Consider the space of oriented p dimensional subspaces of , i.e., the Grassmannian . For any , one can always find an orthonormal basis in such that is a basis in so that its co-volume is

A -form on an open subset of is a calibration of degree if- for every point , the form satisfies for all and such that the contact set is not empty.

One of the applications of calibrations is to provide a bound for the volume of -dimensional submanifolds of . Indeed, the fundamental theorem of calibrations [289] states

Two remarks can motivate why these considerations should have a relation to brane solitons and supersymmetry:

- For static brane configurations with no gauge field excitations and in the absence of WZ couplings, the energy of the brane soliton equals the volume of the brane submanifold embedded in . Thus, bounds on the volume correspond to brane energy bounds, which are related to supersymmetry saturation, as previously reviewed. Indeed, the dynamical field does mathematically describe the map from the world volume into . The above bound can then be re-expressed as where stands for the pullback of the -form .
- There exists an explicit spinor construction of calibrations emphasising the connection between calibrated submanifolds, supersymmetry and kappa symmetry.

Let me review this spinor construction [159, 287]. For , the -form calibration takes the form

where the set () stands for the transverse scalars to the brane parameterising , is a constant real spinor normalised so that and are antisymmetrised products of Clifford matrices in . Notice that, given a tangent -plane , one can write as where the matrix is evaluated at the point to which is tangent. Given the restriction on the values of , It follows that for all . Since is also closed, one concludes it is a calibration. Its contact set is the set of -planes for which this inequality is saturated. Using Eq. (239), the latter is equally characterised by the set of -planes for which Because of Eq. (241) and the fact that , the solution space to this equation is always half the dimension of the spinor space spanned by for any given tangent -plane . However, this solution space generally varies as varies over the contact set, so that the solution space of the set is generally smaller.So far the discussion involved no explicit supersymmetry. Notice, however, that the matrix in Eq. (240) matches the kappa symmetry matrix for branes in the static gauge with no gauge field excitations propagating in Minkowski. This observation allows us to identify the saturation of the calibration bound with the supersymmetry preserving condition (214) derived from the gauge fixing analysis of kappa symmetry.

Let me close the logic followed in Section 4 by pointing out a very close relation between the supersymmetry algebra and kappa symmetry that all my previous considerations suggest. Consider a single infinite flat M5-brane propagating in Minkowski and fix the extra gauge symmetry of the PST formalism by (temporal gauge). The kappa symmetry matrix (157) reduces to

where all indices stand for world space M5 indices. Notice that the structure of this matrix is equivalent to the one appearing in Eq. (216) for by identifying Even though, this was only argued for the M5-brane and in a very particular background, it does provide some preliminary evidence for the existence of such connection. In fact, a stronger argument can be provided by developing a phase space formulation of the kappa symmetry transformations that allows one to write the supersymmetry anticommutator as [278] This result has not been established in full generality but it agrees with the flat space case [165] and those non-flat cases that have been analysed [438, 437]. I refer the reader to [278] where they connect the functional form in the right-hand side of Eq. (245) with the kappa symmetry transformations for fermions in its Hamiltonian form.The connection between calibrations, supersymmetry and kappa symmetry goes beyond the arguments given above. The original mathematical notion of calibration was extended in [277, 278] relaxing its first condition . Physically, this allowed one to include the presence of non-trivial potential energies due to background fluxes coming from the WZ couplings. Some of the applications derived from this notion include [231, 229, 230, 373, 139]. Later, the notion of generalised calibration was introduced in [344], where it was shown to agree with the notion of calibration defined in generalised Calabi–Yau manifolds [267] following the seminal work in [298]. This general notion allows one to include the effect of non-trivial magnetic field excitations on the calibrated submanifold, but still assumes the background and the calibration to be static. Some applications of these notions in the physics literature can be found in [344, 377, 413]. More recently, this formalism was generalised to include electric field excitations [376], establishing a precise correspondence between generic supersymmetric brane configurations and generalised geometry.

Living Rev. Relativity 15, (2012), 3
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