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.
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.
Stable forms
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 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, 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 stability is equivalent to nondegeneratacy. What about p=3? For large n3-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 dimension36-20=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 a complex structure on. Given the stable : 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 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. As application to string theory, the famous OSV conjecture used Hitchin functional in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the holonomy 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 superconformalfield theory. Hitchin functional gives one of the bases of it.
