Gravity and Strings

162 N = 1, 2, d = 4 supergravities
general considerations. Now we expect the theory to have N(N − 1)/2SO(2) gauge fields
that we can still label A ij µ. Since the Z ij sare central, we do not expect the gravitinos to be
charged under the gauge fields, although they will be invariant under some sort of constant
SO(N) rotations. One may want to make the theory invariant under the local version of
these SO(N) rotations, gauging them, and then one would recover the gauged supergravities
(hence the name) we obtained by gauging the N-extended AdS superalgebra.
Now, to perform the Wigner–Inönü contraction, we need to choose a spinor representation
of SO(2, 3). There are two such representations, which are called electric and magnetic
representations, which are explicitly worked out in Appendix B.2.1. They are equivalent in
the sense that they are related by a similarity transformation and, obviously, they are just
two of an infinite family of equivalent representations. These two are, however, of special
interest. If we contract using the electric representation, we obtain, for the anticommutator
of two supercharges,
{Q α i , Q β j }=iδ ij γ a C −1 αβ Pa − i C −1 αβ Z ij , (5.63)
whereas, if we contract using the magnetic representation, we obtain
{Q α i , Q β j }=iδ ij γ a C −1αβ −1
Pa − γ5 C αβ ij
Z . (5.64)
As we advanced, the first surprise is that the central charges occur in this anticommutator,
but nowhere else. The second surprise is that the central charges occur in two different
ways. From the Poincaré point of view, in the electric case the Z ij s are scalars whereas in
the magnetic case they are pseudoscalars. How should we interpret these charges? If we
construct supergravity theories gauging the “electric” superalgebra, we will have to associate
gauge potentials with the Z ij s, which will be, then, interpreted as electric charges,
in agreement with their scalar nature. In the magnetic case, the Z ij s should be interpreted
as magnetic charges. The similarity transformation that relates the electric and magnetic
AdS4 representations becomes a chiral–dual transformation that rotates electric into magnetic
charges and vice-versa. In fact, we can write the most general anticommutator of the
supercharges including both kinds of charges of the most general N-extended Poincaré
superalgebra, 8
[Mab, Mcd] =−MebƔv(Mcd) e a − MaeƔv(Mcd) e b,
[Pa, Mbc] =−PeƔv(Mbc) e a,
α i Q , Mab = Ɣs(Mab) α β Qβ i ,
{Q α i , Qβ j }=iδijγ aC −1αβ Pa − i C−1αβ
ij −1 Q − γ5 C αβ ij P ,
(5.65)
and this anticommutator (and the full superalgebra) will be invariant under the chiral–dual
(electric–magnetic-duality) transformations which we expect to be symmetries of the Nextended
Poincaré supergravity theories, but not of the N-extended AdS supergravities.
8 In a Weyl basis, the electric and magnetic charges are combined into a single complex central charge matrix.