Hitchin functional



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



What are now called Hitchin functionals are functions/functionals on spaces of differential n-forms on manifolds of dimension 6 or 7, for n=3n = 3 and/or n=4n =4. They are such that their critical points characterize holomorphic 3-forms of complex manifolds in dimension 6, as associative 3-forms of G2-manifolds in dimension 7.


In 6 dimensions

Let XX be a closed oriented smooth manifold of dimension 6.

For Ω 3Γ(T *X)\Omega \in \wedge^3 \Gamma(T^* X) a differential 3-form on XX, write

  • |λΩ|( 6Γ(T *X)) 2{\vert \lambda \Omega \vert} \in (\wedge^6 \Gamma(T^* X))^{\otimes 2}

  • |λΩ| 6Γ(T *X)\sqrt{\vert \lambda \Omega \vert} \in \wedge^6 \Gamma(T^* X).

The Hitchin function on 3-forms is the function

3Γ(T *X) \wedge^3 \Gamma(T^* X) \to \mathbb{R}

which sends 3-form to the integration of differential forms of this 6-form over XX

Ω X|λ(Ω)|. \Omega \mapsto \int_X \sqrt{\vert \lambda (\Omega) \vert} \,.

In 7 dimensions

Let XX be 7-dimensional.

Let Ω 3Γ(T *X)\Omega \in \wedge^3 \Gamma(T^* X) be a stable differential form?. This determines a Riemannian metric g Ω\g_\Omega on XX. Write Ω\star_\Omega for the corresponding Hodge star operator. The Hitchin functional now is the function that takes stable forms to

XΩ ΩΩ. \int_X \Omega \wedge \star_\Omega \Omega \,.


In 6 dimensions

A differential 3-form Ω\Omega such that λ(Ω)<0\lambda(\Omega) \lt 0 is a critical point of the above functional precisely if there is the structure of a complex manifold on XX such that Ω\Omega is the real part of a non-vanishing holomorphic 3-form.

This is (Hitchin, theorem 13).

In 7 dimensions



The original articles are

Reviews with a perspective on the role of the functionals in physics/gravity/string theory/topological M-theory include

Last revised on January 9, 2013 at 05:44:54. See the history of this page for a list of all contributions to it.