nLab
Hitchin functional

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

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.

Definition

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 \,.

Properties

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

(…)

References

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.