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
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
In either case, one defines , in terms of an antisymmetric charge conjugation matrix
satisfying
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
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 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:
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
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
The set of global symmetries includes supersymmetry , bosonic translations
and Lorentz
transformations
. The field infinitesimal transformations are
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]
The full effective action can be written as [91]
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 action (136) is also invariant under global super-Poincaré transformations
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].
http://www.livingreviews.org/lrr-2012-3 |
Living Rev. Relativity 15, (2012), 3
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