In all effective actions under consideration, the set of degrees of freedom includes scalars and it may include some gauge field , whose dependence is always through the gauge invariant combination 22. The set of kappa symmetry transformations will universally be given by
The structure of the transformations (143) is universal, but the details of the kappa symmetry matrix depend on the specific theory, as described below. A second universal feature, associated with the projection nature of kappa symmetry transformations, i.e., , is the correlation between the brane charge density and the sign of in Eq. (143). More specifically, any brane effective action will have the structure23.
The effective action describing a single M2-brane in an arbitrary 11-dimensional background is formally the same as in Eq. (136)different 11-dimensional backgrounds is encoded in the different couplings described by the supervielbein and 3-form superfields.
The action (147) is manifestly 3d-diffeomorphism invariant. It was shown to be kappa invariant under the transformations (143), without any gauge field, whenever the background superfields satisfy the constraints reviewed in Appendix A.2, i.e., whenever they are on-shell, for a kappa symmetry matrix given by 
Proceeding in an analogous way for Dp-branes, their effective action in an arbitrary type IIA/B background isall contributions coming from the wedge product of this sum and the Taylor expansion of that saturate the world volume dimension. Notice all information on the background spacetime is encoded in the superfields , , and the set of RR potentials .
The action (149) is dimensional diffeomorphic invariant and it was shown to be kappa invariant under the transformations (143) for in [141, 93] when the kappa symmetry matrix equalson-shell, i.e., satisfies the constraints reviewed in Appendix A.1. In the expressions above stands for the pullback of the bulk tangent space Clifford matrices , readers can find an extension of the results reviewed here when the background includes a mass parameter, i.e., it belongs to massive IIA .
The six-dimensional diffeomorphic and kappa symmetry invariant M5-brane  is a formal extension of the bosonic one
The latter statement uses the terminology of Batalin and Vilkovisky  and it is a direct consequence of its projective nature, since the existence of the infinite chain of transformationsinfinite tower of ghosts when attempting to follow the Batalin–Vilkovisky quantisation procedure, which is also suitable to handle the first remark above. Thus, covariant quantisation of kappa invariant actions is a subtle problem. For detailed discussions on problems arising from the regularisation of infinite sums and dealing with Stueckelberg type residual gauge symmetries, readers are referred to [326, 325, 254, 223, 84].
It was later realised, using the Hamiltonian formulation, that kappa symmetry does allow covariant quantisation provided the ground state of the theory is massive . The latter is clearly consistent with the brane interpretation of these actions, by which these vacua capture the half-BPS nature of the (massive) branes themselves24.
For further interesting kinematical and geometrical aspects of kappa symmetry, see [449, 167, 166] and references therein.
Living Rev. Relativity 15, (2012), 3
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