Gravity and Strings

19.1 String-theory extended objects from duality 521
ences on D-branes are the second volume of Polchinski’s book [779] and [64, 604, 780]
and Johnson’s book [605].
19.1 String-theory extended objects from duality
We have already met some of the extended objects of string theory: (fundamental) strings
(F1) and, in type-II and type-I theories, Dp-branes (Dp) with p = 0, 2, 4, 6, 8 for the type-
IIA theory, p = 1, 3, 5, 7, 9 for the type-IIB theory, and p = 1, 5 for the type-I theory.
Although Dp-branes have masses proportional to the inverse string coupling constant, their
existence has been inferred from the perturbative formulation of string theory, which was
reviewed in Chapter 14. It is not surprising that T duality, a perturbative string duality,
does not require the existence of any new extended objects in the theory: it just relates
Dp-branes to D(p ± 1)-branes and fundamental strings to fundamental strings in different
states. 1 These relations are represented from the viewpoint of the associated classical solutions
in Figure 19.4.1, which also contains many other relations and new objects required
by duality. 2
We have stressed, however, that non-perturbative S dualities require in general the existence
of new non-perturbative states dual to the ones present in the perturbative spectrum.
N = 2B, d = 10 SUEGRA has a global SL(2, R) symmetry and it was proposed in [583]
that this symmetry of the effective action reflects an S duality between type-IIB superstring
theories, which would be related by the discrete subgroup SL(2, Z), asdiscussed in Section
17.2. Let us consider systematically what the implications of the existence of this S duality,
fundamental strings, and Dp-branes (with p odd) in type-IIB superstring theory are.
Extended objects from type-IIB S duality. Fundamental strings couple to the KR 2-form potential
ˆB ˆµˆν, which is interchanged with the RR 2-form Ĉ (2)
ˆµˆν to which D1-branes (D-strings)
couple, by the S-duality transformation S = η.Thus, the S dual of the fundamental string is
the D-string and the two objects form an S-duality doublet, as represented in Figure 19.4.1.
This is not new information, but it fits nicely in the conjectured S duality.
Let us now consider Dp-branes beyond the D-string. The D3-brane couples to the 4-form
potential with self-dual field strength, which transforms into itself under S duality. Thus,
the D3-brane is an S-duality singlet. The D5-brane couples to Ĉ (6) , which is the electric–
magnetic dual of Ĉ (2) (D5-branes are the electric–magnetic duals of D-strings, but not their
S duals). Under S duality Ĉ (6) must transform into ˆB (6) , the electric–magnetic dual of ˆB.
We have to add to the string spectrum the 5-brane (the solitonic 5-brane, NS 5-brane,orS5brane)
that couples to ˆB (6) that we mentioned on page 510. The D5-brane and the S5-brane
1 To the standard Dp-branes with p ≥ 0wecanadd a IIB D(−1)-brane, the D-instanton, with zero worldvolume
directions and ten transverse Euclidean directions. It can be obtained by T dualizing the D0-brane in
the Euclidean time direction.
2 Less-conventional objects whose existence is also implied by string dualities are represented in Figure
19.4.1. On the other hand, the T duality between fundamental strings is represented in Figure 19.4.1
as T duality between gravitational-wave solutions and fundamental-string solutions, which we know represent
momentum and winding string states (Section 15.3). Waves and KK6 monopoles can be viewed as
electric–magnetic duals.