Gravity and Strings

568 The extended objects of string theory
considered as bound states with zero binding energy (or marginally bound states) and their
existence depends on whether it is possible to impose the simultaneous annihilation of that
state by the supercharges that annihilate those associated with each individual brane.
As we saw, the annihilation of a p-brane state by a given set of supercharges is entirely
equivalent to the action of a projector Pp of the generic form Eq. (19.117) on a spinor,
Ppɛ = 0. Then, the existence of a supersymmetric state composed of a p- andap ′ -brane
depends on the compatibility of the respective projectors Pp and Pp ′:itwill exist if
[Pp, Pp ′] = 0, (19.143)
and the state will preserve a quarter of the supersymmetries.
This equation depends on p and p ′ but also on the spatial orientation of the branes. A
general analysis is complicated because of the different Op that occur in the projectors.
Let us consider a simple example first: two p-branes of the same kind, S5A for simplicity,
extended along five Cartesian coordinates (so they are either parallel or orthogonal).
It is relatively easy to see that the two associated projectors commute if the number of
relative transverse dimensions (those which are parallel to one brane and transverse to the
other) is 0 mod 4, which leads to the allowed (supersymmetric) intersections S5 ⊥ S5(3)
and S5 ⊥ S5(1), which are included in Table 19.4. For Dp-branes Op = iI, iƔ11,σ 1 , iσ 2
depend on p mod 4 and the analysis of intersections between Dp-branes gives the allowed
intersection Dp ⊥ D(p + 4)(p) and, with a little more effort, the other cases in the table
[52, 346].
This analysis, which is essentially based on the spacetime supersymmetry algebra, allows
the study of more complicated intersections involving more branes [115] or non-orthogonal
intersections (branes at angles [128, 143, 182, 852, 904]). The inclusion of another brane
is allowed if its associated projector commutes with the other ones. The amount of unbroken
supersymmetry is generically halved each time a brane is included, except in the case
in which the projector of the additional brane does not impose any new constraint on the
spinor. The canonical example is that of a D5 in the directions 12345 and an S5B in the directions
12678, so they intersect in two directions. Their associated projectors PD5(12345)
and PS5B(12679) (in the obvious notation) commute, and
PD5(12345)PS5B(12679)ɛ ∼ PD3(129)ɛ = 0, (19.144)
and, thus, including a D3 in the directions 129 does not break any additional supersymmetry
(for any sign of the charge). This property gives rise to the phenomenon of D3-brane
creation when a D5 and an S5B cross [65, 107, 290, 370, 497]. Adding a fourth brane may
break all or no additional supersymmetry, depending on its charge’s sign (Section 20.1).
Another case in which no additional constraint is imposed is when we add a brane of the
same kind, but rotated by a supersymmetry-preserving angle (“branes at angles”) [143]. We
have no space to review this important case and we refer the reader to the literature.
19.6.3 Intersecting-brane solutions
The worldvolume and spacetime arguments that we have reviewed above suggest that classical
solutions of the low-energy effective string/M theories describing intersecting-branes