Abstract
Special p-forms are forms which have components \({\varphi_{\mu_1\dots\mu_p}}\) equal to +1, −1 or 0 in some orthonormal basis. A p-form \({\varphi\in \Lambda^p\mathbb{R}^d}\) is called democratic if the set of nonzero components \({\{\varphi_{\mu_1\dots\mu_p}\}}\) is symmetric under the transitive action of a subgroup of \({{\rm O}(d,\mathbb{Z})}\) on the indices {1, . . . , d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P ≥ p and D ≥ d. In particular, we display a remarkable nested structure of special forms including a U(3)-invariant 2-form in six dimensions, a G2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form Ω in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form.
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References
Corrigan E., Devchand C., Fairlie D.B., Nuyts J.: First order equations for gauge fields in spaces of dimension greater than four. Nucl. Phys. B214, 452–464 (1983)
Fairlie D.B., Nuyts J.: Spherically symmetric solutions of gauge theories in eight-dimensions. J. Phys. A17, 2867–2872 (1984)
Brihaye Y., Devchand C., Nuyts J.: Selfduality for eight-dimensional gauge theories. Phys. Rev. D32, 990–994 (1985)
Devchand C., Nuyts J.: Super self-duality for Yang-Mills fields in dimensions greater than four. JHEP 12, 020 (2001)
Ward R.S.: Completely solvable gauge field equations in dimension greater than four. Nucl. Phys. B236, 381–396 (1984)
Fubini S., Nicolai H.: The octonionic instanton. Phys. Lett. B155, 369–372 (1985)
Capria M.M., Salamon S.M.: Yang-Mills fields on quaternionic spaces. Nonlinearity 1, 517–530 (1988)
Nitta T.: Vector bundles over quaternionic Kähler manifolds. Tohoku Math. J. 40, 425–440 (1988)
Tian G.: Gauge theory and calibrated geometry. I. Ann. Math. 151(2), 193–268 (2000)
Alekseevsky D.V., Cortes V., Devchand C.: Yang-Mills connections over manifolds with Grassmann structure. J. Math. Phys. 44, 6047–6076 (2003)
Devchand C., Nuyts J., Weingart G.: Special graphs. Int. J. Geom. Meth. Mod. Phys. 3, 1011–1018 (2006)
Acknowledgements
One of us (J.N.) thanks the Belgian Fonds National de la Recherche Scientifique for travel support, the Max-Planck-Institut für Mathematik in Bonn as well as Prof. Hermann Nicolai and the Max-Planck-Institut für Gravitationsphysik in Potsdam for hospitality.
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Communicated by A. Connes
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Devchand, C., Nuyts, J. & Weingart, G. Matryoshka of Special Democratic Forms. Commun. Math. Phys. 293, 545–562 (2010). https://doi.org/10.1007/s00220-009-0939-5
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DOI: https://doi.org/10.1007/s00220-009-0939-5