Gravity and Strings

5.4 Extended supersymmetry algebras 163
The main reason for this is that we do not know how to generalize electric–magneticduality
transformations to the non-Abelian setting and also that, in the gauged supergravity
theories, the gravitinos are electrically charged with respect to the gauge vectors but there
are no additional fields magnetically charged with respect to them.
The above result opens up the possibility that there are more general central charges in
the anticommutator of two supercharges that we have not considered at the beginning. We
consider this interesting possibility in the next section.
5.4.1 Central extensions
According to the Haag–Lopuszański–Sohnius theorem, [496], the above anticommutator is
the most general allowed if we impose the condition that our theory is Poincaré-invariant.
Let us, therefore, not require Poincaré invariance. It turns out that any (Poincaré orAdS)
superalgebra can be extended by including “central charges” with n antisymmetric Lorentz
indices and two SO(N) indices Z ij
a1···an [538]. Generically, they appear in the anticommutator
of two supercharges in the form
1 a1···an −1
γ C
n!
αβ ij
Za1···an , (5.66)
with the factor being necessary in order to have the right Hermiticity properties (which can
be a γ5 only in Poincaré superalgebras). These are not central charges in the strict sense
because they do not commute with the Lorentz generators. In fact, consistency implies
kl
Zc1···cn , Mab =−nƔv(Mab) e kl
[c1 Z|e|c2···cn] . (5.67)
The new central charge will be symmetric or antisymmetric in the SO(N) indices depending
on whether γ a1···an −1 C αβ is symmetric or antisymmetric in αβ since the full anticommutator
has to be symmetric under the simultaneous interchange of αβ and ij.
In four dimensions (and similarly in any dimensionality) it is easy to determine the symmetry
of the possible terms:
C −1 , γ5C −1 , γ5γaC −1 , γabcC −1 , γabcdC −1 , (5.68)
are antisymmetric. In fact the second and the fifth and the third and the fourth matrices are
related by Eq. (B.94). The symmetric matrices are
γaC −1 , γabC −1 , γ5γabC −1 , γ5γabcC −1 . (5.69)
The first and the fourth and the second and the third matrices are related by Eq. (B.94).
The most general anticommutator of the two central charges in d = 4 will, therefore, be
{Q α i , Q β j }=iδ ij γ a C −1αβ Pa + i C −1αβ [ij] −1
Z + γ5 C αβ ˜Z [ij]
+ γ a C −1 αβ Z (ij)
a
+ i γ5γ a C −1 αβ Z [ij]
a
+ i γ ab C −1αβ (ij)
Z ab + γ5γ ab C −1αβ ˜Z (ij)
ab . (5.70)
It is equally easy to determine the most general anticommutator of two supercharges in
the AdS case, but in this case the Jacobi identities do not allow for any central charge.
We are now going to study the two simplest examples of extended Poincaré and AdS
supergravity.