These considerations also apply to the dimensional supermultiplets describing the physical brane degrees of freedom propagating in , since these correspond to supersymmetric field theories in . The main difference in the GS formulation of brane effective actions is that it is spacetime itself that must be formulated in a manifestly supersymmetric way. By the same argument used in global supersymmetric theories, one would be required to work in a 10- or 11-dimensional superspace, with standard bosonic coordinates and the addition of fermionic ones , whose representations will depend on the dimension of the bosonic submanifold. There are two crucial points to appreciate for our purposes

- the superspace coordinates will become the brane dynamical degrees of freedom , besides any additional gauge fields living on the brane;
- the couplings of the latter to the fixed background where the brane propagates must also be described in a manifestly spacetime supersymmetric way. The formulation achieving precisely that is the superspace formulation of supergravity theories [491].

Both these points were already encountered in our review of the GS formulation for the superstring. The same features will hold for all brane effective actions discussed below. After all, both strings and branes are different objects in the same theory. Consequently, any manifestly spacetime supersymmetric and covariant formulation should refer to the same superspace. It is worth emphasising the world volume manifold with local coordinates remains bosonic in this formulation. This is not what occurs in standard superspace formulations of supersymmetric field theories. There exists a classically equivalent formulation to the GS one, the superembedding formulation that extends both the spacetime and the world volume to supermanifolds. Though I will briefly mention this alternative and powerful formulation in Section 8, I refer readers to [460].

As in global supersymmetric theories, supergravity superspace formulations involve an increase in the number of degrees of freedom describing the spacetime dynamics (to preserve supersymmetry covariance). Its equivalence with the more standard component formalism is achieved through the satisfaction of a set of non-trivial constraints imposed on the supergravity superfields. These guarantee the on-shell nature of the physical superfield components. I refer the reader to a brief but self-contained Appendix A where this superspace formulation is reviewed for type IIA/B and supergravities, including the set of constraints that render them on-shell. These will play a very important role in the self-consistency of the supersymmetric effective actions I am about to construct.

Instead of discussing the supersymmetric coupling to an arbitrary curved background
at once, my plan is to review the explicit construction of supersymmetric D-brane and
M2-brane actions propagating in Minkowski spacetime, or its superspace extension,
super-Poincaré.^{20}
The logic will be analogous to that presented for the superstring. First, I will construct these
supersymmetric and kappa invariant actions without using the superspace formulation, i.e., using a more
explicit component approach. Afterwards, I will rewrite these actions in superspace variables, pointing in
the right direction to achieve a covariant extension of these results to curved backgrounds in
Section 3.5.

In this section, I am aiming to describe the propagation of D-branes in a fixed Minkowski target space
preserving all spacetime supersymmetry and being world volume kappa symmetry invariant. Just as for bosonic
open strings, the gauge field dependence was proven to be of the DBI form by explicit open superstring
calculations [482, 389, 87].^{21}

Here I follow the strategy in [9]. First, I will construct a supersymmetric invariant DBI action, building on the superspace results reported in Section 2. Second, I will determine the WZ couplings by requiring both supersymmetry and kappa symmetry invariance. Finally, as in our brief review of the GS superstring formulation, I will reinterpret the final action in terms of superspace quantities and their pullback to world volume hypersurfaces. This step will identify the correct structure to be generalised to arbitrary curved backgrounds.

Let me first set my conventions. The field content includes a set of dimensional world volume scalar fields describing the embedding of the brane into the bulk supermanifold. Fermions depend on the theory under consideration

- type IIA superspace involves two fermions of different chiralities , i.e., , where . I describe them jointly by a unique fermion , satisfying .
- type IIB superspace contains two fermions of the same chirality (positive by assumption), . The index is an internal index keeping track of the doublet structure on which Pauli matrices act.

In either case, one defines , in terms of an antisymmetric charge conjugation matrix satisfying

with satisfying the standard Clifford algebra with mostly plus eigenvalues. I am not introducing a special notation above to refer to the tangent space, given the flat nature of the bulk. This is not accurate but will ease the notation below. I will address this point when reinterpreting our results in terms of a purely superspace formulation.Let me start the discussion with the DBI piece of the action. This involves couplings to the NS-NS bulk sector, a sector that is also probed by the superstring. Thus, both the supervielbein and the NS-NS 2-form were already identified to be

in type IIA, whereas in type IIB one replaces by . The DBI action will therefore be invariant under the spacetime supersymmetry transformations if both, the induced world volume metric and the gauge invariant 2-form, , are. These are defined by where stands for the pullback of the superspace 2-form into the worldvolume, i.e., . Since is quasi-invariant under (105), one chooses so that , guaranteeing the invariance of the action (104) since the set of 1-forms are supersymmetric invariant.Let me consider the WZ piece of the action

Since invariance under supersymmetry allows total derivatives, the Lagrangian can be characterised in terms of a -form satisfying Thus, mathematically, must be constructed out of supersymmetry invariants .The above defines a cohomological problem whose solution is not guaranteed to be kappa invariant. Since our goal is to construct an action invariant under both symmetries, let me first formulate the requirements due to the second invariance. The strategy followed in [9] has two steps:

- First, parameterise the kappa transformation of the bosonic fields in terms of an arbitrary . Experience from supersymmetry and kappa invariance for the superparticle and superstring suggest Notice, is chosen to remove the exact form coming from the kappa symmetry variation of , i.e., .
- Second, kappa symmetry must be able to remove half of the fermionic degrees of freedom. Thus, as in the superstring discussion, one expects to involve some non-trivial projector. This fact can be used to conveniently parameterise the kappa invariance of the total Lagrangian. The idea in [9] was to parameterise the DBI kappa transformation as requiring the WZ kappa transformation to be In this way, the kappa symmetry variation of the full Lagrangian equals This is guaranteed to vanish choosing , given the projector nature of .

The question is whether , and exist satisfying all the above requirements. The explicit construction of these objects was given in [9]. Here, I simply summarise their results. The WZ action was found to be

where is the pullback of the field strength of the RR gauge potential , as defined in Eq. (521). Using , this can be written as [293] in type IIA, whereas in type IIB [329] Two observations are in order:- is indeed manifestly supersymmetric, since it only depends on supersymmetric invariant quantities, but is quasi-invariant. Thus, when computing the algebra closed by the set of conserved charges, one can expect the appearance of non-trivial charges in the right-hand side of the supersymmetry algebra. This is a universal feature of brane effective actions that will be conveniently interpreted in Section 3.6.
- This analysis has determined the explicit form of all the RR potentials as superfields in superspace. This was achieved by world volume symmetry considerations, but it is reassuring to check that the expressions found above do satisfy the superspace constraints reported in Appendix A.1. I will geometrically reinterpret the derived action as one describing a Dp-brane propagating in a fixed super-Poincaré target space shortly.

Let me summarise the global and gauge symmetry structure of the full action. The set of gauge symmetries involves world volume diffeomorphisms , an abelian gauge symmetry and kappa symmetry . Their infinitesimal transformations are

where is given in Eq. (112) and was determined in [9] In type IIA, the matrix stands for the world volume form coefficient of , where , while in type IIB, it is given by It was proven in [9] that . This proves equals the identity, as required in our construction.The set of global symmetries includes supersymmetry , bosonic translations and Lorentz transformations . The field infinitesimal transformations are

with given in Eq. (108) and .

Let me consider an M2-brane as an example of an M-brane propagating in super-Poincaré. Given the lessons from the superstring and D-brane discussions, my presentation here will be much more economical.

First, let me describe super-Poincaré as a solution of eleven-dimensional supergravity using the superspace formulation introduced in Appendix A.2. In the following, all fermions will be 11-dimensional Majorana fermions as corresponds to superspace. Denoting the full set of superspace coordinates as with and , the superspace description of super-Poincaré is [165, 144]

It includes the supervielbein and the gauge invariant field strengths and its Hodge dual , defined as Eq. (550) in Appendix A.2.The full effective action can be written as [91]

Notice it depends on the supervielbeins and the three form potential superfields only through their pullbacks to the world volume.Its symmetry structure is analogous to the one described for D-branes. Indeed, the action (136) is gauge invariant under world volume diffeomorphisms and kappa symmetry with infinitesimal transformations given by

The kappa matrix (139) satisfies . Thus, is a projector that will allow one to gauge away half of the fermionic degrees of freedom.The action (136) is also invariant under global super-Poincaré transformations

Supersymmetry quasi-invariance can be easily argued for since is manifestly invariant. Thus, its gauge potential pullback variation will be a total derivative for some spinor-valued two form .It is worth mentioning that just as the bosonic membrane action reproduces the string world-sheet action under double dimensional reduction, the same statement is true for their supersymmetric and kappa invariant formulations [192, 476].

Living Rev. Relativity 15, (2012), 3
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