The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. Hitchin (2000) and Hitchin (2001) 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

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

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

 

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

Properties

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  • 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   is a three-dimensional complex manifold and   is the real part of a non-vanishing holomorphic 3-form, then   is a critical point of the functional   restricted to the cohomology class  . Conversely, if   is a critical point of the functional   in a given comohology class and  , then   defines the structure of a complex manifold, such that   is the real part of a non-vanishing holomorphic 3-form on  .
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   is known, we only have to look at its critical points to find the possible complex structures.

Stable forms

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Action functionals often determine geometric structure[2] on   and geometric structure are often characterized by the existence of particular differential forms on   that obey some integrable conditions.

If an 2-form   can be written with local coordinates

 

and

 ,

then   defines symplectic structure.

A p-form   is stable if it lies in an open orbit of the local   action where n=dim(M), namely if any small perturbation   can be undone by a local   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  , is of the order of  , grows more fastly than the dimension of   which is  . But there are some very lucky exceptional case, namely,  , when dim  , dim  . Let   be a stable real 3-form in dimension 6. Then the stabilizer of   under   has real dimension 36-20=16, in fact either   or  .

Focus on the case of   and if   has a stabilizer in   then it can be written with local coordinates as follows:

 

where   and   are bases of  . Then   determines an almost complex structure on  . Moreover, if there exist local coordinate   such that   then it determines fortunately a complex structure on  .

Given the stable  :

 .

We can define another real 3-from

 .

And then   is a holomorphic 3-form in the almost complex structure determined by  . Furthermore, it becomes to be the complex structure just if   i.e.   and  . This   is just the 3-form   in formal definition of Hitchin functional. These idea induces the generalized complex structure.

Use in string theory

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Hitchin functionals arise in many areas of string theory. An example is the compactifications of the 10-dimensional string with a subsequent orientifold projection   using an involution  . In this case,   is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates   is given by

 

The potential function is the functional  , where J is the almost complex structure. Both are Hitchin functionals.Grimm & Louis (2005)

As application to string theory, the famous OSV conjecture Ooguri, Strominger & Vafa (2004) used Hitchin functional in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the   holonomy Dijkgraaf et al. (2005) argued about topological M-theory and in the   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 Witten (2007). Hitchin functional gives one of the bases of it.

Notes

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  1. ^ For explicitness, the definition of Hitchin functional is written before some explanations.
  2. ^ For example, complex structure, symplectic structure,   holonomy and   holonomy etc.

References

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  • Hitchin, Nigel (2000). "The geometry of three-forms in six and seven dimensions". arXiv:math/0010054.
  • Hitchin, Nigel (2001). "Stable forms and special metric". arXiv:math/0107101.
  • Grimm, Thomas; Louis, Jan (2005). "The effective action of Type IIA Calabi-Yau orientifolds". Nuclear Physics B. 718 (1–2): 153–202. arXiv:hep-th/0412277. Bibcode:2005NuPhB.718..153G. CiteSeerX 10.1.1.268.839. doi:10.1016/j.nuclphysb.2005.04.007. S2CID 119502508.
  • Dijkgraaf, Robbert; Gukov, Sergei; Neitzke, Andrew; Vafa, Cumrun (2005). "Topological M-theory as Unification of Form Theories of Gravity". Adv. Theor. Math. Phys. 9 (4): 603–665. arXiv:hep-th/0411073. Bibcode:2004hep.th...11073D. doi:10.4310/ATMP.2005.v9.n4.a5. S2CID 1204839.
  • Ooguri, Hiroshi; Strominger, Andrew; Vafa, Cumran (2004). "Black Hole Attractors and the Topological String". Physical Review D. 70 (10): 6007. arXiv:hep-th/0405146. Bibcode:2004PhRvD..70j6007O. doi:10.1103/PhysRevD.70.106007. S2CID 6289773.
  • Witten, Edward (2007). "Conformal Field Theory In Four And Six Dimensions". arXiv:0712.0157 [math.RT].