# nLab definite form

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

singular 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

Given a vector space $V$ and an element $\phi$ in the exterior product $\wedge^p V^\ast$ (a $p$-covector), then a differential p-form $\omega$ on a smooth manifold $X$ whose tangent spaces look like $V$ is called definite on $\phi$ (Bryant 05, section 3.1.1) or stable at each point (Hitchin, p. 3) if at each point $x \in X$ the restriction $\omega|_x \in \wedge^p T^\ast X \simeq \wedge^p V^\ast$ is equal to $\phi$, up to a general linear transformation.

The existence of a definite form implies a G-structure on $X$ for $G$ the stabilizer subgroup of $\phi$.

A class of examples of definite forms are the 3-forms on G2-manifolds, these are definite on the “associative 3-form” on $\mathbb{R}^7$.

The higher prequantization of a definition form is a definite globalization of a WZW term.

## Definition

Given a vector space $V$ and a stable form $\phi \in \wedge^p V^\ast$ (hence a form whose orbit under the general linear group $GL(V)$ is an open subspace in $\wedeg^p V$), and given a smooth manifold modeled on the vector space $V$, then a differential form $\omega \in \Omega^p(X)$ is definite on $\phi$ if at each point it is in this open orbit.

## Examples

### $G_2$-manifolds

• Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)

• Robert Bryant, Some remarks on $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf