Gravity and Strings

19.2 String-theory extended objects from effective-theory solutions 541
either as a scalar (and then its Hodge dual, the RR 7-form Ĉ (7) ,has to be treated as a pseudotensor)
or as a pseudoscalar (and the RR 7-form Ĉ (7) has to be treated as a tensor). 11 The
second option (which is also the one we adopted for consistency when we defined the magnetic
RR potentials) was chosen by the authors of [435], who performed the Wick rotation
using the RR 7-form and treating it as a tensor. If Ĉ (0) is a pseudoscalar then it acquires an
extra factor of i in the Wick rotation: Ĉ (0) = i ¯ Ĉ (0)
¯S =
ˆg 2 B
16πG (10)
N
d 10 ¯ ˆx
| ¯ ˆg E|
¯ˆRE + 1
. The Euclidean action becomes
ˆϕ
e2
2
(∂e −ˆϕ ) 2 − (∂ ¯ Ĉ (0) ) 2
, (19.69)
Observe that the two scalars contribute with different signs to the action. Their “energy–
momentum” tensors will appear with opposite signs in the Einstein equation. Thus, one
can obtain a solution with flat spacetime by taking the derivatives of the two scalars to be
equal, up to a global sign. Then one need only solve the scalar equations. The D-instanton
solution takes the following form in the (unmodified) Einstein and string frames [435]:
d ¯ ˆs 2 E = e− ˆϕ 0 2 d ¯x 2
10 , d ¯ ˆs 2 s = H 1 2
Di d ¯x 2
10 ,
e −2 ˆϕ = e −2 ˆϕ0 H −2
Di ,
HDi = 1 + hDi
.
|¯x 8
10|
¯ Ĉ (0) =±e −ˆϕ0
H −1
Di − 1 ,
(19.70)
The value of hDi is the extrapolation to p =−1ofthe value of hDp Eq. (19.65). This
value can also be obtained via a T-duality relation with the D0-brane.
At first sight there is a singularity at ρ =|¯x 10|=0 (in the string frame). However, the
string metric is invariant under the reparametrization
ρ = h 1 4
Di / ˜ρ, (19.71)
which shows that, in the limit ρ → 0, one finds another asymptotically flat region identical
to the one at ρ →∞. The metric, therefore, describes a sort of Euclidean wormhole joining
the two asymptotically flat regions and is regular everywhere.
The value of the Euclidean action of the D-instanton can be calculated. In the modified
Einstein frame, and normalized in our conventions, it takes the value
I = 2π/gB = qDi/gB. (19.72)
The action Eq. (19.68) appears in other contexts in d = 10 dimensions. We have met it,
for instance, as a truncation of the N = 4, d = 4SUEGRA theory that arises in the toroidal
reduction of the heterotic-string effective action Eq. (16.160), but it appears in many other
reductions, as shown in [666]. It is possible to find instanton solutions for all of them that
are almost identical with the D-instanton solution, differing only in the harmonic function
which, in d > 2 dimensions, will be H = 1 + h/|xd| d−2 . The string-frame geometry will
11 This difference between these two options can be seen as the reason why Wick rotations and Hodge duality
do not commute.