# Hitchin functional

The **Hitchin functional** is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. Template:Harvtxt and Template:Harvtxt 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.

## Formal definition

This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.^{[1]}

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.

## Properties

- 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 Template:Harvs and Template:Harvs 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.

## Stable forms

Action functionals often determine geometric structure^{[2]} on and geometric structure are often characterized by the existence of particular differential forms on that obey some integrable conditions.

If an *m*-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. So any *1*-form that don't vanish everywhere is stable; *2*-form (or *p*-form when *p* is even) stability is equivalent to nondegeneratacy.

What about *p*=3? For large *n* *3*-form is difficult because the dimension of , , grows more firstly than the dimension of , . But there are some very lucky exceptional case, namely, , when dim , dim . Let be a stable real *3*-form in dimension *6*. Then the stabilizer of under has real dimension *36-20=16*, in fact either or .

Focus on the case of and if has a stabilizer in then it can be written with local coordinates as follows:

where and are bases of . Then determines an almost complex structure on . Moreover, if there exist local coordinate such that then it determines fortunately an complex structure on .

We can define another real *3*-from

And then is a holomorphic *3*-form in the almost complex structure determined by . Furthermore, it becomes to be the complex structure just if i.e.
and . This is just the *3*-form in formal definition of *Hitchin functional*. These idea induces the generalized complex structure.

## Use in string theory

Hitchin functionals arise in many areas of string theory. An example is the compactifications of the 10-dimensional string with a subsequent orientifold projection using an involution . In this case, is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates is given by

The potential function is the functional , where J is the almost complex structure. Both are Hitchin functionals.Template:Harvtxt

As application to string theory, the famous OSV conjecture Template:Harvtxt used *Hitchin functional* in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the holonomy Template:Harvtxt argued about topological M-theory and in the holonomy topological F-theory might be argued.

More recently, E. Witten claimed the mysterious superconformal field theory in six dimensions, called 6D (2,0) superconformal field theory Template:Harvtxt. Hitchin functional gives one of the bases of it.

## Notes

## References

- Template:Cite arXiv
- Template:Cite arXiv
- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

- Template:Cite arXiv
- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}