Gravity and Strings

372 Unbroken supersymmetry
conserved quantities (the d components of the momentum and the d(d − 1)/2 components
of the angular momentum). QFTs in Minkowski spacetime are constructed preserving
Poincaré symmetry and the quanta of the fields will be particles defined by the
values of the invariants that can be constructed with the conserved quantities (mass and
spin).
It is natural to associate solutions with a maximal number of unbroken symmetries with
possible vacuum states of the QFT. These states will be annihilated by the operators associated
with these symmetries in the quantum theory. In GR with no cosmological constant, the
only maximally symmetric solution is the Minkowski spacetime (ten isometries in d = 4
dimensions). With a (negative) positive cosmological constant, the Minkowski metric is
not a solution and the only maximally symmetric solutions are the (anti-)de Sitter spacetimes
whose isometry group (SO(2,3)) SO(1,4) is also ten-dimensional. These are the only
maximally symmetric solutions of GR.
It is possible to define field theories in (anti-)de Sitter spacetime, but it is also possible
(albeit unusual) to do it in spacetimes with fewer isometries, except in higher dimensions:
for instance, we have studied in Chapter 11 Kaluza–Klein vacua that are the products of
d-dimensional Minkowski spacetime and a circle whose isometry group is considerably
smaller than that of (d + 1)-dimensional Minkowski spacetime, which is spontaneously
broken by the choice of vacuum. The spectrum of the KK theory is determined by the
unbroken symmetry group, and it is the spectrum of a d-dimensional theory with gravity.
The name spontaneous compactification could be applied to this and other cases in which
there is a classical solution that we associated with a vacuum in which the spacetime is a
product of a lower-dimensional spacetime and a compact space.
We can also consider other solutions of GR that asymptotically approach one of the three
vacua we just mentioned. As we have stressed repeatedly, solutions of this kind represent
isolated systems in GR. We can use the Abbott–Deser formalism of Section 6.1.2 to find
the values of the d(d + 1)/2 conserved quantities of those spacetimes which are associated
with the isometries of the vacuum (even if the solutions themselves do not have any isometry).
If we associate with the systems described by the asymptotically vacuum solutions
states of a QFT built over the associated vacuum state, then the generators of the symmetry
algebra have a well-defined action on them. 2 On the other hand, only the vacuum state
is annihilated by all those generators, corresponding to its invariance under all the isometries.
In particular, the vacuum state will be annihilated by the energy operator, and thus
(if we restrict ourselves to states with non-negative energy) it will be the state with minimal
energy. This point is problematic in de Sitter spacetimes, which compromises their
stability.
This association of solutions that approach asymptotically a vacuum and states of a quantum
theory on which the generators of the vacuum isometries act is a very fruitful point of
view that we will use extensively. It can be extended to less-symmetric vacua, defining its
ownclass of asymptotic behavior.
We are now ready to extend this concept to the supersymmetry context.
2 It should be stressed that this can be done for all the states corresponding to spacetimes with the same
asymptotic behavior. We cannot compare the energies of, say, asymptotically flat and asymptotically anti-de
Sitter spacetimes.