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

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

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. is a three-dimensional complex manifold is the real part of a non-vanishing holomorphic is a critical point restricted to the cohomology class defines.

• The proof of the theorem in Hitchin's articles and 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 that obey some integrable conditions.

If an 2 can be written with local coordinates

and

1. ,

defines symplectic structure.

A p is stable 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 be a stable real 3-form in dimension 6 has real dimension 36-20=16.

then it can be written with local coordinates as follows:

determines an almost complex structure then it determines fortunately a complex structure.

:

1. .

We can define another real 3-from

1. .

is a holomorphic 3 is just the 3 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 using an involution is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates is given by

, where J is the almost complex structure. Both are Hitchin functionals.

As application to string theory, the famous OSV conjecture used Hitchin functional holonomy argued about topological M-theory 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