Gravity and Strings

416 String theory
Table 14.1. A brane scan taking into account only scalar supermultiplets. N is
given by the quotient between the numbers given and M.
p =−1 p = 0 p = 1 p = 2 p = 3 p = 4 p = 5
d M d (d − 1) (d − 2) (d − 3) (d − 4) (d − 5) (d − 6)
2 1 8 4
3 2 12 8 4
4 4 16 12 8 4
5 8 16 8
6 8 16 8
7 16 16
8 16 32 16
9 16 32 16
10 16 32 16
11 32 32
and it can be solved in the cases represented in Table 14.1 [13]. There are five series of
solutions. Four of them (with NM = 32, 16, 8, and 4) are associated with the four division
algebras O, Q, C, and R, respectively. These series correspond to objects related by double
dimensional reduction,aswewill see.
There is another way to understand this result: if there is linearly realized worldvolume
supersymmetry, the worldvolume fields must fit in (p + 1)-dimensional scalar supermultiplets.
Each solution in the table corresponds to a scalar multiplet. There is agreement with
the fact that scalar multiplets exist only in up to six dimensions.
For us, it is interesting that spacetime supersymmetric string actions could in principle
be constructed in d = 3, 4, 6, and 10, provided that the action has κ-symmetry. Green and
Schwarz showed in [472] that the ten-dimensional action previously constructed by them in
[471] has κ-symmetry. The Green–Schwarz (GS) action is not just a straightforward generalization
of the superparticle action Eq. (14.28), because the kinetic term would not be
κ-symmetric by itself. The key to κ-symmetry is the addition of a (super-)Wess–Zumino
(WZ) term: the integral of a 2-form 2 such that 3 = d2 is (target) Poincaré- and supersymmetry
invariant [528]: 10
2 =−idX µ δ a µ ∧ ( ¯θ 1 Ɣaθ 1 − ¯θ 2 Ɣadθ 2 ) + ( ¯θ 1 Ɣadθ 1 ) ∧ ( ¯θ 2 Ɣ a dθ 2 ). (14.33)
The GS action is then given by
S=− T
d
2
2ξ √ |γ |γ ij (∂i X µ δa µ − i ¯θ I γ a∂iθ I )(∂ j X νδb ν − i ¯θ J γ b∂ jθ J
)ηab + T 2.
(14.34)
10 According to Table 14.1, this is an N = 2 theory, with two minimal (Majorana–Weyl spinors with 16 real
components) ten-dimensional spinors, θ 1 and θ 2 , with equal or opposite chiralities: type-IIB and type-IIA
strings, respectively. These theories can also be described with the RNS theory, but only after quantization.