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.

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: where \Omega is a 3-form and * denotes the Hodge star operator.

Properties

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 2-form \omega can be written with local coordinates and then \omega 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 non-degeneracy. What about p=3? For large n 3-form is difficult because the dimension of, is of the order of n^3, grows more fastly than the dimension of which is n^2. But there are some very lucky exceptional case, namely, n=6, when dim, dim. Let \rho be a stable real 3-form in dimension 6. Then the stabilizer of \rho under has real dimension 36-20=16, in fact either or. Focus on the case of and if \rho has a stabilizer in then it can be written with local coordinates as follows: where and e_i are bases of T^*M. Then \zeta_i determines an almost complex structure on M. Moreover, if there exist local coordinate such that then it determines fortunately a complex structure on M. Given the stable : We can define another real 3-from And then is a holomorphic 3-form in the almost complex structure determined by \rho. Furthermore, it becomes to be the complex structure just if d\Omega=0 i.e. d\rho=0 and. This \Omega is just the 3-form \Omega 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 \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 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 G_2 holonomy 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. Hitchin functional gives one of the bases of it.