Gravity and Strings

510 Extended objects
We have introduced K0, the asymptotic value of K at infinity (assuming that the metric
gµν is asymptotically flat, at least in the directions transverse to the p-brane), so T(p) is the
physical p-brane tension 7 and usually is proportional to K0.
The coupling to K can always be changed or eliminated by a Weyl rescaling of the
spacetime metric. It is important to make clear in which Weyl conformal reference frame
we are writing the p-brane action. There are two special frames that can always be defined.
We use the fundamental string worldsheet action to illustrate the definitions.
(Fundamental) p-brane frames. These are defined as the Weyl conformal frames in which
the p-brane action does not couple to the scalar K .Atthe same time, all the terms in the
action for the spacetime fields should carry the same K -dependent factor [337].
For instance, by definition, the fundamental string worldsheet action Eq. (15.31) does
not depend on the dilaton when it is written in the string conformal frame. At the same
time, all the terms in the action for the spacetime fields that couple to the string Eq. (15.1)
carry the same e −2φ factor.
Dual p-brane frames. These are the frames in which the electric-magnetic dual ˜p-brane
would be fundamental. In the fundamental string case in d dimensions the KR 2-form is
dual to a (d − 4)-form potential C with field strength K = (d − 3)∂C and the dual object
has ˜p = d − 5(a5-brane in d = 10). On Poincaré dualizing the KR 2-form, we obtain
S =
d d x |g|e −2φ
R − 4(∂φ) 2 + (−1)(d−4)
2 · (d − 3)! e4φG 2
. (18.49)
g2 16πG (d)
N
Thus, gµv, the string metric, is not the metric to which the dual (d − 5)-brane naturally
couples. On performing a conformal transformation,
g = (1)−(d−5)g(d−5), (18.50)
and imposing that the dilaton factor is the same for all terms in the action, one obtains
and
S =
g2 16πG (d)
N
(1)−(d−5) = e 4
d−4 φ , (18.51)
d d x |g(d−5)|e 4
d−4 φ
R + (−1)(d−4)
2 · (d − 3)! G2
. (18.52)
In the frame g(d−5), bydefinition, the NG (d − 5)-brane action has no dilaton factors.
Then, going back to the string frame, we find
d d−4 ξ e −2φ |gij|. (18.53)
S (d−5)
NG [X µ (ξ)] =−T(d−5)
The dual of the fundamental string has, then, a tension proportional to g −2 (g being the
string coupling constant), as a typical solitonic object. The dual object to the fundamental
7 This is called in the literature effective tension. See footnote 4 on page 435.