Hitchin functional

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,

where \Omega 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 M is a three-dimensional complex manifold and \Omega is the real part of a non-vanishing holomorphic 3-form, then \Omega is a critical point of the functional \Phi restricted to the cohomology class [\Omega] \in H^3(M,R). Conversely, if \Omega is a critical point of the functional \Phi in a given comohology class and \Omega \wedge * \Omega , then \Omega defines the structure of a complex manifold, such that \Omega is the real part of a non-vanishing holomorphic 3-form on M.

:The proof of the theorem in Hitchin's articles and 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 2-form \omega can be written with local coordinates :\omega=dp_1\wedge dq_1+\cdots+dp_m\wedge dq_m and :d\omega=0, then \omega defines symplectic structure.

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 non-degeneracy.

What about p=3? For large n 3-form is difficult because the dimension of \wedge^3(\mathbb{R}^n), is of the order of n^3, grows more fastly than the dimension of GL(n,\mathbb{R}) which is n^2. But there are some very lucky exceptional case, namely, n=6, when dim \wedge^3(\mathbb{R}^6)=20, dim GL(6,\mathbb{R})=36. Let \rho be a stable real 3-form in dimension 6. Then the stabilizer of \rho under GL(6,\mathbb{R}) has real dimension 36-20=16, in fact either SL(3,\mathbb{R})\times SL(3,\mathbb{R}) or SL(3,\mathbb{C}).

Focus on the case of SL(3,\mathbb{C}) and if \rho has a stabilizer in SL(3,\mathbb{C}) then it can be written with local coordinates as follows: :\rho=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3}) where \zeta_1=e_1+ie_2,\zeta_2=e_3+ie_4,\zeta_3=e_5+ie_6 and e_i are bases of T^*M. Then \zeta_i determines an almost complex structure on M. Moreover, if there exist local coordinate (z_1,z_2,z_3) such that \zeta_i=dz_i then it determines fortunately a complex structure on M.

Given the stable \rho\in\Omega^3(M,\mathbb{R}): :\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}).

And then \Omega=\rho+i\tilde{\rho}(\rho) 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 d\tilde{\rho}(\rho)=0. 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

: 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.

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.

Notes

References

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References

1. For explicitness, the definition of ''Hitchin functional'' is written before some explanations.
2. For example, complex structure, symplectic structure, $G\_2$ holonomy and $Spin(7)$ holonomy etc.