Gravity and Strings

312 The Kaluza–Klein black hole
their constant values at infinity). Thus, we can truly say that the above transformation is a
symmetry of the theory.
We could have considered similar transformations for the untilded fields. For instance,
the following transformation leaves the equations of motion invariant:
F ′ = k −2
d−1
+2
0 (k/k0) d−2 ⋆ F, k ′ = k −1 . (11.94)
However, this is not a symmetry of the theory. The above transformation inverts k0. If
we went back to the ˆd theory, we would find that the radius of the compact dimension is
inverted and that the ˆd-dimensional Newton constant does not have the same value.
The transformation Eqs. (11.93) is going to relate electric and magnetic objects in the
same theory. If a quantum theory with electrically and magnetically charged states is going
to make sense, all the possible pairs of electric and magnetic charges must satisfy the Dirac
quantization condition Eq. (8.170). The electric–magnetic-duality symmetry allows us to
generate magnetic charges from electric charges and we want the magnetic charges created
to be compatible with the original electric charges that we have shown the KK theory to
have. We defined the electric and magnetic charges of a solution in Eq. (11.63).
If we start with a field ˜F with electric charge ˜q and perform the electric–magnetic-duality
transformation above, we generate the following magnetic charge:
˜p ′
=−
S 2 ∞
˜F ′
=−
S 2 ∞
d−1
˜k
−2
d−2 ⋆ ˜F =−16πG (4)
˜q, (11.95)
where we have used the definition of ˜q in Eqs. (11.63). Then (ignoring the sign)
˜p ′ ˜q = 16πG (4)
N ˜q2 = 16πG (4)
N n2 /R 2 z , (11.96)
on account of Eqs. (11.88) and (11.32). This quantity will be an integer multiple of 2π if
Rz = 8G (4)
N /|m|, m ∈ Z. (11.97)
The existence of electric–magnetic-duality symmetry (so that each object and its dual
can coexist) requires the radius of the internal dimension to be of the order of the Planck
length.
Similar constraints on the sizes of the internal dimensions or the values of other moduli
can be found in string theory, requiring that each object and its U dual can coexist.
A non-trivial check of U duality is that the constraints on moduli obtained from different
dual object-pairs are consistent. We will see in Section 19.3, for instance, that the coexistence
of all ten-dimensional D-p-branes and their electric–magnetic duals implies the same
condition on the value of the ten-dimensional Newton constant.
We can say that, for values of the compactification radius, the theory can undergo a
duality transformation into another theory, but, for the “self-dual compactification radius,”
the theory enjoys an additional symmetry. U duality will become a symmetry for the “selfdual
values of the moduli.” In this language, there is an enhancement of symmetry at the
N