Hitchin functional

This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract.^[1]

Let ${\displaystyle M}$ be a compact, oriented 6-manifold with trivial canonical bundle. Then the Hitchin functional is a functional on 3-forms defined by the formula:

${\displaystyle \Phi (\Omega )=\int _{M}\Omega \wedge *\Omega ,}$

where ${\displaystyle \Omega }$ is a 3-form and * denotes the Hodge star operator.

• 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 ${\displaystyle M}$ is a three-dimensional complex manifold and ${\displaystyle \Omega }$ is the real part of a non-vanishing holomorphic 3-form, then ${\displaystyle \Omega }$ is a critical point of the functional ${\displaystyle \Phi }$ restricted to the cohomology class ${\displaystyle [\Omega ]\in H^{3}(M,R)}$. Conversely, if ${\displaystyle \Omega }$ is a critical point of the functional ${\displaystyle \Phi }$ in a given comohology class and ${\displaystyle \Omega \wedge *\Omega <0}$, then ${\displaystyle \Omega }$ defines the structure of a complex manifold, such that ${\displaystyle \Omega }$ is the real part of a non-vanishing holomorphic 3-form on ${\displaystyle M}$.

The proof of the theorem in Hitchin's articles Hitchin (2000) and Hitchin (2001) is relatively straightforward. The power of this concept is in the converse statement: if the exact form ${\displaystyle \Phi (\Omega )}$ is known, we only have to look at its critical points to find the possible complex structures.

Action functionals often determine geometric structure^[2] on ${\displaystyle M}$ and geometric structure are often characterized by the existence of particular differential forms on ${\displaystyle M}$ that obey some integrable conditions.

If an 2-form ${\displaystyle \omega }$ can be written with local coordinates

${\displaystyle \omega =dp_{1}\wedge dq_{1}+\cdots +dp_{m}\wedge dq_{m}}$

and

${\displaystyle d\omega =0}$,

then ${\displaystyle \omega }$ defines symplectic structure.

A p-form ${\displaystyle \omega \in \Omega ^{p}(M,\mathbb {R} )}$ is stable if it lies in an open orbit of the local ${\displaystyle GL(n,\mathbb {R} )}$ action where n=dim(M), namely if any small perturbation ${\displaystyle \omega \mapsto \omega +\delta \omega }$ can be undone by a local ${\displaystyle GL(n,\mathbb {R} )}$ 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 ${\displaystyle \wedge ^{3}(\mathbb {R} ^{n})}$, is of the order of ${\displaystyle n^{3}}$, grows more fastly than the dimension of ${\displaystyle GL(n,\mathbb {R} )}$ which is ${\displaystyle n^{2}}$. But there are some very lucky exceptional case, namely, ${\displaystyle n=6}$, when dim ${\displaystyle \wedge ^{3}(\mathbb {R} ^{6})=20}$, dim ${\displaystyle GL(6,\mathbb {R} )=36}$. Let ${\displaystyle \rho }$ be a stable real 3-form in dimension 6. Then the stabilizer of ${\displaystyle \rho }$ under ${\displaystyle GL(6,\mathbb {R} )}$ has real dimension 36-20=16, in fact either ${\displaystyle SL(3,\mathbb {R} )\times SL(3,\mathbb {R} )}$ or ${\displaystyle SL(3,\mathbb {C} )}$.

Focus on the case of ${\displaystyle SL(3,\mathbb {C} )}$ and if ${\displaystyle \rho }$ has a stabilizer in ${\displaystyle SL(3,\mathbb {C} )}$ then it can be written with local coordinates as follows:

${\displaystyle \rho ={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}+{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})}$

where ${\displaystyle \zeta _{1}=e_{1}+ie_{2},\zeta _{2}=e_{3}+ie_{4},\zeta _{3}=e_{5}+ie_{6}}$ and ${\displaystyle e_{i}}$ are bases of ${\displaystyle T^{*}M}$. Then ${\displaystyle \zeta _{i}}$ determines an almost complex structure on ${\displaystyle M}$. Moreover, if there exist local coordinate ${\displaystyle (z_{1},z_{2},z_{3})}$ such that ${\displaystyle \zeta _{i}=dz_{i}}$ then it determines fortunately a complex structure on ${\displaystyle M}$.

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

${\displaystyle \rho ={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}+{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})}$.

We can define another real 3-from

${\displaystyle {\tilde {\rho }}(\rho )={\frac {1}{2}}(\zeta _{1}\wedge \zeta _{2}\wedge \zeta _{3}-{\bar {\zeta _{1}}}\wedge {\bar {\zeta _{2}}}\wedge {\bar {\zeta _{3}}})}$.

And then ${\displaystyle \Omega =\rho +i{\tilde {\rho }}(\rho )}$ is a holomorphic 3-form in the almost complex structure determined by ${\displaystyle \rho }$. Furthermore, it becomes to be the complex structure just if ${\displaystyle d\Omega =0}$ i.e. ${\displaystyle d\rho =0}$ and ${\displaystyle d{\tilde {\rho }}(\rho )=0}$. This ${\displaystyle \Omega }$ is just the 3-form ${\displaystyle \Omega }$ in formal definition of Hitchin functional. These idea induces the generalized complex structure.