Gravity and Strings

246 The Reissner–Nordström black hole
and it transforms in the vector representation of the duality group, a subgroup of GL(2, R):
˜F = M F,
a
M =
−b
b cos ξ
=±λ
a − sin ξ
sin ξ
.
cos ξ
(8.132)
In this form we see that the duality group consists of rescalings and O(2) rotations of F.
Observe that, if we integrate the Hodge dual of the duality vector ⋆ F over a2-sphere at
infinity we obtain a charge vector whose first component is 16πG (4)
N q,inour conventions.
The second component will be, by definition, the magnetic charge p:
⋆ F =
≡ 16πG (4)
q
N q, q =
. (8.133)
S 2 ∞
16πG (4)
N q
p
p/(16πG (4)
N )
Although this transformation looks very simple written in terms of the electromagnetic
field strength Fµν,itisvery non-local in terms of the true field variable Aµ.Tosee this, we
simply have to use Eq. (8.31) to obtain an explicit relation between Aµ and the dual vector
field õ:
1
ρσ
ν
ɛµν
õ(x) =− dλλx √ ∂ρ Aσ (λx). (8.134)
|g|
0
This non-locality is, at the same time, what makes this duality transformation interesting
and the source of problems. To start with, the replacement of F by ⋆F is not a symmetry
of the Maxwell action because ( ⋆F) 2 =−F2 . The reason for this is that the transformation
should be done on the right variable, namely the vector field, but this is difficult to do.
Another possibility is to write an action that really is a functional of the field strength.
On this action, the above replacement can be performed and gives the right results. This
procedure is called Poincaré duality and we explain it in detail in Section 8.7.1.
Let us now see what modifications the coupling to gravity Eq. (8.58) produces. The main
difference is that we now have one more equation (Einstein’s). For our purposes, it is useful
to rewrite it in this form (see Section 1.6 and Eq. (1.126)):
Gµν − Fµ ρ Fνρ + ⋆ Fµ ρ⋆
Fνρ = 0, (8.135)
or, using the duality vector,
Gµν − Fµ ρ T Fνρ = 0, (8.136)
which makes it clear that only the O(2) subgroup leaves the Einstein equation invariant.
Out of this O(2) group, the parity transformation clearly belongs to a different class (if we
had N vector fields, it would belong to the O(N) group that rotates the vectors amongst
themselves). Thus, the classical electric–magnetic-duality group of the Einstein–Maxwell
theory is actually SO(2).
We are studying an Abelian theory without matter and therefore it has no coupling constant.
However, we could think of this U(1) gauge symmetry as part of a bigger, non-
Abelian, broken symmetry group and introduce a (dimensionless in natural units in d = 4)
coupling constant g that appears as a g −2 factor in front of F 2 in the action and that we will
not reabsorb into a rescaling of the vector field. The appropriate duality vector the integral