Consider any supergravity theory having bosonic and fermionic
degrees of freedom. It is
consistent with the equations of motion to set
. The question of whether the configuration
preserves supersymmetry reduces to the study of whether there exists any supersymmetry transformation
preserving the bosonic nature of the on-shell configuration, i.e.,
, without transforming
, i.e.,
. Since the structure of the local supersymmetry transformations in supergravity is
This argument is general and any condition derived from it is necessary. Thus, one is instructed to analyse the
condition before solving the equations of motion. As a particular example, and to make
contact with the discussions in Section 3.1.1, consider
supergravity. The only fermionic
degrees of freedom are the gravitino components
. Their supersymmetry transformation
is [466
]
The same question for brane effective actions is treated in a conceptually analogous way. The subspace
of bosonic configurations defined by
is compatible with the brane equations of motion.
Preservation of supersymmetry requires
. The total transformation
is given by
We are interested in deriving a general condition for any bosonic configuration to preserve
supersymmetry. Since not all fermionic fields are physical, working on the subspace
is not
precise enough for our purposes. We must work in the subspace of field configurations being both physical
and bosonic [85
].This forces us to work at the intersection of
and some kappa symmetry gauge
fixing condition. Because of this, I find it convenient to break the general argument into two
steps.
I will refer to Eq. (214) as the kappa symmetry preserving condition. It was first derived in [85]. This is
the universal necessary condition that any bosonic on-shell brane configuration
must satisfy to
preserve some supersymmetry.
Brane |
Bosonic kappa symmetry matrix |
M2-brane |
|
M5-brane |
|
|
|
IIA Dp-branes |
|
IIB Dp-branes |
|
In Table 5, I evaluate all kappa symmetry matrices in the subspace of bosonic configurations
for future reference. This matrix encodes information
Just as in supergravity, any solution to Eq. (214) involves two sets of conditions, one on the space of
configurations
and one on the amount of supersymmetries. More precisely,
The first set will turn out to be BPS equations, whereas the second will determine the amount of supersymmetry preserved by the combined background and probe system.
http://www.livingreviews.org/lrr-2012-3 |
Living Rev. Relativity 15, (2012), 3
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