Gravity and Strings

19.5 String-theory extended objects from superalgebras 557
19.5 String-theory extended objects from superalgebras
In Chapter 5 we introduced supergravities as the gauge theories of the supersymmetry algebras.
These contain a great deal of information about the global and local symmetries of
each supergravity theory. When we studied four-dimensional Poincaré-extended supersym-
metry algebras, we saw that, associated with each of the possible “electric” central charges
Q ij , there was an SO(2) gauge potential A ij
µ whose gauge symmetry is generated by Qij .
They contributed to the gauge superpotential with a term 1
2 Aij µ Qij . The “magnetic” central
charges could be associated with the electric–magnetic dual potentials, which are not independent.
The central charges could be associated with electric and magnetic charges of
supergravity solutions in Chapter 13 and the superalgebra could be used to see whether the
solutions preserved any supersymmetries.
This correspondence between central charges and Abelian potentials holds for “quasi-
central charges” with Lorentz indices as well, and with each charge Z (p)
a1···ap we can associate
in the supergravity theory a (p + 1)-form potential A (p+1) that transforms under
Abelian gauge transformations. They contribute to the gauge superpotential with a term
(1/p!)A (p+1) µ a1···ap (p)
Z a1···ap. The electric–magnetic dual ( ˜p + 1)-form potential is associ-
( ˜p)
ated with a Z a1···a ˜p quasi-central charge that must also be present in the superalgebra. It
is clear that these quasi-central charges must be associated with p-brane solutions of the
supergravity theory [59] and that the superalgebra can be used to study their unbroken
supersymmetries.
This is a very powerful tool that can be used to determine which objects/states may exist
in a supergravity theory knowing just which quasi-central charges are algebraically allowed
in the anticommutator of the supercharges of a given superalgebra. 20 Here we are going to
write the superalgebras of the string/M-theory effective actions (supergravities) and we are
going to study some examples, following in part [905] and starting with the algebra of
d = 11 supergravity.
The superalgebra of d = 11 superalgebra (also known as M superalgebra)admits quasicentral
charges of ranks 1, 2, 5, 6, 9, and 10. The last three values are just the duals of the
first three. Therefore, the M superalgebra is usually written in the form
ˆQ α , ˆQ β
= c( ˆƔ â Cˆ −1 ) αβ c2 ˆP + â 2! ( ˆƔ â ˆb
Cˆ −1 ) αβ Z ˆ (2) c5
+
â ˆb 5! ( ˆƔ â1···â5Cˆ −1 ) αβ Z ˆ (5)
, (19.113)
â1···â5
with constants c, c2, and c5 that are convention-dependent and immaterial for our discussion.
21 We immediately recognize the momentum and the charges associated with the M2and
M5-branes. The gravitational wave is associated with the momentum, but what is the
charge associated with the KK monopole (KK7M)? Furthermore, is there a charge for the
KK9M? As a matter of fact, as we have stressed repeatedly, these KK-branes are not states
20 We found all the possibilities in d = 4inSection 5.4.1. In higher dimensions the analysis is almost identical.
21 This formula can be interpreted as a decomposition of a symmetric bi-spinor into Lorentz tensors. A consistency
check is provided by the counting of independent components on both sides of the equation:
33 × 32/2. Physically, this formula should be understood as an inventory of possibilities: for instance, the
superalgebra of N = 1, d = 10 supergravity admits quasi-central charges of ranks 1 and 5, but, physically,
we expect on the r.h.s. one rank-5 and two rank-1 charges: momentum and the string charge. The counting
on the two sides gives different results, but physically it is correct.