Special Graphs

A special p-form is a p-form which, in some orthonormal basis {e µ }, has components ϕ µ1...µp = ϕ(e µ1 ,. .. , e µp) taking values in {−1, 0, 1}. We discuss graphs which characterise such forms. A constant p-form ϕ in a d-dimensional Euclidean space is a calibration if for any p-dimensional subspace spanned by a set of orthonormalised vectors e with equality holding for at least one subspace. Let U be an oriented p–dimensional subspace of R d with oriented metric volume vol U. The set of all such subspaces is the oriented Grassmannian Gr p R d. A calibration ϕ ∈ Λ p R d is thus a p–form with the property that the function ϕ : Gr p R d −→ R associated to ϕ and defined by U → ϕ(U) := ϕ, vol U takes values in [−1, 1] ⊂ R with at least one of the two extremal values ±1 being achieved. The p-planes U for which ϕ(U) = ±1 are said to be calibrated by ϕ. Almost all examples of calibrations known are invariant under a group G ⊂ O(R d) large enough so that it is relatively simple to check the calibration condition directly. Interestingly most of these examples, in particular the calibrations char-acterising special holonomy manifolds, for instance the G 2 –invariant Cayley 3-form in seven dimensions, defined by the structure constants of the imaginary octonions, and the Spin(7)-invariant 4-forms in eight dimensions are special forms: