The geometry of three-forms in six and seven dimensions

By N. Hitchin. 2000

241 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ac889c3553f252f74414bd7d47950c2e0fbb36d3

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In Applied Mathematics

In Computer Graphics and Computer-Aided Design

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Citations in the corpus

Hitchin functionals in N = 2 supergravity (2005, background): “Following Hitchin [48, 51] we will show, first, that there is a natural special Kähler structure on the space of so called “stable” spinors χǫ, and, second, how this also gives a special Kähler structure on the space of stable forms χ.”
M theory and singularities of exceptional holonomy manifolds (2004, unknown): “Stable forms are defined as follows [30].”
The effective action of type II Calabi‐Yau orientifolds (2004, background): “This submanifold continues to be a product of two components M̃K×M̃Q.”
SU(3) × SU(3) compactification and mirror duals of magnetic fluxes (2006, background): “One can get a second real form by derivation of the Hitchin function: χ̂±(χ) := −∂H(χ±)/∂χ±.”
The baryonic branch of Klebanov-Strassler solution: a supersymmetric family of SU(3) structure backgrounds (2004, background): “3Using the fact that by scaling the fermion one can shift away W5 one could relate this to the Hitchin’s variational problem [21]- indeed what we have here is an example of a family of closed 3-forms and the CY metric should come out of the minimization of the volume functional.”
The gauge dual of Romans mass (2009, background): “This implies that each ReΩa should determine the whole of Ωa (see for example [34]); hence one gets Ω1 = Ω2.”
The effective theory of type IIA AdS4 compactifications on nilmanifolds and cosets (2008, background): “where H(Imt) is the Hitchin functional [ 65 , 69, 30].”
Generalized N=1 orientifold compactifications and the Hitchin functionals (2006, background): “A possible mechanism to generate a non-trivial potential for these fields is the inclusion of background fluxes arising as vacuum expectation values of field strengths in the supergravity theory [11].”
M-theory, exceptional generalised geometry and superpotentials (2008, methodology): “In the type II case we also find [11, 20] that e−K is proportional to the Hitchin functional [1, 63].”
Topological M-theory as unification of form theories of gravity (2004, methodology): “The notion of genericity we have in mind is known as stability, as described in [42] and reviewed below.”
Moduli stabilization from fluxes in a simple IIB orientifold (2002, background): “R Ramadas for bringing [31] to his attention and for related discussions.”
Short-time behaviour of a modified Laplacian coflow of G2-structures (2012, background): “As shown by Hitchin [16], for a closed G2-structure ϕ, V attains a critical point within the fixed cohomology class of ϕ whenever d ∗ ϕ = 0, that is, if and only if ϕ defines a torsion-free G2-structure.”
Almost parallel structures (2001, background): “The stabilizer of both φ and ψ is the same subgroup SL(3,C) of GL(6,R); such structures have been investigated in (Hitchin, 2001).”