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Concept
Hitchin functional
Résumé
The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. and are the original articles of the Hitchin functional.
As with Hitchin's introduction of generalized complex manifolds, this is an example of a mathematical tool found useful in mathematical physics.
This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.
Let be a compact, oriented 6-manifold with trivial canonical bundle. Then the Hitchin functional is a functional on 3-forms defined by the formula:
where is a 3-form and * denotes the Hodge star operator.
The Hitchin functional is analogous for six-manifold to the Yang-Mills functional for the four-manifolds.
The Hitchin functional is manifestly invariant under the action of the group of orientation-preserving diffeomorphisms.
Theorem. Suppose that is a three-dimensional complex manifold and is the real part of a non-vanishing holomorphic 3-form, then is a critical point of the functional restricted to the cohomology class . Conversely, if is a critical point of the functional in a given comohology class and , then defines the structure of a complex manifold, such that is the real part of a non-vanishing holomorphic 3-form on .
The proof of the theorem in Hitchin's articles and is relatively straightforward. The power of this concept is in the converse statement: if the exact form is known, we only have to look at its critical points to find the possible complex structures.
Action functionals often determine geometric structure on and geometric structure are often characterized by the existence of particular differential forms on that obey some integrable conditions.
If an 2-form can be written with local coordinates
and
then defines symplectic structure.
A p-form is stable if it lies in an open orbit of the local action where n=dim(M), namely if any small perturbation can be undone by a local action.
Source officielle
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