First, one works with the bosonic truncation . The background, in Cartesian coordinates, involves the metric
Half-BPS branes should correspond to vacuum configurations in these field theories describing infinite branes breaking the isometry group to and preserving half of the supersymmetries. Geometrically, these configurations are specified by the brane location. This is equivalent to first splitting the scalar fields into longitudinal and transverse directions, setting the latter to constant values (the transverse brane location). Second, one identifies the world volume directions with the longitudinal directions, . The latter can also be viewed as fixing the world volume diffeomorphisms to the static gauge. This information can be encoded as an array
It is easy to check that the above is an on-shell configuration given the structure of the Euler–Lagrange equations and the absence of non-trivial couplings except for the induced world volume metric , which equals in this case.
To analyse the supersymmetry preserved, one must solve Eq. (214). Notice that in the static gauge and in the absence of any further excitations, the induced gamma matrices equal
All these configurations have an energy density equaling the brane tension since the Hamiltonian constraint is always solved byany excitation above the infinite brane configuration would increase the energy. From the world-volume perspective, the solution is a vacuum, and consequently, it is annihilated by all sixteen world-volume supercharges. These are precisely the ones solving the kappa symmetry preserving condition (214).
Living Rev. Relativity 15, (2012), 3
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