To argue this, analyse the field content of these vector supermultiplets. These include a set of scalar fields and a gauge field in dimensions, describing physical polarisations. Thus, the total number of massless bosonic degrees of freedom is

Since Dp-branes propagate in 10 dimensions, any covariant formalism must involve a set of 10 scalar fields , transforming like a vector under the full Lorentz group . This is the same situation we encountered for the superstring. As such, it should be clear the extra bosonic gauge symmetries required to remove these extra scalar fields are dimensional diffeomorphisms describing the freedom in embedding in . Physically, the Dirichlet boundary conditions used in the open string description did fix these diffeomorphisms, since they encode the brane location in .

What about the fermionic sector? The discussion here is entirely analogous to the superstring one. This is because spacetime supersymmetry forces us to work with two copies of Majorana–Weyl spinors in 10 dimensions. Thus, matching the eight on-shell bosonic degrees of freedom requires the effective action to be invariant under a new fermionic gauge symmetry. I will refer to this as kappa symmetry, since it will share all the characteristics of the latter for the superstring.

First, target space covariance requires the background to allow a superspace formulation in
dimensions^{10}.
Such formulation involves a single copy of Majorana fermions, which gives rise to a pair of
Majorana–Weyl fermions, matching the superspace formulation for the superstring described in
Section 2. Majorana spinors have real components, which are further reduced to 16
due to the Dirac equation. Thus, a further gauge symmetry is required to remove half of these fermionic
degrees of freedom, matching the eight bosonic on-shell ones. Once again, kappa symmetry will be required
to achieve this goal.

What about the M5-brane? The fermionic discussion is equivalent to the M2-brane one. The bosonic one must contain a new ingredient. Indeed, geometrically, there are only five scalars describing the transverse M5-brane excitations. These do not match the eight on-shell fermionic degrees of freedom. This is reassuring because there is no scalar supermultiplet in dimensions with such number of scalars. Interestingly, there exists a tensor supermultiplet in dimensions whose field content involves five scalars and a two-form gauge potential with self-dual field strength. The latter involves 6-2 choose 2 physical polarisations, with self-duality reducing these to three on-shell degrees of freedom. To keep covariance and describe the right number of polarisations, the theory must be invariant under gauge transformations for the 2-form gauge potential. I will later discuss how to keep covariance while satisfying the self-duality constraint.

Concerning vector supermultiplets with scalars in dimensions, the results are summarised in Table 2. Note that the last column describes the field content of all Dp-branes, starting from the D0-brane and finishing with the D9 brane filling in all spacetime. Thus, the field content of all Dp-branes matches with the one corresponding to the different vector supermultiplets in dimensions. This point agrees with the open string conformal field theory description of D branes.

Finally, there is just one interesting tensor multiplet with scalars in six dimensions, corresponding to the aforementioned M5 brane, among the six-dimensional tensor supermultiplets listed in Table 3.

By construction, an effective action written in terms of these on-shell degrees of freedom can neither be spacetime covariant nor invariant (in the particular case when branes propagate in Minkowski, as I have assumed so far). Effective actions satisfying these two symmetry requirements involve the addition of both extra, non-physical, bosonic and fermionic degrees of freedom. To preserve their non-physical nature, these supersymmetric brane effective actions must be invariant under additional gauge symmetries

- world volume diffeomorphisms, to gauge away the extra scalars,
- kappa symmetry, to gauge away the extra fermions.

Branes carry energy, consequently, they gravitate. Thus, one expects to find gravitational configurations (solitons) carrying the same charges as branes solving the classical equations of motion capturing the effective dynamics of the gravitational sector of the theory. The latter is the effective description provided by type IIA/B supergravity theories, describing the low energy and weak coupling regime of closed strings, and supergravity. The purpose of this section is to argue the existence of the same world-volume degrees of freedom and symmetries from the analysis of massless fluctuations of these solitons, applying collective coordinate techniques that are a well-known notion for solitons in standard, non-gravitational, gauge theories.

In field theory, given a soliton solving its classical equations of motion, there exists a notion of
effective action for its small excitations. At low energies, the latter will be controlled by massless
excitations, whose number matches the number of broken symmetries by the background soliton [243]^{12}.
These symmetries are global, whereas all brane solitons are on-shell configurations in supergravity, whose
relevant symmetries are local. To get some intuition for the mechanism operating in our case, it is
convenient to review the study of the moduli space of monopoles or instantons in abelian gauge theories.
The collective coordinates describing their small excitations include not only the location of the
monopole/instanton, which would match the notion of transverse excitation in our discussion given the
pointlike nature of these gauge theory solitons, but also a fourth degree of freedom associated with the
breaking of the gauge group [431, 288]. The reason the latter is particularly relevant to us is because,
whereas the first set of massless modes are indeed related to the breaking of Poincaré invariance,
a global symmetry in these gauge theories, the latter has its origin on a large gauge
transformation.

This last observation points out that the notion of collective coordinates can generically be associated with large gauge transformations, and not simply with global symmetries. It is precisely in this sense how it can be applied to gravity theories and their soliton solutions. In the string theory context, the first work where these ideas were applied was [127] in the particular set-up of 5-brane solitons in heterotic and type II strings. It was later extended to M2-branes and M5-branes in [332]. In this section, I follow the general discussion in [6] for the M2, M5 and D3-branes. These brane configurations are the ones interpolating between Minkowski, at infinity, and AdS times a sphere, near their horizons. Precisely for these cases, it was shown in [236] that the world volume theory on these branes is a supersingleton field theory on the corresponding AdS space.

Before discussing the general strategy, let me introduce the on-shell bosonic configurations to be analysed below. All of them are described by a non-trivial metric and a gauge field carrying the appropriate brane charge. The multiple M2-brane solution, first found in [198], is

Here, and in the following examples, describe the longitudinal brane directions, i.e., for the M2-brane, whereas the transverse Cartesian coordinates are denoted by , . The solution is invariant under and is characterised by a single harmonic function in The structure for the M5-brane, first found in [273], is analogous but differs in the dimensionality of the tangential and transverse subspaces to the brane and in the nature of its charge, electric for the M2-brane and magnetic for the M5-brane below In this case, and . The isometry group is and again it is characterised by a single harmonic function in The D3-brane, first found in [195], similarly has a non-trivial metric and self-dual five form RR field strength with isometry group . It is characterised by a single harmonic function in All these brane configurations are half-BPS supersymmetric. The subset of sixteen supercharges being preserved in each case is correlated with the choice of sign in the gauge potentials fixing their charges. I shall reproduce this correlation in the effective brane action in Section 3.5.Let me first sketch the argument behind the generation of massless modes in supergravity theories, where all relevant symmetries are gauge, before discussing the specific details below. Consider a background solution with field content , where labels the field, including its tensor character, having an isometry group . Assume the configuration has some fixed asymptotics with isometry group , so that . The relevant large gauge transformations in our discussion are those that act non-trivially at infinity, matching a broken global transformation asymptotically , but differing otherwise in the bulk of the background geometry

In this way, one manages to associate a gauge transformation with a global one, only asymptotically. The idea is then to perturb the configuration by such pure gauge, and finally introduce some world volume dependence on the parameter , i.e., . At that point, the transformation is no longer pure gauge. Plugging the transformation in the initial action and expanding, one can compute the first order correction to the equations of motion fixing some of the ambiguities in the transformation by requiring the perturbed equation to correspond to a massless normalisable mode.In the following, I explain the origin of the different bosonic and fermionic massless modes in the world volume supermultiplets discussed in Section 3.1 by analysing large gauge diffeomorphisms, supersymmetry and abelian tensor gauge transformations.

where indices stand for the chirality of the fermionic zero modes. In particular, for the M2 brane it describes negative eight dimensional chirality of the 11-dimensional spinor , while for the M5 and D3 branes, it describes positive six-dimensional and four-dimensional chirality.

Thus, using purely effective field theory techniques, one is able to derive the spectrum of massless excitations of brane supergravity solutions. This method only provides the lowest order contributions to their equations of motion. The approach followed in this review is to use other perturbative and non-perturbative symmetry considerations in string theory to determine some of the higher-order corrections to these effective actions. Our current conclusion, from a different perspective, is that the physical content of these theories must be describable in terms of the massless fields in this section.

Living Rev. Relativity 15, (2012), 3
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