nLab calibration

Contents

Context

Riemannian geometry

Riemannian geometry

Applications

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

A degree-$p$ calibration of an oriented Riemannian manifold $(X,g)$ is a differential p-form $\omega \in \Omega^p(X)$ with the property that

1. it is closed $d \omega = 0$;

2. evaluated on any oriented $p$-dimensional subspace of any tangent space of $X$, it is less than or equal to the induced degree-$p$ volume form, with equality for at least one choice of subspace.

A Riemannian manifold equipped with such a calibration is also called a calibrated geometry (Harvey-Lawson 82) or similar.

A calibrated submanifold of a manifold with calibration is an oriented submanifold such that restricted to each of its tangent spaces $\omega$ equals the induced volume form of the submanifold there.

Properties

Minimal volume submanifolds

Any calibrated submanifold $\Sigma \hookrightarrow X$ minimizes volume in its homology class.

For Let $\tilde \Sigma \hookrightarrow X$ be a homologous submanifold. Then Stokes theorem together with the condition that $d \phi = 0$ implies that the integration of differential forms of $\phi$ over $\Sigma$ equals that over $\tilde \Sigma$. The defining conditions on calibrations and on calibrated submanifolds then imply the inequality

$vol(\Sigma) \stackrel{cal\,subm}{=} \int_\Sigma \phi \stackrel{Stokes}{=} \int_{\tilde \Sigma} \phi \stackrel{calib}{\leq} \int_{\tilde \Sigma} d vol = vol(\tilde \Sigma) \,.$

Calibrations from spinors

under construction

For suitable $n$ and $p$, and given a real spin representation of $Spin(n)$, then the Cartesian space $\mathbb{R}^n$ with its canonical Riemannian structure becomes $p$-calibrated with the calibration form being

$\omega_{\epsilon} \coloneqq (\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon) \, e^{a_1} \wedge \cdots \wedge e^{a_p}$

where

1. $\{e^a\}$ denotes the canonical linear basis of differential 1-forms;

2. $\epsilon$ is a non-vanishing spinor;

3. $\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon$ is the canonical bilinear pairing which in components is given by evaluating $\epsilon$ in the quadratic form given by multiplying the skew-symmetrized product of $p$ of the representation matrices $\Gamma^a$ of the Clifford algebra with the charge conjugation matrix $C$.

For instance for $n = 7$ and $p = 3$ then this gives the associative 3-form calibration.

More generally for $X$ an $n$-dimensional Riemannian manifold with a covariantly constant spinor $\epsilon$, then under suitable conditions applying this construction in each tangent space gives a calibration.

Examples

General

The original articles are

The relation to Killing spinors goes back to