nLab Hitchin functional

Idea

What are now called Hitchin functionals are functions/functionals on spaces of differential n-forms on manifolds of dimension 6 or 7, for $n = 3$ and/or $n =4$. They are such that their critical points characterize holomorphic 3-forms of complex manifolds in dimension 6, as associative 3-forms of G₂-manifolds in dimension 7.

Definition

In 6 dimensions

Let $X$ be a closed oriented smooth manifold of dimension 6.

For $\Omega \in \wedge^3 \Gamma(T^* X)$ a differential 3-form on $X$, write

• ${\vert \lambda \Omega \vert} \in (\wedge^6 \Gamma(T^* X))^{\otimes 2}$

• $\sqrt{\vert \lambda \Omega \vert} \in \wedge^6 \Gamma(T^* X)$.

The Hitchin function on 3-forms is the function

$$\wedge^3 \Gamma(T^* X) \to \mathbb{R}$$

which sends 3-form to the integration of differential forms of this 6-form over $X$

$$\Omega \mapsto \int_X \sqrt{\vert \lambda (\Omega) \vert} \,.$$

In 7 dimensions

Let $X$ be 7-dimensional.

Let $\Omega \in \wedge^3 \Gamma(T^* X)$ be a stable differential form?. This determines a Riemannian metric $\g_\Omega$ on $X$. Write $\star_\Omega$ for the corresponding Hodge star operator. The Hitchin functional now is the function that takes stable forms to

$$\int_X \Omega \wedge \star_\Omega \Omega \,.$$

Properties

In 6 dimensions

A differential 3-form $\Omega$ such that $\lambda(\Omega) \lt 0$ is a critical point of the above functional precisely if there is the structure of a complex manifold on $X$ such that $\Omega$ is the real part of a non-vanishing holomorphic 3-form.

This is (Hitchin, theorem 13).

In 7 dimensions

(…)

Related concepts

• Calabi-Yau manifold, G₂-manifold

• topological M-theory

References

The original articles are

• Nigel Hitchin, The geometry of three-forms in six and seven dimensions, (arXiv:math/0010054)

• Nigel Hitchin, Stable forms and special metrics (arXiv:math/0107101)

Reviews with a perspective on the role of the functionals in physics/gravity/string theory/topological M-theory include

• Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke, Cumrun Vafa, Topological M-theory as Unification of Form Theories of Gravity (arXiv:hep-th/0411073v2)