Hitchin functional

The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. Template:Harvtxt and Template:Harvtxt 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.[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.

Properties

• The Hitchin functional is analogous for six-manifold to the Yang-Mills functional for the four-manifolds.
The proof of the theorem in Hitchin's articles Template:Harvs and Template:Harvs 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.

Stable forms

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 m-form ${\displaystyle \omega }$ can be written with local coordinates

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

and

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

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 nondegeneratacy.

What about p=3? For large n 3-form is difficult because the dimension of ${\displaystyle \wedge ^{3}(\mathbb {R} ^{n})}$, ${\displaystyle n^{3}}$, grows more firstly than the dimension of ${\displaystyle GL(n,\mathbb {R} )}$, ${\displaystyle n^{2}}$. 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 ${\displaystyle SL(3,\mathbb {C} )\cap SL(3,\mathbb {C} )}$.

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

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

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.

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 ${\displaystyle \kappa }$ using an involution ${\displaystyle \nu }$. In this case, ${\displaystyle M}$ is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates ${\displaystyle \tau }$ is given by

${\displaystyle g_{ij}=\tau {\text{im}}\int \tau i^{*}(\nu \cdot \kappa \tau ).}$

The potential function is the functional ${\displaystyle V[J]=\int J\wedge J\wedge J}$, where J is the almost complex structure. Both are Hitchin functionals.Template:Harvtxt

As application to string theory, the famous OSV conjecture Template:Harvtxt used Hitchin functional in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the ${\displaystyle G_{2}}$ holonomy Template:Harvtxt argued about topological M-theory and in the ${\displaystyle Spin(7)}$ 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 Template:Harvtxt. Hitchin functional gives one of the bases of it.

Notes

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

References

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