# 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.

Let $M$ be a compact, oriented 6-manifold with trivial canonical bundle. Then the Hitchin functional is a functional on 3-forms defined by the formula:

$\Phi (\Omega )=\int _{M}\Omega \wedge *\Omega ,$ ## Properties

• The Hitchin functional is analogous for six-manifold to the Yang-Mills functional for the four-manifolds.
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 $\Phi (\Omega )$ 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 on $M$ and geometric structure are often characterized by the existence of particular differential forms on $M$ that obey some integrable conditions.

If an m-form $\omega$ can be written with local coordinates

$\omega =dp_{1}\wedge dq_{1}+\cdots +dp_{m}\wedge dq_{m}$ and

$d\omega =0$ ,

A p-form $\omega \in \Omega ^{p}(M,\mathbb {R} )$ is stable if it lies in an open orbit of the local $GL(n,\mathbb {R} )$ action where n=dim(M), namely if any small perturbation $\omega \mapsto \omega +\delta \omega$ can be undone by a local $GL(n,\mathbb {R} )$ 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.

$\rho ={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}+{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})$ $\rho ={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}+{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})$ .

We can define another real 3-from

${\tilde {\rho }}(\rho )={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}-{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})$ .

## 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 $\kappa$ using an involution $\nu$ . In this case, $M$ is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates $\tau$ is given by

$g_{ij}=\tau {\text{im}}\int \tau i^{*}(\nu \cdot \kappa \tau ).$ The potential function is the functional $V[J]=\int J\wedge J\wedge J$ , 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 $G_{2}$ holonomy Template:Harvtxt argued about topological M-theory and in the $Spin(7)$ 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.