Supersymmetry and calibrations

The link to supersymmetry goes via spinors, like so much else in life.

A basic fact in the representation theory of the spin group is that the tensor product of two spinorial representations decomposes into exterior powers of the vector representation. This is familiar to anyone (un)lucky enough to have ever used a Fierz identity.

This means that in a spin manifold, one can obtain a number of differential forms out of spinor fields. In particular "squaring" in this way a spinor field which is parallel (i.e., covariantly constant) with respect to the Levi-Civita connection, the resulting forms will also be parallel, and in particular closed. After normalising them so that they have unit comass, they give rise to calibrations. (It is possible to obtain calibrations from spinors which are not parallel, but which obey some other first-order equation.)

Now consider one such calibration p-form ω obtained from a spinor field ε. A plane Π is calibrated by ω precisely when

Π ⋅ ε = ε,

where the ⋅ means the action of the Clifford algebra on spinors.

This condition, repeated for all tangent planes Π, is precisely what is required for the embedding of a p-brane in the spacetime to preserve some supersymmetry &mdash a fact discovered by Becker2 and Strominger. The underlying supersymmetric theory in this case is a Green-Schwarz-type action for the p-brane. The above condition is actually that the supersymmetry variation can be compensated by a local κ-symmetry transformation.

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