Gravity and Strings

18.1 Generalities 505
their equations of motion directly in the action (at least in this simple σ -model with no WZ
term). These are
and their solution is
k(I )µDi X µ = 0, (18.21)
C I i =−h IJ k(J) µ ∂i X ν gµν, (18.22)
where we have defined the metric h IJ (assumed to be invertible, which is not true in general,
butisinmany cases of interest),
In this case, we have on-shell
Observe that the matrix
h IJ = kI µ kJ ν gµν, h IJ h JK = δ I K . (18.23)
Di X µ = g µ ν − h IJ k(I ) µ
k(J) ν ∂i X ν . (18.24)
µ ν ≡ g µ ν − h IJ k(I ) µ µ
k(J) ν , ν ν ρ = µ ρ, (18.25)
projects onto the space orthogonal to the orbits of the isometry group:
µ νk(I ) ν = 0, ∀ I = 1,...,r, (18.26)
so r directions simply disappear from the σ -model action. We will see this more clearly
when we perform the target-space (direct) dimensional reduction of the gauged σ -model.
After eliminating the auxiliary vector fields (assuming that this was possible), the gauged
σ -model takes the form
S =− T(p)
2
d p+1 ξ |γ | γ ij ∂i X µ ∂ j X ν µν − (p − 1) , (18.27)
since gρσρ µσ ν = µν. Inthe case of just one isometry, we have seen in Section 11.2
that, in adapted coordinates, ˆ ˆµˆν is zero except for the ( dˆ − 1) × ( dˆ − 1) submatrix ˆµν
which is the metric in dˆ − 1 dimensions. The above σ -model (adding hats everywhere)
does not depend on the isometric coordinate Z and reduces simply to a σ -model with
( dˆ − 1)-dimensional target space.
In this simple case, then, the gauged σ -model with d-dimensional target space is actually
a σ -model with (d − 1)-dimensional target space in disguise, written in d-dimensional
covariant language. In more general cases it is not possible to eliminate completely the
non-physical degrees of freedom (such as Z), but the σ -model still has d − r degrees of
freedom.
Let us now consider the coupling of p-branes to other spacetime fields. The simplest and
more natural ones are the couplings to scalar fields (which act as local coupling “constants”)
and to (p + 1)-form potentials, the fields p-branes can be charged with respect to. Let us
start with (p + 1)-form potentials.