Gravity and Strings

444 The string effective action and T duality
and, on substituting this into the D ˆp-brane effective action and integrating over ζ ,weobtain
the effective action of a Dp-brane moving in d spacetime dimensions:
S =−TD ˆp2πℓ|c|k0
|M|. (15.59)
d p+1 ξe −(φ−φ0) (k/k0) 1 2 [1 − U +
i (M−1 )ikU −
k ] 1 2
Direct dimensional reduction of the Dp-brane effective action. Now we use hats and
primes (indicating that we are in the T-dual situation) for the spacetime fields which we
are going to reduce in the direction parametrized by z ′ , and primes but no hats for the
(p + 1)-dimensional worldvolume fields. We split the spacetime fields as in Eq. (15.54),
where
ˆH ′
ij ≡ˆg′ ij + ˆB ′ ij + 2πα′ F ′
ij = M′ ij
−′ +′
− Ui U j , (15.60)
M ′ ij ≡ Hij − k ′−2 B ′ i B′ j − B′ [i A′ j] ,
±′
U i ≡ (k′ F ′
i ± k′−1B ′ i ),
′
F i ≡ ∂i Z ′ + A ′ i .
(15.61)
M ′ ±′
ij , U i , and F i
i are exactly the Buscher T duals of the unprimed ones plus the relation
Z ′ = ˆVζ . (15.62)
Now, it only remains to see that the action we obtain after this reduction is equivalent to
Eq. (15.59). First, we rewrite ˆH ′
ij as the product of two matrices,
ˆH ′
ij = M′ ik [δkj − M ′−1 −′ +′
ik Uk U j ], (15.63)
the second of which has p times the eigenvalue +1 (for each of the p vectors orthogonal to
U +′ ) and one time the eigenvalue 1 − U +′ −′
U
i M′−1
ij
j for the eigenvector M ′−1 U −′ , and
det( ˆH ′
ij ) = det(M′ +′
ij ) [1 − U i (M ′−1 )ikU −′
k ]. (15.64)
Writing e −( ˆφ ′ − ˆφ ′ 0 ) = e −(φ′ −φ ′ 0 ) (k ′ /k ′ 0 )− 1 2 , the action takes the form
S =−T ′ Dp
d p+1 ξe −(φ−φ0) ′ ′
(k /k 0 ) − 1 2 [1 − U +′
i (M ′−1 )ikU −′
k ] 1
2 |M ′ |. (15.65)
This action is identical to Eq. (15.59) after application of Buscher’s rules if we identify
T ′ Dp = TD(p+1)2πℓ|c|k 1 2
0 = 2π RzTp+1. (15.66)
The Dp-brane tension Tp should be essentially independent of the compactification radius
(since it takes the same value in uncompactified spacetimes). It can depend solely on
ℓs and g. Since the string coupling constants of T-dual theories are related by Eqs. (14.61)
and (14.62) and since it has units of mass by p-volume, for all p
TDp ∼ 1
ℓ p+1 . (15.67)
s g
We will see later on other methods by which to find the same result and the proportionality
constant. The g −1 dependence of the tension of these objects gives them unique status,
intermediate between standard solitons, whose mass is proportional to g −2 ,andthe fundamental
objects that appear in the perturbative spectrum, with masses (tensions) independent