Gravity and Strings

of whose dual over S 2 ∞
is 16πG(4)
N
8.7 Electric-magnetic duality 247
q is now
F ≡
g −2 F
⋆ F
. (8.137)
In terms of this duality vector, the Einstein equation can be rewritten as follows:
Gµν + Fµ ρ T
η ⋆ Fνρ = 0, η≡
0 1
−1 0
, (8.138)
and it is invariant under Sp(2, R) ∼ SL(2, R). Now, itcan be checked that, out of the full
group, and allowing for transformations of g, only the following transformations (rescalings
and Z2 duality rotations and their products) are consistent with the duality-vector constraint:
a 0
M = , g
0 1/a
′ = a−1g,
0 1
M = , g
−1 0
′ (8.139)
= 1/g.
Now we see the main reason why this duality is interesting: if the coupling constant g of the
original theory is large so perturbation theory cannot be used and non-perturbative states
become light, then the coupling constant of the dual theory g ′ = 1/g is small and can be
used to do perturbative expansions and the dual theory gives a better description of the same
phenomena and states. In particular, magnetic monopoles are typical non-perturbative states
of gauge theories with masses proportional to 1/g 2 and become perturbative, electrically
charged states of the dual theory.
Although, originally, electric–magnetic duality arose as a symmetry of the theory, a better
point of view is that it is a relation, a mapping, between two theories that describe the
same degrees of freedom in different ways. One of them can describe better one region of
the moduli space 24 than can the other. Dualities in which the coupling constant is inverted
and perturbative (weak-coupling) and non-perturbative (strong-coupling) regimes are related
go by the name of S dualities. Electric–magnetic duality in the Maxwell theory is
the simplest example. Perturbative dualities such as the O(N) rotation between the N vector
fields that we considered in Section 8.3 go by the name of T dualities, atleast in the
string-theory context. In some string theories (type II) the two kinds of dualities are part of
a bigger duality group (which is not just the direct product of the S and T duality groups)
which is called the U duality group [583].
A last comment on semantics: when talking about duality, there are always certain ambiguities
in the use of the word “theory.” Two theories that are dual are two different descriptions
of the same physical system and many physicists would say that they are, therefore,
the same “theory” written in different variables. We would like to call them different “theories”
describing the same reality. Both points of view are legitimate and are similar to the
active and passive points of view in symmetry transformations.
24 The coupling constant g and other parameters necessary to describe completely a theory are usually called
moduli. The space in which they take values is the moduli space of the theory.