Gravity and Strings

19.6 Intersections 565
Table 19.4. Elementary intersections of ten-dimensional
extended objects.
F1 S5, F1 ⊥ Dp(0),
S5 ⊥ S5(1), S5⊥ S5(3), S5⊥ Dp(p − 1) (p > 1),
Dp ⊥ Dp ′ (m), p + p ′ = 4 + 2m,
W F1, W S5, W Dp,
KK6 ⊥ Dp(p − 2)
Table 19.5. Elementary intersections of 11-dimensional extended objects.
M2 ⊥ M2(0), M2⊥ M5(1), M5⊥ M5(1), M5⊥ M5(3),
W M2, W M5,
KK7M M2, KK7M ⊥ M2(0), KK7M M5, KK7M ⊥ M5(1), KK7M ⊥ M5(3),
W KK, W ⊥ KK7M(2), W⊥ KK7M(4)
M2 ⊥ M5(1). The dimensional reduction along the intersection corresponds to
F1A ⊥ D4(0), which was discussed above.
3. A solution describing the supersymmetric M5 worldvolume soliton associated with
the intersection M5 ⊥ M5(3) was constructed in [568]. The worldvolume gauge field
is here the dual of an embedding scalar. These are present in any p-brane with
p < d − 1 and are (p − 1)-forms. They indicate the possibility of two p-branes intersecting
over a (p − 2)-brane.
Indeed, on T-dualizing the D1 ⊥ D3(0) in a direction parallel to the D3 and perpendicular
to the D1, we find D2 ⊥ D2(0). Tduality in directions transverse to both
branes generates another sequence of possible intersections, Dp ⊥ Dp(p − 2).
Had we dualized in a direction perpendicular to the D3 and parallel to the D1, we
would have generated D0 ⊥ D4(0) and then further T dualities would have generated
the sequence Dp ⊥ Dp + 4(p).
It is clear that we can go on generating new intersections via dualities. The results,
in terms of supergravity solutions (including gravitational waves and KK monopoles
[116, 669]), are summarized in Tables 19.4 and 19.5. Some of the intersections (named
overlaps in [421]) cannot be associated with excited worldvolume fields. They arise, in
fact, in degenerate limits of intersections involving more than two branes. For instance, the
M5 ⊥ M5(1) intersection corresponds to an M2 ending on two M5s in a limit in which
these become infinitely close and the M2 disappears.
As we have mentioned, these intersections, seen as excited worldvolume configurations
(branes within branes), always preserve some supersymmetry. Actually, in general