Gravity and Strings

13.2 Supersymmetric vacua and residual (unbroken) supersymmetries 373
13.2 Supersymmetric vacua and residual (unbroken) supersymmetries
In general, the solutions of a supergravity theory are not invariant under any of the (infinite)
supersymmetry transformations that leave the theory invariant. Those which are invariant
under some (always a finite number of residual or unbroken supersymmetries) aresaid to
be supersymmetric, BPS,orBPS-saturated. 3
Schematically, the local supersymmetry transformations take the form
δɛ B ∼ ɛ F,
δɛ F ∼ ∂ɛ + Bɛ,
(13.6)
for boson (B) and fermion (F) fields. We are interested in purely bosonic solutions since
these are the ones that correspond to classical solutions. 4 They are also solutions of the
bosonic action that one obtains by setting to zero all the fermion fields of the supergravity
theory, because this is always a consistent truncation. These bosonic actions are just wellknown
actions of GR coupled to matter fields (for instance, the Einstein–Maxwell theory
in the N = 2, d = 4 supergravity case).
According to the general definition, a bosonic solution will be supersymmetric if the
above transformations vanish for some infinitesimal supersymmetry parameter ɛ(x).Inthe
absence of fermion fields, the bosonic fields are always invariant, and it is necessary only
that the supersymmetry transformations of the fermion fields vanish:
δκ F ∼ ∂ɛ + Bɛ = 0. (13.7)
From the superspace point of view, this can be seen as invariance under an infinitesimal
super-reparametrization. Thus, by analogy with GR, this is called the Killing spinor equation
and its solutions can be seen as the product of an infinitesimal anticommuting number
ɛ and a finite commuting spinor κ called a Killing spinor that also satisfies the above equation.
There is a different Killing spinor equation for each supergravity theory but, since we
have defined it for purely bosonic configurations, it can be used without any reference to
supergravity or fermion fields.
What is the symmetry group generated by the Killing spinors? Clearly, it has to be
a finite-dimensional supergroup of which the Killing spinors are the fermionic generators.
The supergroup is part of the infinite-dimensional supergroup of superspace superreparametrizations
that includes all the local supersymmetry transformations, GCTs, etc.
However, where are the bosonic generators?
In the case of the isometry group of a metric, the structures of the finitedimensional
group and of the algebra of its generators are inherited from those of the
3 We focus on local supersymmetries, although it is evidently possible to define unbroken supersymmetry in
theories that are invariant only under global supersymmetry. For instance, in the context of super-Yang–Mills
theory, the Bogomol’nyi–Prasad–Sommerfield (BPS) limit of the ’t Hooft–Polyakov monopole discussed in
Section 9.2.3 has some unbroken supersymmetries. This is why supersymmetric solutions are sometimes
called BPS solutions. The reason why they are called BPS-saturated will be explained when we discuss
supersymmetry, Bogomol’nyi, orBPS bounds.
4 We observe only macroscopic bosonic fields in nature. However, technically, we could equally well consider
non-vanishing fermionic fields. Also, we can generate fermionic fields by performing supersymmetry
transformations on purely bosonic solutions.