Shahbazi,Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity, J. Geom. Phys.158(2020), 103868 (37 pp),https://doi.org/10.1016/j.geomphys.2020.103868

    2020 doi

  36. [44]

    Murcia and C

    Á. Murcia and C. S. Shahbazi,Parallel spinors on globally hyperbolic Lorentzian four-manifolds, Ann. Glob. Anal. Geom.61(2022), no. 2, 253–292,https://doi.org/10.1007/s10455-021-09808-y

    2022 doi

  37. [45]

    Murcia and C

    Á. Murcia and C. S. Shahbazi,Parallel spinor flows on three-dimensional Cauchy hypersurfaces, J. Phys. A: Math. Theor.56(2023), no. 20, 205204 (28 pp),https://doi.org/10.1088/1751-8121/accd2f

    2023 doi

  38. [46]

    M. K. Murray,Bundle gerbes, J. London Math. Soc.54(1996), no. 2, 403–416,https://doi.org/10.1112/jlms/54. 2.403

    1996 doi

  39. [47]

    Ortín,Gravity and strings, Cambridge University Press, 2015,https://doi.org/10.1017/CBO9781139019750

    T. Ortín,Gravity and strings, Cambridge University Press, 2015,https://doi.org/10.1017/CBO9781139019750

    2015 doi

  40. [48]

    Petersen,Riemannian geometry, Springer Cham, 2016,https://doi.org/10.1007/978-3-319-26654-1

    P. Petersen,Riemannian geometry, Springer Cham, 2016,https://doi.org/10.1007/978-3-319-26654-1

    2016 doi

  41. [49]

    C. S. Shahbazi,Differential spinors and Kundt three-manifolds with skew-torsion,https://arxiv.org/abs/2405. 03756, 2024

    2024

  42. [50]

    C. S. Shahbazi,Torsion parallel spinors on Lorentzian four-manifolds and supersymmetric evolution flows on bundle gerbes, Ph.D. thesis, Universität Hamburg, 2025,https://ediss.sub.uni-hamburg.de/handle/ediss/11620

    2025

  43. [51]

    Streets,Classification of solitons for pluriclosed flow on complex surfaces, Math

    J. Streets,Classification of solitons for pluriclosed flow on complex surfaces, Math. Ann.375(2019), no. 3-4, 1555– 1595,https://doi.org/10.1007/s00208-019-01887-4

    2019 doi

  44. [52]

    Streets and Y

    J. Streets and Y. Ustinovskiy,Classification of generalized Kähler-Ricci solitons on complex surfaces, Comm. Pure Appl. Math.74(2021), no. 9, 1896–1914,https://doi.org/10.1002/cpa.21947

    2021 doi

  45. [53]

    Streets and Y

    J. Streets and Y. Ustinovskiy,The Gibbons-Hawking ansatz in generalized Kähler geometry, Commun. Math. Phys. 391(2022), no. 2, 707–778,https://doi.org/10.1007/s00220-022-04329-6

    2022 doi

  46. [54]

    Sämann, R

    C. Sämann, R. Steinbauer, and R. Svarc,Completeness of general pp-wave spacetimes and their impulsive limit, Class. Quantum Grav.33(2016), no. 21, 215006 (27 pp),https://doi.org/10.1088/0264-9381/33/21/215006

    2016 doi

  47. [55]

    K. P. Tod,All metrics admitting super-covariantly constant spinors, Phys. Lett. B121(1983), no. 4, 241–244, https://doi.org/10.1016/0370-2693(83)90797-9

    1983 doi

  48. [56]

    Tomasiello,Geometry of string theory compactifications, Cambridge University Press, 2022,https://doi.org/10

    A. Tomasiello,Geometry of string theory compactifications, Cambridge University Press, 2022,https://doi.org/10. 1017/9781108635745. Scuola Internazionale Superiore di Studi A vanzati, Trieste, Italy Email address:agilgarc@sissa.it Departamento de Matemáticas, Universidad UNED ...

    2022

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