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 structure

Notice this is equivalent to requiring , a property that is just reflecting the half-BPS nature of these branes. It can be shown that The choice of is correlated to the distinction between a brane and an anti-brane. Both are supersymmetric, but preserve complementary supercharges. This ambiguity explains why some of the literature has apparently different conventions, besides the possibility of working with different Clifford algebra realisations

The effective action describing a single M2-brane in an arbitrary 11-dimensional background is formally the same as in Eq. (136)

with the same definitions for the induced metric and the pull back 3-form . The information regarding 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 [90]

where is the pullback of the curved supervielbein to the world volume.

Proceeding in an analogous way for Dp-branes, their effective action in an arbitrary type IIA/B background is

It is understood that and is defined using the same notation as in Eq. (521), i.e., as a formal sum of forms, so that the WZ term picks all 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 equals

and the background is on-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 and stands for the wedge product of r of these 1-forms. In [94], readers can find an extension of the results reviewed here when the background includes a mass parameter, i.e., it belongs to massive IIA [434].

The six-dimensional diffeomorphic and kappa symmetry invariant M5-brane [45] is a formal extension of the bosonic one

where all pullbacks refer to superspace. This is kappa invariant under the transformations (143) for , including the extra transformation law for the auxiliary scalar field introduced in the PST formalism. These transformations are determined by the kappa symmetry matrix where and the vector fields and are defined by

- the algebra of -transformations only closes on-shell,
- -symmetry is an infinitely reducible symmetry.

The latter statement uses the terminology of Batalin and Vilkovisky [52] and it is a direct consequence of its projective nature, since the existence of the infinite chain of transformations

gives rise to an infinite 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 [327]. 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
themselves^{24}.

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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