# nLab Hitchin functional

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

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 $n = 3$ and/or $n =4$. 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 $X$ be a closed oriented smooth manifold of dimension 6.

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

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

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

The Hitchin function on 3-forms is the function

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

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

$\Omega \mapsto \int_X \sqrt{\vert \lambda (\Omega) \vert} \,.$

### In 7 dimensions

Let $X$ be 7-dimensional.

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

$\int_X \Omega \wedge \star_\Omega \Omega \,.$

## Properties

### In 6 dimensions

A differential 3-form $\Omega$ such that $\lambda(\Omega) \lt 0$ is a critical point of the above functional precisely if there is the structure of a complex manifold on $X$ such that $\Omega$ 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