Gravity and Strings

8.1 Coupling a scalar field to gravity and no-hair theorems 217
F µ =∇ µ ϕ.Aswill be explained later for the electric charge, this is just the continuity
equation for the current F µ and suggests the definition of scalar charge
d 3 µ∇ µ ϕ, (8.12)
V
which will be locally conserved. The conservation of this current is associated via Noether’s
theorem with the invariance of the action under constant shifts of the scalar.
Second, the Bianchi-type identity ∂[µ∂ν]ϕ = 0can be rewritten in the form ∇µF µνρ = 0,
where we have defined the completely antisymmetric tensor F µνρ = (1/ √ |g|)ɛ µνρσ ∂σ ϕ.
With this definition it is possible to show that the line integral
1
3!
d
γ
1 µνρ F µνρ =
γ
dϕ, (8.13)
along the curve γ is conserved. Observe that, if γ is closed, the integral will only be different
from zero if ϕ is multivalued, for instance if ϕ is an axion (a pseudoscalar) that takes values
in a circle.
How should we interpret these charges? We will see later in this chapter that the electromagnetic
field Aµ has a natural coupling to the worldline of a particle with electric charge
q given by Eq. (8.53). The particle’s electric charge is given by the surface integral over a
sphere S 2 of the Hodge dual of the electromagnetic-field-strength 2-form Fµν. The particle’s
magnetic charge is given by the surface integral over a sphere S 2 of the electromagneticfield-strength
2-form. The electric charge is conserved due to the equation of motion and
the magnetic charge is conserved due to the Bianchi identity. A topologically nontrivial
configuration of the field is needed in order to have magnetic charge.
Potentials that are differential forms of higher rank couple to the worldvolumes of extended
objects: a (p + 1)-form potential A(p+1) naturally couples to p-dimensional objects
with a (p + 1)-dimensional worldvolume (we will explain how this comes about in Chapter
18). The electric charge is the integral over the sphere S d−(p+2) transverse to the object’s
worldvolume of the Hodge dual of the (p + 2)-form field strength F(p+2) = dA(p+1). The
magnetic charge would be the electric charge of the dual (d − p − 4)-dimensional object,
charged under the dual potential whose field strength is the Hodge dual of F(p+2).
Looking now at the above charges, we immediately realize that the charge defined in
Eq. (8.13) is the charge of a one-dimensional object (string) and the former Eq. (8.12) is
the charge of a “−1-dimensional object.” Such an object would be an instanton, defined in
Euclidean space and with zero-dimensional worldvolume. Then “charge conservation” is
not a concept to be applied to it. In both cases ϕ has to be a pseudoscalar.
Observe that, indeed, a line integral as Eq. (8.13) cannot measure a point-like charge because
we could continuously contract the loop γ to a point without meeting the singularity
at which the charge rests. The line integral has to have a non-vanishing linking number
with the one-dimensional object, which has to have either infinite length or the topology of
S 1 ; otherwise the integral would be zero by the same argument. The behavior of the scalar
field has to be ϕ ∼ ln ρ, where ρ measures the distance to the one-dimensional object in the
two-dimensional plane orthogonal to it.
Similar arguments apply to the definition Eq. (8.12) and ϕ ∼ 1/ρ 2 , where now ρ measures
the distance to the instanton in the four-dimensional Euclidean space.