Gravity and Strings

346 The Kaluza–Klein black hole
It is given by the action
11.5.3 Example 3: an SL(2, R)/SO(2) σ-model
ˆS =
d ˆd
ˆx |ˆg|
ˆR + 1
∂ ˆτ∂ˆτ ∗
2 (Im(τ)) 2
. (11.245)
This action is invariant under global SL(2, R) fractional-linear transformations of ˆτ =â +
ie −ˆϕ and, in particular, under constant shifts ˆτ →ˆτ + b, which act only on the real part.
Furthermore, we have argued that, in many cases, we should consider as equivalent two
values of τ related by SL(2, Z) transformations, which in particular means that a lives in a
circle of unit length. In this case, the σ -model metric is regular for finite values of ˆϕ.
The general recipe of GDR tells us to use the Ansatz
ˆτ(ˆx) = τ(x) + N
z,
2πℓ
(11.246)
and we obtain the action
S = d d x
|g| k R − 1
4k2 F 2 + 1
2 (∂ϕ)2 + 1
2 (Da)2 − 1
2 N
k 2 2πℓ
−2 e −2ϕ
,
(11.247)
where
Dµa = ∂µa + N
2πℓ Aµ, (11.248)
and there is invariance under the massive U(1) gauge transformations
δ Aµ = ∂µ, δa = N
. (11.249)
2πℓ
Global SL(2, R) invariance is now clearly broken and the KK U(1) gauge invariance is
also broken by the standard Stückelberg mechanism.
Let us look for a vacuum solution that will have Aµ = 0 and a = a0, aconstant. The
action for the remaining fields, in the Einstein frame, is
S =
d d x |g|
R + 2(∂χ) 2 + 1
2 (∂ϕ)2 − 1
2 N
e 2 2πℓ
−2
2 d−1
d−2 χ−2ϕ
. (11.250)
The potential for the remaining scalars has no lower bound, but we can still look for
(d − 2)-brane solutions. First, we diagonalize the potential by redefining the scalars:
χ ′
d − 1 2 ϕ
= χ +
3d − 5 2(3d − 5) 2
d − 2
,
ϕ ′
2 =−
2
d − 1 ϕ
χ +
2(3d − 5) 3d − 5 2
d − 2
,
(11.251)