Gravity and Strings

344 The Kaluza–Klein black hole
m φ
φ
^
A
^
(d−2)
A A
(d−3) (d−2)
Fig. 11.2. This diagram represents two different ways of obtaining the same result: generalized dimensional
reduction and “dual” standard dimensional reduction.
Now we dualize into a scalar field the (d − 2)-form potential: we add the term
1
(d − 1)(d − 1)!
d d x ɛF(d−1)∂ϕ, (11.235)
and eliminate F(d−1) by substituting into the action its equation of motion
F(d−1) = (−1) (d−1) k ⋆ Dϕ, (11.236)
obtaining the same result as with GDR. The two possible routes by which to arrive at the
same d-dimensional theory are represented in Figure 11.2.
Thus, the standard recipe for GDR is just a way to take into account all the fields
and degrees of freedom that can arise in the dimensional reduction. The new degrees of
freedom are discrete degrees of freedom described by a (d − 1)-form potential or by the
dual variable that can take the values Nm/ℓ, N ∈ Z and are associated with a choice of
vacuum.
Now, with the form  ( ˆd−2) we can associate a ( ˆd − 3)-brane. If one dimension is compact,
there are two possibilities: either one of the dimensions of the brane is wrapped around
the compact dimension or none is. From the d-dimensional point of view, the first configuration
looks like a ( ˆd − 4) = (d − 3)-brane and the second like a ( ˆd − 3) = (d − 2)-brane.
The ( ˆd − 3) = (d − 2)-brane has no dynamics and has only one degree of freedom: its
charge (or mass, which is usually proportional), which is the mass parameter that appears
in the d-dimensional action. The mass parameters are to be considered fields, although one
can equally consider them as expectation values of those fields. In this language we can say
that our vacuum contains a (d − 2)-brane. 31
The charge of the ( ˆd − 3)-brane can be associated with the monodromy of ˆϕ:
⋆
q ∼ ˆF ( ˆd−1) ∼
d ˆϕ ∼ mN. (11.237)
31 We have said that it actually does not in this academic example, although it will in more general cases.