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The Hitchin functional is a
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concept with applications in string theory that was introduced by the British
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Nigel Hitchin Nigel James Hitchin FRS (born 2 August 1946) is a British mathematician working in the fields of differential geometry, gauge theory, algebraic geometry, and mathematical physics. He is a Professor Emeritus of Mathematics at the University of ...
. and are the original articles of the Hitchin functional. As with Hitchin's introduction of
generalized complex manifold In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures we ...
s, this is an example of a mathematical tool found useful in
mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
.


Formal definition

This is the definition for 6-manifolds. The definition in Hitchin's article is more general, but more abstract. Let M be a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
6-
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
with trivial
canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...
. Then the Hitchin functional is a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...
on 3-forms defined by the formula: : \Phi(\Omega) = \int_M \Omega \wedge * \Omega, where \Omega is a 3-form and * denotes the
Hodge star In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
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 Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of orientation-preserving
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
s. * Theorem. Suppose that M is a three-dimensional
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
and \Omega is the real part of a non-vanishing
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
3-form, then \Omega is a critical point of the functional \Phi restricted to the
cohomology class In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
Omega Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. Th ...
\in H^3(M,R). Conversely, if \Omega is a critical point of the functional \Phi in a given comohology class and \Omega \wedge * \Omega < 0, then \Omega defines the structure of a complex manifold, such that \Omega is the real part of a non-vanishing holomorphic 3-form on M. :The proof of the theorem in Hitchin's articles and is relatively straightforward. The power of this concept is in the converse statement: if the exact form \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 structureFor example, complex structure, symplectic structure, G_2 holonomy and Spin(7) holonomy etc. on M and geometric structure are often characterized by the existence of particular differential forms on M that obey some integrable conditions. If an ''2''-form \omega can be written with local coordinates :\omega=dp_1\wedge dq_1+\cdots+dp_m\wedge dq_m and :d\omega=0, then \omega defines ''
symplectic structure Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...
''. A ''p''-form \omega\in\Omega^p(M,\mathbb) is ''stable'' if it lies in an open orbit of the local GL(n,\mathbb) action where n=dim(M), namely if any small perturbation \omega\mapsto\omega+\delta\omega can be undone by a local GL(n,\mathbb) 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 \wedge^3(\mathbb^n),is of the order of n^3, grows more fastly than the dimension of GL(n,\mathbb) which is n^2. But there are some very lucky exceptional case, namely, n=6, when dim \wedge^3(\mathbb^6)=20, dim GL(6,\mathbb)=36. Let \rho be a stable real ''3''-form in dimension ''6''. Then the stabilizer of \rho under GL(6,\mathbb) has real dimension ''36-20=16'', in fact either SL(3,\mathbb)\times SL(3,\mathbb) or SL(3,\mathbb). Focus on the case of SL(3,\mathbb) and if \rho has a stabilizer in SL(3,\mathbb) then it can be written with local coordinates as follows: :\rho=\frac(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar\wedge\bar\wedge\bar) where \zeta_1=e_1+ie_2,\zeta_2=e_3+ie_4,\zeta_3=e_5+ie_6 and e_i are bases of T^*M. Then \zeta_i determines an
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
on M. Moreover, if there exist local coordinate (z_1,z_2,z_3) such that \zeta_i=dz_i then it determines fortunately a complex structure on M. Given the stable \rho\in\Omega^3(M,\mathbb): :\rho=\frac(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar\wedge\bar\wedge\bar). We can define another real ''3''-from :\tilde(\rho)=\frac(\zeta_1\wedge\zeta_2\wedge\zeta_3-\bar\wedge\bar\wedge\bar). And then \Omega=\rho+i\tilde(\rho) is a holomorphic ''3''-form in the almost complex structure determined by \rho. Furthermore, it becomes to be the complex structure just if d\Omega=0 i.e. d\rho=0 and d\tilde(\rho)=0. This \Omega is just the ''3''-form \Omega in formal definition of ''Hitchin functional''. These idea induces the
generalized complex structure In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic structure. Generalized complex structures we ...
.


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 In theoretical physics orientifold is a generalization of the notion of orbifold, proposed by Augusto Sagnotti in 1987. The novelty is that in the case of string theory the non-trivial element(s) of the orbifold group includes the reversal of th ...
projection \kappa using an
involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...
\nu. In this case, M is the internal 6 (real) dimensional Calabi-Yau space. The couplings to the complexified Kähler coordinates \tau is given by : g_ = \tau \text \int \tau i^*(\nu \cdot \kappa \tau). The potential function is the functional V = \int J \wedge J \wedge J, where J is the
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not comple ...
. Both are Hitchin functionals. As application to string theory, the famous OSV conjecture used ''Hitchin functional'' in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the G_2 holonomy argued about topological M-theory and in the 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 . Hitchin functional gives one of the bases of it.


Notes


References

* * * * * * {{DEFAULTSORT:Hitchin Functional Complex manifolds String theory