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The geometry of three-forms in six and seven dimensions

Nigel Hitchin
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Abstract

We study the special algebraic properties of alternating 3-forms in 6 and 7 dimensions and introduce a diffeomorphism-invariant functional on the space of differential 3-forms on a closed manifold M in these dimensions. Restricting the functional to closed forms in a fixed cohomology class, we find that a critical point which is generic in a suitable sense defines in the 6-dimensional case a complex threefold with trivial canonical bundle and in 7 dimensions a Riemannian manifold with holonomy G2. This approach gives a direct method of finding a local moduli space, with its special geometry, for these structures.

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Nigel Hitchin. 2000-10-05. The geometry of three-forms in six and seven dimensions. https://arxiv.org/abs/math/0010054

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