The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin.
As with Hitchin's introduction of generalized complex manifolds, this is an example of a mathematical tool found useful in mathematical physics.
The definition in Hitchin's article is more general, but more abstract.
be a compact, oriented 6-manifold with trivial canonical bundle.
is a 3-form and * denotes the Hodge star operator.
Action functionals often determine geometric structure[2] on
and geometric structure are often characterized by the existence of particular differential forms on
can be written with local coordinates and then
is stable if it lies in an open orbit of the local
action where n=dim(M), namely if any small perturbation
So any 1-form that don't vanish everywhere is stable; 2-form (or p-form when p is even) stability is equivalent to non-degeneracy.
For large n 3-form is difficult because the dimension of
, grows more fastly than the dimension of
But there are some very lucky exceptional case, namely,
be a stable real 3-form in dimension 6.
has real dimension 36-20=16, in fact either
then it can be written with local coordinates as follows: where
Moreover, if there exist local coordinate
then it determines fortunately a complex structure on
is a holomorphic 3-form in the almost complex structure determined by
in formal definition of Hitchin functional.
These idea induces the generalized complex structure.
Hitchin functionals arise in many areas of string theory.
An example is the compactifications of the 10-dimensional string with a subsequent orientifold projection
is the internal 6 (real) dimensional Calabi-Yau space.
The couplings to the complexified Kähler coordinates
Both are Hitchin functionals.Grimm & Louis (2005) As application to string theory, the famous OSV conjecture Ooguri, Strominger & Vafa (2004) used Hitchin functional in order to relate topological string to 4-dimensional black hole entropy.
holonomy Dijkgraaf et al. (2005) 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 Witten (2007).