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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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