Gravity and Strings

18.1 Generalities 511
string in d = 10 dimensions is known as the solitonic 5-brane or S5-brane (also called
NS5-brane to distinguish it from the D5-brane that couples to the RR 6-form potential).
It is natural to use in all cases the string frame because it is the theory of strings that we
know how to quantize (even if imperfectly). From the string point of view, we can classify
all the extended objects that we are going to see in terms of the scalar couplings that appear
in their NG actions
Fundamental p-branes They do not couple to the dilaton (to lowest order in α ′ ), which
does not occur in the NG action
S (pf)
NG [X µ (ξ)] =−T(p)
d p+1 ξ |gij|. (18.54)
Their mass is independent of the string coupling constant g = e φ0 (φ0 being the constant
value of the dilaton at infinity). In d = 10 there is only one: the fundamental
string. In d = 11 the M2- and M5-branes can both be considered fundamental.
Solitonic p-branes They couple to the dilaton as follows:
d p+1 ξ e −2φ |gij|. (18.55)
S (ps)
NG [X µ (ξ)] =−T(p)
Their mass is proportional to g −2 , which is typical of standard solitons. In d = 10
there is only one: the S5-brane.
Dirichlet (D) p-branes They couple to the dilaton as follows:
d p+1 ξ e −φ |gij|. (18.56)
S (D)
NG [X µ (ξ)] =−T(p)
Their mass is proportional to g −1 . They are a new type of purely stringy solitons.
They occur only in d = 10 and lower dimensions and couple to RR potentials.
Momentum modes They are charged point-like objects that couple to the KK scalar k as
follows:
SNG[X µ
(ξ)] =−T(0) dξ k −1 |gξξ|. (18.57)
Their mass is proportional to k −1
0 , that is, to the inverse of the radius of the compact
dimension, and they couple to the KK vector. We have met them in several places.
We are going to see that d = 11 momentum modes can be seen as d = 10 D0-branes,
given the relation between the KK scalar k and the dilaton and between the KK vector
and the RR 1-form, Eqs. (16.35).
Winding modes8 They are charged point-like objects that couple to the KK scalar k as
follows:
SNG[X µ
(ξ)] =−T(0) dξ k |gξξ|. (18.58)
8 There are also winding modes associated with other branes wrapped on compact spaces. Here we refer only
to the string winding modes.