4.1 Supersymmetric bosonic configurations and kappa symmetry
To know whether any given on-shell bosonic brane configuration is supersymmetric, and if so, how many
supersymmetries are preserved, one must develop some tools analogous to the ones for bosonic supergravity
configurations. I will review these first.
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
these conditions reduce to . In general, the Clifford valued operator is not higher
than first order in derivatives, but it can also be purely algebraic. Solutions to this equation
- Differential constraints on the subset of bosonic configurations . Given the first-order nature
of the operator , these are simpler than the second-order equations of motion and help
to reduce the complexity of the latter.
- Differential and algebraic constraints on . These reduce the infinite dimensional character
of the original arbitrary supersymmetry transformation parameter to a finite dimensional
subset, i.e., , where the function is uniquely specified by the bosonic
background and the constant spinor typically satisfies a set of conditions ,
where are projectors satisfying and . These are the Killing
spinors of the bosonic background . They can depend on the spacetime point, but they are
no longer arbitrary. Thus, they are understood as global parameters.
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
Solving the supersymmetry preserving condition in the M2-brane and M5-brane backgrounds
determines the Killing spinors of these solutions to be [466, 467]
A similar answer is found for all D-branes in type IIA/B supergravities.
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
where and stand for the kappa symmetry and world volume diffeomorphism infinitesimal
transformations and for any global symmetries different from supersymmetry, which is
generated by the Killing spinors . When restricted to the subspace of bosonic configurations,
one is left with
This is because describes linearly realised symmetries. Thus, kappa symmetry and supersymmetry
transformations do generically not leave the subspace invariant.
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 .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
- Invariance under kappa symmetry. Consider the kappa-symmetry gauge-fixing condition ,
where stands for any field independent projector. This allows us to decompose the original
fermions according to
To preserve the kappa gauge slice in the subspace requires
This determines the necessary compensating kappa symmetry transformation as a function of
the background Killing spinors.
- Invariance under supersymmetry. Once the set of dynamical fermions is properly
defined, we ask for the set of global supersymmetry transformations preserving them
This is equivalent to
once Eq. (211) is taken into account. Projecting this equation into the subspace gives
No further information can be gained by projecting to the orthogonal subspace .
I will refer to Eq. (214) as the kappa symmetry preserving condition. It was first derived in . This is
the universal necessary condition that any bosonic on-shell brane configuration must satisfy to
preserve some supersymmetry.
Table 5: Set of kappa symmetry matrices evaluated in the bosonic subspace of configurations
Bosonic kappa symmetry matrix
In Table 5, I evaluate all kappa symmetry matrices in the subspace of bosonic configurations
for future reference. This matrix encodes information
- on the background, both explicitly through the induced world volume Clifford valued matrices
and the pullback of spacetime fields, such as , or
, but also implicitly through the background Killing spinors solving the supergravity
constraints , which also depend on the remaining background gauge potentials,
- on the brane configuration , including scalar fields and gauge fields, either
or , depending on the brane under consideration.
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,
- a set of constraints among dynamical fields and their derivatives, ,
- a set of supersymmetry projection conditions, , with being projectors, reducing
the dimensionality of the vector space spanned by the original .
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.