Gravity and Strings

11.2 KK dimensional reduction on a circle S 1 307
because that would spoil U(1) gauge invariance. The scalar acts as a sort of local coupling
constant. In particular, its presence modifies the mass of the particle, which is no longer the
coefficient in front of the action: if the metric is asymptotically flat and K0 is the constant
value at infinity of K , then the mass is the coefficient in front of the action times K0. We
have already taken this into account in writing Eq. (11.66).
KK modes and also string-theory objects called “winding modes” and “D0-branes” that
we will study are examples of “K particles.” The former couple to the inverse of the KK
scalar, i.e. K = k−1 ,asweare immediately going to see. Winding modes couple to k directly,
K = k, and D0-branes couple to the dilaton e−φ in string theory.
Although we are going to explain this procedure (called direct dimensional reduction)
in full detail, it is worth stressing that we are not going to prove that the two actions are
completely equivalent. Rather, what we are going to prove is that all the solutions of the
first action are of the form of those of the second one for some value of the mass and charge.
If we take only one specific value of the mass and charge, we are reducing the system to
some sector with a given, fixed, momentum in the internal direction.
Our starting point is the action of a point-like massless particle given in Eq. (3.258),
which we rewrite here for convenience:
ˆS[ ˆX ˆµ (ξ), γ (ξ)] =− p
2
dξ γ − 1 2 ˆg ˆµˆν( ˆX) ˙ ˆX ˆµ ˙ ˆX ˆν . (11.68)
This action is usually said to be invariant under GCTs. In fact it is just covariant, since
one goes from one metric to a different (even if physically equivalent) one. This happens
typically when the action depends on potentials instead of field strengths. The infinitesimal
transformations giving ˜δS = 0 are
˜δ ˆX ˆµ = ˆX ′ˆµ − ˆX ˆµ =ˆɛ ˆµ ( ˆX),
˜δ ˆg ˆµˆν =ˆg ′
ˆµˆν ( ˆX ′ ) −ˆg ˆµˆν( ˆX) =−2 ˆgˆλ( ˆµ ∂ˆν)ˆɛ ˆλ .
Let us now consider infinitesimal displacements in the direction ˆɛ ˆµ ,
δˆɛ ˆX ˆµ =ˆɛ ˆµ ,
δˆɛ ˆg ˆµˆν =ˆg ˆµˆν( ˆX ′ ) −ˆg ˆµˆν( ˆX) =ˆɛ ˆλ ∂ˆλ ˆg ˆµˆν.
Using the formulae in Chapter 1, we find that the change of the action is now
δˆɛ ˆS =− p
2
(11.69)
(11.70)
dξγ − 1
2 Lˆɛ ˆg
˙ˆX ˆµˆν
ˆµ ˙ˆX ˆν . (11.71)
Thus, the action is invariant if and only if ˆɛ ˆµ =ˆɛ ˆk ˆµ , ˆɛ being an infinitesimal constant
parameter and ˆk ˆµ being a Killing vector. In other words, if the metric admits an isometry,
the above action is invariant under the above symmetry and there is a conserved quantity,
namely the momentum in the ˆk ˆµ direction:
ˆP =−pγ − 1 2 ˆk ˆµ ˆX ˆµ ,
˙ ˆP = 0. (11.72)