nLab
calibration

Contents

Context

Riemannian geometry

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

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

  1. it is closed dω=0d \omega = 0;

  2. evaluated on any oriented pp-dimensional subspace of any tangent space of XX, it is less than or equal to the induced degree-pp 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 ω\omega equals the induced volume form of the submanifold there.

Properties

Minimal volume submanifolds

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

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

vol(Σ)=calsubm Σϕ=Stokes Σ˜ϕcalib Σ˜dvol=vol(Σ˜). 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 nn and pp, and given a real spin representation of Spin(n)Spin(n), then the Cartesian space n\mathbb{R}^n with its canonical Riemannian structure becomes pp-calibrated with the calibration form being

ω ϵ(ϵ¯Γ a 1a 2a pϵ)e a 1e a p \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}\{e^a\} denotes the canonical linear basis of differential 1-forms;

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

  3. ϵ¯Γ a 1a 2a pϵ\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon is the canonical bilinear pairing which in components is given by evaluating ϵ\epsilon in the quadratic form given by multiplying the skew-symmetrized product of pp of the representation matrices Γ a\Gamma^a of the Clifford algebra with the charge conjugation matrix CC.

(e.g. Dadok-Harvey 93).

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

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

Examples

References

General

The original articles are

The relation to Killing spinors goes back to

See also

  • Wikipedia, Calibrated geometry

  • Jason Dean Lotay, Calibrated submanifolds and the Exceptional geometries, 2005 (pdf)

Application in string theory

Discussion in string theory/M-theory includes the following.

Last revised on April 2, 2019 at 13:33:49. See the history of this page for a list of all contributions to it.