nLab calibration

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 $C$.

(e.g. Dadok-Harvey 93).

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

• The globalization of the associative 3-form of a G₂-manifold is a calibration. A calibrated submanifold in this case is also called an associative submanifold.

• The Cayley 4-form on Spin(7)-manifolds.

References

General

The original articles are

• Reese Harvey, H. Blaine Lawson, Calibrated geometries, Acta Mathematica July 1982, Volume 148, Issue 1, pp 47-157

• Reese Harvey, Calibrated geometries, Proceeding of the ICM 1983 (pdf)

The relation to Killing spinors goes back to

• Reese Harvey, Spinors and Calibrations, Academic Press, 1990 (publisher)

• Jiri Dadok, Reese Harvey, Calibrations and spinors, Acta Mathematica 1993, Volume 170, Issue 1, pp 83-120

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.

• Gary Gibbons, George Papadopoulos, Calibrations and Intersecting Branes (arXiv:hep-th/9803163)

• Jerome Gauntlett, Neil Lambert, Peter West, Branes and Calibrated Geometries, Commun.Math.Phys. 202 (1999) 571-592 (arXiv:hep-th/9803216)

• George Papadopoulos, Jan Gutowski, AdS Calibrations, Phys.Lett.B462:81-88,1999 (arXiv:hep-th/9902034)

• Jan Gutowski, George Papadopoulos, Paul Townsend, Supersymmetry and generalized calibrations, Phys.Rev.D60:106006, 1999 (arXiv:hep-th/9905156)

• Jan Gutowski, S. Ivanov, George Papadopoulos, Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class, Asian Journal of Mathematics 7 (2003), 39-80 (arXiv:0205012)