Calibrations and spinors

Intersecting brane geometries

“…Although computing the comass of a form ϕ is generally a difficult problem, it simplifies tremendously when ϕ can be constructed by squaring spinors. The cleaner results are in seven and eight dimensions [31,25] and more generally in 8k dimensions [15]; but similar results can also be obtained in eleven dimensions with Lorentzian signature [3]. Remarkably, it is the elevenand eight-dimensional cases which arise in the study of intersecting branes [3,4].…”

“…Complex lagrangian geometry. Let Ξ denote the complex lagrangian calibration given by (15). The corresponding endomorphism Ξ satisfies the characteristic polynomial…”

Supersymmetry and generalized calibrations

“…Lumps are also relevant to static M5-brane configurations. For this we place a M5-brane in a background of the form E (1,5) × M 5 where the M 5 metric is Ricci flat and, for preservation of supersymmetry, has holonomy in SU (2). It follows that M 5 = E × M 4 , with a direct product metric such that the M 4 metric has holonomy in SU (2).…”

“…, 4) be coordinates on M 4 . Consider the manifold E 4 × M 4 where E 4 is the subspace of E (1,5) with constant x 0 and x 5 . This is a Kähler manifold with Kähler 2-form…”

Spinor spaces of Clifford algebras

“…Denote G = A(2,4)(1 +/3(2,4)). The following elements e 1 G,• • • , e6 , G, w(2, 4)G, form a basis of the pinor space V(2,4) of CQ(2,4). Then a2 = (e7 + es)ejO, i= 1, • • • , 6 , a7 = (e7 + es)G, as = (e7 + es)w( 2 , 4)G, ai+s = (e7e8 -1 )eiO, i = 1,.. ,6, a15 = (e7e8-1)G, a16 = ( e7es-1) w(2,4)G, form a basis of the pinor space V(3,5) of C(3,5).…”