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| The '''Hitchin functional''' is a [[mathematics|mathematical]] concept with applications in [[string theory]] that was introduced by the British [[mathematician]] [[Nigel Hitchin]]. {{harvtxt|Hitchin|2000}} and {{harvtxt|Hitchin|2001}} are the original articles of the Hitchin functional. | | The name of the author is Garland. After becoming out of my job for years I became a production and distribution officer but I plan on altering it. The [http://Www.Dmv.org/buy-sell/auto-warranty/extended-warranty.php favorite pastime] extended car [http://Www.Dmv.org/buy-sell/auto-warranty/extended-warranty.php warranty] for my kids and me is taking part in crochet and now I'm attempting to earn money with auto warranty it. Kansas is exactly where her house is but she needs [http://www.duelingkitchens.com.au/portfolio/do-it-yourself-auto-repair-tricks-and-tips/ extended auto warranty] to move because of her family members.<br><br>My web page [http://www.link2connect.com/profile-764935/info/ extended auto warranty] - [http://www.friscowebsolutions.com/UserProfile/tabid/379/userId/203/Default.aspx www.friscowebsolutions.com] |
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| As with Hitchin's introduction of [[generalized complex manifold]]s, this is an example of a mathematical tool found useful in [[mathematical physics]].
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| == 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.<ref>For explicitness, the definition of ''Hitchin functional is written before some explanations.</ref>
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| Let <math>M</math> be a [[compact space|compact]], [[oriented]] 6-[[manifold]] with trivial [[canonical bundle]]. Then the '''Hitchin functional''' is a [[functional (mathematics)|functional]] on [[differential form|3-forms]] defined by the formula:
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| : <math>\Phi(\Omega) = \int_M \Omega \wedge * \Omega,</math> | |
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| where <math>\Omega</math> is a 3-form and * denotes the [[Hodge star]] operator.
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| == Properties ==
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| * The Hitchin functional is analogous for six-manifold to the [[Yang-Mills]] functional for the four-manifolds.
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| * The Hitchin functional is manifestly [[invariant (mathematics)|invariant]] under the [[group action|action]] of the [[group (mathematics)|group]] of orientation-preserving [[diffeomorphism]]s.
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| * '''Theorem.''' Suppose that <math>M</math> is a three-dimensional [[complex manifold]] and <math>\Omega</math> is the real part of a non-vanishing [[holomorphic]] 3-form, then <math>\Omega</math> is a [[critical point (mathematics)|critical point]] of the functional <math>\Phi</math> restricted to the [[cohomology class]] <math>[\Omega] \in H^3(M,R)</math>. Conversely, if <math>\Omega</math> is a critical point of the functional <math>\Phi</math> in a given comohology class and <math>\Omega \wedge * \Omega < 0</math>, then <math>\Omega</math> '''defines''' the structure of a complex manifold, such that <math>\Omega</math> is the real part of a non-vanishing holomorphic 3-form on <math>M</math>.
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| :The proof of the theorem in Hitchin's articles {{harvs|txt|last=Hitchin|year=2000}} and {{harvs|txt|last=Hitchin|year=2001}} is relatively straightforward. The power of this concept is in the converse statement: if the exact form <math>\Phi(\Omega)</math> is known, we only have to look at its critical points to find the possible complex structures.
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| == Stable forms ==
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| Action functionals often determine geometric structure<ref>For example, complex structure, symplectic structure, <math>G_2</math> holonomy and <math>Spin(7)</math> holonomy etc.</ref> on <math>M</math> and geometric structure are often characterized by the existence of particular differential forms on <math>M</math> that obey some integrable conditions.
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| If an ''m''-form <math>\omega</math> can be written with local coordinates
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| :<math>\omega=dp_1\wedge dq_1+\cdots+dp_m\wedge dq_m</math>
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| and
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| :<math>d\omega=0</math>,
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| then <math>\omega</math> defines ''[[symplectic manifold|symplectic structure]]''.
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| A ''p''-form <math>\omega\in\Omega^p(M,\mathbb{R})</math> is ''stable'' if it lies in an open orbit of the local <math>GL(n,\mathbb{R})</math> action where n=dim(M), namely if any small perturbation <math>\omega\mapsto\omega+\delta\omega</math> can be undone by a local <math>GL(n,\mathbb{R})</math> 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.
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| What about ''p''=3? For large ''n'' ''3''-form is difficult because the dimension of <math>\wedge^3(\mathbb{R}^n)</math>, <math>n^3</math>, grows more firstly than the dimension of <math>GL(n,\mathbb{R})</math>, <math>n^2</math>. But there are some very lucky exceptional case, namely, <math>n=6</math>, when dim <math>\wedge^3(\mathbb{R}^6)=20</math>, dim <math>GL(6,\mathbb{R})=36</math>. Let <math>\rho</math> be a stable real ''3''-form in dimension ''6''. Then the stabilizer of <math>\rho</math> under <math>GL(6,\mathbb{R})</math> has real dimension ''36-20=16'', in fact either <math>SL(3,\mathbb{R})\times SL(3,\mathbb{R})</math> or <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math>.
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| Focus on the case of <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math> and if <math>\rho</math> has a stabilizer in <math>SL(3,\mathbb{C})\cap SL(3,\mathbb{C})</math> then it can be written with local coordinates as follows:
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| :<math>\rho=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3})</math>
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| where <math>\zeta_1=e_1+ie_2,\zeta_2=e_3+ie_4,\zeta_3=e_5+ie_6</math> and <math>e_i</math> are bases of <math>T^*M</math>. Then <math>\zeta_i</math> determines an [[almost complex structure]] on <math>M</math>. Moreover, if there exist local coordinate <math>(z_1,z_2,z_3)</math> such that <math>\zeta_i=dz_i</math> then it determines fortunately an [[complex manifold|complex structure]] on <math>M</math>.
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| Given the stable <math>\rho\in\Omega^3(M,\mathbb{R})</math>:
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| :<math>\rho=\frac{1}{2}(\zeta_1\wedge\zeta_2\wedge\zeta_3+\bar{\zeta_1}\wedge\bar{\zeta_2}\wedge\bar{\zeta_3})</math>.
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| We can define another real ''3''-from
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| :<math>\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})</math>.
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| And then <math>\Omega=\rho+i\tilde{\rho}(\rho)</math> is a holomorphic ''3''-form in the almost complex structure determined by <math>\rho</math>. Furthermore, it becomes to be the complex structure just if <math>d\Omega=0</math> i.e. <math>d\rho=0</math>
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| and <math>d\tilde{\rho}(\rho)=0</math>. This <math>\Omega</math> is just the ''3''-form <math>\Omega</math> in formal definition of ''Hitchin functional''. These idea induces the [[generalized complex structure]]. | |
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| == Use in string theory ==
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| Hitchin functionals arise in many areas of string theory. An example is the [[compactification (physics)|compactifications]] of the 10-dimensional string with a subsequent [[orientifold]] projection <math>\kappa</math> using an [[Involution (mathematics)|involution]] <math>\nu</math>. In this case, <math>M</math> is the internal 6 (real) dimensional [[Calabi-Yau space]]. The couplings to the complexified [[Kähler manifold|Kähler coordinates]] <math>\tau</math> is given by
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| : <math>g_{ij} = \tau \text{im} \int \tau i^*(\nu \cdot \kappa \tau).</math>
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| The potential function is the functional <math>V[J] = \int J \wedge J \wedge J</math>, where J is the [[almost complex structure]]. Both are Hitchin functionals.{{harvtxt|Grimm|Louis|2004}}
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| As application to string theory, the famous OSV conjecture {{harvtxt|Ooguri|Strominger|Vafa|2004}} used ''Hitchin functional'' in order to relate topological string to 4-dimensional black hole entropy. Using similar technique in the <math>G_2</math> holonomy {{harvtxt|Dijkgraaf|Gukov|Neitake|Vafa|2004}} argued about [[Topological string theory#Topological M-theory|topological M-theory]] and in the <math>Spin(7)</math> holonomy topological F-theory might be argued.
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| More recently, [[E. Witten]] claimed the mysterious superconformal field theory in six dimensions.{{harvtxt|Witten|2007}}
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| == Notes ==
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| <references/>
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| == References ==
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| *{{citation|title=The geometry of three-forms in six and seven dimensions
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| |journal=
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| |publisher=
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| |volume=| issue =|year= 2000
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| |doi=
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| |pages=
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| |url=http://arxiv.org/abs/math/0010054
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| |authorlink=Nigel Hitchin
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| |first=Nigel |last=Hitchin}}
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| *{{citation|title=Stable forms and special metric
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| |journal=
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| |publisher=
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| |volume=| issue =|year= 2001
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| |doi=
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| |pages=
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| |url=http://arxiv.org/abs/math/0107101
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| |authorlink=Nigel Hitchin
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| |first=Nigel |last=Hitchin}}
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| *{{citation|title=The effective action of Type IIA Calabi-Yau orientifolds
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| |author=Thomas Grimm; Jan Louis
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| |journal=Nuclear Physics B
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| |publisher=Institut für Theoretische Physik
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| |volume=718 | issue=1–2 |year= 2005
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| |doi=10.1016/j.nuclphysb.2005.04.007
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| |pages=153–202 |url=http://arxiv.org/abs/hep-th/0412277}}
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| *{{citation|title=Topological M-theory as Unification of Form Theories of Gravity
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| |author=Robert Dijikgraaf; Sergei Gukov; Andrew Neitzke & Cumrun Vafa
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| |journal=
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| |publisher=
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| |volume=| issue =|year= 2004
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| |doi=
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| |pages=
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| |url=http://arxiv.org/pdf/hep-th/0411073}}
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| *{{citation|title=Black Hole Attractors and the Topological String
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| |journal=
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| |publisher=
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| |volume=| issue =|year= 2004
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| |doi=
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| |pages=
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| |url=http://arxiv.org/abs/hep-th/0405146
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| |author=Hiroshi Ooguri, Andrew Strominger & Cumran Vafa }}
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| *{{citation|title=Conformal Field Thoery In Four And Six Dimensions
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| |first=Edward |last=Witten
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| |journal=
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| |publisher=
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| |volume=| issue =|year= 2007
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| |doi=
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| |pages=
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| |url=http://arxiv.org/abs/0712.0157v2
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| }}
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| {{DEFAULTSORT:Hitchin Functional}}
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| [[Category:Complex manifolds]]
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| [[Category:Symplectic manifolds]]
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| [[Category:String theory]]
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