3.4 Supersymmetric brane effective actions in Minkowski
In the study of global supersymmetric field theories, one learns the superfield formalism is the most
manifest way of writing interacting manifestly-supersymmetric Lagrangians . One extends the
manifold to a supermanifold through the addition of Grassmann fermionic coordinates
. Physical fields become components of superfields , the natural objects
in this mathematical structure defined as finite polynomials in a Taylor-like expansion
that includes auxiliary (non-dynamical) components allowing one to close the supersymmetry algebra
off-shell. Generic superfields do not transform irreducibly under the super-Poincaré group. Imposing
constraints on them, i.e., , gives rise to the different irreducible supersymmetric representations.
For a standard reference on these concepts, see .
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
- 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 .
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
Figure 5: Kappa symmetry and world volume diffeomorphisms allow one to couple the brane degrees
of freedom to the superfield components of supergravity in a manifestly covariant and supersymmetric
way. Invariance under kappa symmetry forces the background to be on-shell. The gauge fixing of
these symmetries connects the GS formulation with world volume supersymmetry, whose on-shell
degrees of freedom match the Goldstone modes of the brane supergravity configurations.
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
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,
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
3.4.1 D-branes in flat superspace
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].
Here I follow the strategy in . 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 ,
- 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
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
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
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  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
Notice, is chosen to remove the exact form coming from the kappa symmetry variation of ,
- 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  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 . 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 
in type IIA, whereas in type IIB 
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 
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  that . This proves equals the identity, as required in our
The set of global symmetries includes supersymmetry , bosonic translations and Lorentz
transformations . The field infinitesimal transformations are
with given in Eq. (108) and .
Geometrical reinterpretation of the effective action: the supersymmetric action was constructed out of the
supersymmetric invariant forms . These can be reinterpreted as the pullback of
10-dimensional superspace tensors to the brane world volume. To see this, it is convenient to
introduce the explicit supervielbein components , defined in Appendix A.1, where the index M
stands for curved superspace indices, i.e., , and the index A for tangent flat superspace
indices, i.e., . In this language, the super-Poincaré supervielbein components equal
manifest that all Clifford matrices act in the tangent space, as they should. The components (128)
allow us to rewrite all couplings in the effective action as pullbacks
of the background superfields , and to the brane world volume. Furthermore, the
kappa symmetry transformations (112) and (122) also allow a natural superspace description as
where the kappa symmetry matrix is nicely repackaged
in terms of the induced Clifford algebra matrices and the gauge invariant tensor
whereas stands for the wedge product of the 1-forms .
Summary: We have constructed an effective action describing the propagation of Dp-branes in
10-dimensional Minkowski spacetime being invariant under dimensional diffeomorphisms,
10-dimensional supersymmetry and kappa symmetry. The final result resembles the bosonic action (47) in
that it is written in terms of pullbacks of the components of the different superfields ,
and encoding the non-trivial information about the non-dynamical background where the
brane propagates in a manifestly supersymmetric way. These superfields are on-shell supergravity
configurations, since they satisfy the set of constraints listed in Appendix A.1. It is this set of features that
will allow us to generalise these couplings to arbitrary on-shell superspace backgrounds in Section 3.5,
while preserving the same kinematic properties.
3.4.2 M2-brane in flat superspace
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
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 
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].