The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin.^[1]

As with Hitchin's introduction of generalized complex manifolds, this is an example of a mathematical tool found useful in theoretical physics.

Formal definition [link]

This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.

Let Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi(\Omega) = \int_M \Omega \wedge * \Omega,

where Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega

is a 3-form and * denotes the Hodge star operator.

Properties [link]

• The Hitchin functional is analogous 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 Failed to parse (Missing texvc executable; please see math/README to configure.): M

is a three-dimensional complex manifold and Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega
is the real part of a non-vanishing holomorphic 3-form, then Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega
is a critical point of the functional Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi
restricted to the cohomology class Failed to parse (Missing texvc executable; please see math/README to configure.): [\Omega] \in H^3(M,R)

. Conversely, if Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega

is a critical point of the functional Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi
in a given comohology class and Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega \wedge * \Omega < 0

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega

defines the structure of a complex manifold, such that Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega
is the real part of a non-vanishing holomorphic 3-form on Failed to parse (Missing texvc executable; please see math/README to configure.): M

.

The proof of the theorem in Hitchin's article ^[1] is relatively straightforward. The power of this concept is in the converse statement: if the exact form Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi(\Omega)

is known, we only have to look at its critical points to find the possible complex structures.

Use in string theory [link]

Hitchin functionals arise in many areas of string theory. An example is the compactifications of the 10-dimensional string with a subsequent orientifold projection Failed to parse (Missing texvc executable; please see math/README to configure.): \kappa

using an involution Failed to parse (Missing texvc executable; please see math/README to configure.): \nu

. In this case, Failed to parse (Missing texvc executable; please see math/README to configure.): M

is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): \tau
is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): g_{ij} = \tau \text{im} \int \tau i^*(\nu \cdot \kappa \tau).

The potential function is the functional Failed to parse (Missing texvc executable; please see math/README to configure.): V[J] = \int J \wedge J \wedge J , where J is the almost complex structure. Both are Hitchin functionals.^[2]

Notes [link]

1. ^ ^{a b} The original article by Hitchin https://arxiv.org/abs/math/0010054
2. ^ Hitchin functional in orientifold projections https://arxiv.org/abs/hep-th/0412277