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High Energy Physics - Theory, hep-th,Mathematical Physics, math-ph,Mathematics, Mathematical Physics, math.MP,
Abstract:
Using (partial) curvature flows and the transitive action of subgroups of
O(d,Z) on the indices {1,...,d} of the components of the Yang-Mills curvature
in an orthonormal basis, we obtain a nested system of equations in successively
higher dimensions d, each implying the Yang-Mills equations on d-dimensional
Riemannian manifolds possessing special geometric structures. This `matryoshka'
of self-duality equations contains the familiar self-duality equations on
Riemannian 4-folds as well as their generalisations on complex K\"ahler 3-folds
and on 7- and 8-dimensional manifolds with G_2 and Spin(7) holonomy. The
matryoshka allows enlargement (`oxidation') to a remarkable system in 12
dimensions invariant under Sp(3). There are hints that the underlying geometry
is related to the sextonions, a six-dimensional algebra between the quaternions
and octonions.
