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
The effective action describing a single M2-brane in an arbitrary 11-dimensional background is formally the
same as in Eq. (136)
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
]
Proceeding in an analogous way for Dp-branes, their effective action in an arbitrary type IIA/B background is
It is understood that 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
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
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
themselves24.
For further interesting kinematical and geometrical aspects of kappa symmetry, see [449, 167, 166] and references therein.
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Living Rev. Relativity 15, (2012), 3
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