Gravity and Strings

504 Extended objects
Let us assume now that the metric admits an isometry group generated by the Killing
vectors k(I ) µ (X), I = 1,...,r,
[k(I ), k(J)] = f IJ K k(K ), (18.12)
and let us consider the following infinitesimal transformations (which are not infinitesimal
GCTs):
X µ → X µ ′ = X µ + ηI k(I ) µ ,
gµν(X) → gµν(X ′ ) = gµν(X) + ηI k(I ) λ (18.13)
∂λgµν,
where the ηI s are constant infinitesimal parameters. The variation of the action is
δηS =−T(p) d p+1 ξ |γ |η I γ ij ∂i X µ ∂ j X ν ∇(µ|k(I )|ν)
and vanishes if (as is assumed) the k(I ) satisfy the Killing equation ∇(µ|k(I )|ν) = 0.
Now we want to gauge this symmetry. The infinitesimal transformations will be
δη X µ = η I (ξ)k(I ) µ (X),
δηgµν(X) = η I (ξ)k(I ) λ ∂λgµν .
Observe that the Killing vectors transform as follows:
δηk(I ) = η (J) k(J) ν∂νk(I ) µ = η (J) µ
k(J), k(I ) + k(I ) ν∂νk(J) µ
= η (J) f JI K k(K ) µ + η (J) ∂νk(J) µ k(I ) ν .
(18.14)
(18.15)
(18.16)
To make the σ -model invariant under the above local transformations, it suffices to replace
the partial derivative of the worldvolume scalars X µ by the covariant derivative
Di X µ = ∂i X µ + C I ik(I ) µ , (18.17)
where we have introduced the non-dynamical worldvolume vector fields C (I ) i which transform
as standard gauge potentials:
δηC I i =− ∂iη I + f JK I C J iη K =−Diη I . (18.18)
The covariant derivative defined above transforms covariantly, that is, with no derivatives
of the gauge parameter:
δηDi X µ = η J ∂νk(J) µ Di X ν . (18.19)
The gauged σ -model action (without WZ term) then takes the form
S =− T(p)
2
d p+1 ξ √ |γ | γ ij Di X µ D j X ν gµν − (p − 1) . (18.20)
Since the worldvolume vector fields C a i are not dynamical (their derivatives do not occur
in the action), they play the role of Lagrange multipliers and may be eliminated by using