Gravity and Strings

558 The extended objects of string theory
of the uncompactified d = 11 theory and appear only after compactification. Still, we can
include them in the d = 11 superalgebra using dual charges and vectors k a , and l a that
project the charges in the compact direction. The two terms that correspond to the KK7M
and the KK9M and should be added are [666]
+ c6
6! ( ˆƔ â1···â6Cˆ −1 ) αβ ˆ
Z (7)
â1···â7
k â7
c9
+
9! ( ˆƔ â1···â9Cˆ −1 ) αβ ˆ
Z (8)
â1···â8
l . (19.114)
â9
The dimensional reduction of the M algebra should give the N = 2A, d = 10 superalgebra
with a charge for each of the known objects of this theory. We need only reduce the vector
indices (as we did in the reduction of d = 11 supergravity). Each of the standard quasicentral
charges gives two in d = 10: ˆPâ = ˆPâ, ˆP z = Z ˆ (0) , Z ˆ (2)
â ˆb = Z ˆ (2)
â ˆb , Z ˆ (2)
âz = Z ˆ (1)
â ,etc. The
non-standard ones give rise to three, for instance Z ˆ (7)
â1···â7 = Z ˆ (7)
â1···â7 , Z ˆ (7)
â1···â6z = Z ˆ (6)
, and
â1···â6
ˆ Z (6)
, corresponding, respectively, to the KK7A, the D6, and the KK6A. The
Z (7)
â1···zâ6 = ˆ
â1···â6
KK6A is the standard KK monopole. The KK7A is the solution one obtains by reducing
the KK7M (the M-theory KK monopole) in a genuine transverse direction (the harmonic
function is smeared by the usual procedure and then one solves for the vector field in the
metric [691]).
The result of the reduction is the N = 2A, d = 10 superalgebra generalized with the
inclusion of KK-brane charges:
ˆQ α , ˆQ β
= c( ˆƔ â ˆ
C −1 ) αβ ˆPâ +
+
n=2,5,6
n=0,1,4,8
cn
n! ( ˆƔ â1···ânCˆ −1 ) αβ ˆ
cn
n! ( ˆƔ â1···ân ˆƔ11 ˆ
C −1 ) αβ Z (n)
â1···ân
Z (n)
â1···ân
+ c5
5! ( ˆƔ â1···â5 ˆƔ11 ˆ C −1 ) αβ Z ˆ (6) ˆk â1···â5â6
â6
c6
+
6! ( ˆƔ â1···â6Cˆ −1 ) αβ Z ˆ (7) ˆl â1···â6â7
â7
+ c8
8! ( ˆƔ â1···â8Cˆ −1 ) αβ Z ˆ (7) c9
ˆmâ8 + â1···â7 9! ( ˆƔ â1···â9Cˆ −1 ) αβ Z ˆ (8)
ˆnâ9 . (19.115)
â1···â8
Let us now turn to the N = 2B, d = 10 superalgebra. It contains an SO(2) pair of chiral
supercharges labeled by i, j = 1, 2 and the charges that appear on the r.h.s. of their anticommutator
carry a pair symmetric or antisymmetric in these indices. The allowed ranks
for antisymmetric indices are 3 and 7 and those for symmetric indices are 1,5, and 9. The
charges with antisymmetric indices are proportional to σ 2 and those with symmetric indices
can be decomposed into a basis of symmetric 2 × 2 matrices: I,σ1 and σ 3 . The charges proportional
to σ 2 and I are invariant under SO(2), and charges proportional to σ 1 and σ 3 form
SO(2) doublets. Combining the latter into symmetric traceless charges denoted by (ij), the
algebra is usually written in the form
ˆQ i α , ˆQ
j β
= cδ ij ( ˆƔ â Cˆ −1 ) αβ ˆPâ + c1( ˆƔ â Cˆ −1 ) αβ Z ˆ (1)(ij)
+ c3
3! (σ 2 ) ij ( ˆƔ â1â2â3 Cˆ −1 ) αβ ˆ
+ c5
5! δij ( ˆƔ â1···â5 ˆ
C −1 ) αβ ˆ
Z (3)
â1â2â3
Z (5)
â1···â5
â
c5
+
5! ( ˆƔ â1···â5Cˆ −1 ) αβ ˆ
Z (5)(ij)
â1···â5
, (19.116)
where it is understood that the r.h.s. has to be projected over the positive-chirality subspace.