Gravity and Strings

360 Dilaton and dilaton/axion black holes
When a = 0 this transformation is just the electric–magnetic-duality transformation
of the dilaton model ϕ ′ =−ϕ.
2. The action is invariant under the first two kinds of transformations: the rescalings of
τ can be compensated by opposite rescalings of F:
F ′ = 1
F, (12.49)
α
and the shifts of a simply change the action by a total derivative β √ |g|F ⋆ F. This is
just an Abelian version of the Peccei–Quinn symmetry. In the Euclidean non-Abelian
SU(2) case the total derivative is proportional to a topological invariant; namely the
second Chern class defined in Eq. (9.18) that takes integer values. If the Euclidean
action is properly normalized, the Peccei–Quinn transformation simply shifts it by β
times an integer, which results in a phase change in the integrand of the path integral.
Thus, the classical continuous Peccei–Quinn symmetry is broken to Z since the only
transformations that leave the path integral invariant are those with β = 2πn, n ∈ Z.
This is one of the quantum effects 6 that breaks SL(2, R) to SL(2, Z), the group of
S duality.
3. The equations of motion (but not the action) of the whole theory are also invariant
under SO(2) rotations. To see this (to check invariance under the whole SL(2, R)), it
is convenient to define the SL(2, R)-dual ˜F of the vector-field strength F:
˜Fµν ≡ e −2ϕ ⋆ Fµν + aFµν. (12.50)
The Maxwell equation is now the Bianchi identity of the S-dual field strength:
∇µ ⋆ ˜F µν = 0. (12.51)
It is convenient to define two S-duality vectors F and F,
⋆F
F ≡ ,
F
F ≡ e −ϕ
˜F
V F = ,
F
(12.52)
where V is the upper-triangular unimodular matrix that we defined in Eq. (11.208)
that satisfies VV T = M. F transforms covariantly under S ∈ SL(2, R):
F ′ = S F. (12.53)
The two components of this vector are not independent, but are related by a constraint
that involves τ. This constraint must be preserved by S and one can check that this
happens if, and only if, τ transforms according to Eqs. (11.205) and (11.206). The
transformation τ ′ =−1/τ interchanges the two components of the duality vector. For
avanishing axion field, this is the discrete electric–magnetic-duality transformation
of the dilaton-gravity model.
6 The other one is charge quantization.