Gravity and Strings

410 String theory
γij. The so-called Polyakov action 5 reads
SP[X µ (ξ), γij(ξ)] =− T
d
2
2ξ √ |γ | γ ij∂i X µ ∂ j X νgµν(X). (14.5)
The general variation of this action is given by
δSP = T d
2ξ √ |γ | δX µ
2 ν ij
gµν ∇ X + γ ∂i X ρ∂ j X σ Ɣρσ ν (g)
− T
d
2
2ξ √ |γ | δγ ij∂i X µ ∂ j X ν − 1
2γijγ kl∂k X µ ∂l X νgµν
− T di δX µ ∂i X νgµν. (14.6)
∂
Since there is no kinetic term for the worldsheet metric, its equation of motion just gives
the Rosenfeld energy–momentum tensor of the worldsheet fields X µ and tells us that it
must be zero. This equation is just a primary constraint that we can use to eliminate γij in
the Polyakov action: on writing it in the form
γkl = 2gkl/gi i , (14.7)
and substituting this into the Polyakov action we recover the Nambu–Goto action. Observe
that the equation of motion of the worldsheet metric γij tells us only that it is proportional
to the induced metric gij,but, in just two worldsheet dimensions, it is impossible to determine
the proportionality coefficient gi i = γ ijgij because this equation of motion (and,
hence, the energy–momentum tensor) is (off-shell) traceless. This property is related to an
additional symmetry of the Polyakov action for strings: invariance under Weyl rescalings
of the worldsheet metric,
γij → 2 (ξ)γij. (14.8)
This symmetry plays a crucial role in the quantization of the Polyakov action, allowing
one to gauge away the worldsheet metric completely and consistently: in two dimensions
it is always possible to put the metric in the conformal gauge γij ∝ ηij by using
reparametrizations. Then, with a Weyl rescaling, we can always obtain γij = ηij. This
symmetry is, however, potentially broken by anomalies that impose further restrictions
on the spacetime dimensions, metric, etc., in order to have consistent string theories (see
e.g. [832]).
Let us now consider the equation of motion of the X µ s. The boundary term can be nonvanishing
if we consider the propagation of open strings, with free endpoints and the topology
of a line segment (the alternative is to consider closed strings, with the topology of a
circle). In order to eliminate it, we have to impose special boundary conditions (BCs) for
open strings. There are two main possibilities (for each disconnected piece of the boundary
∂ (n) and for each embedding coordinate X µ )
5 It was Polyakov who first quantized it and its supersymmetric version (Eq. 14.23) using the path-integral
formalism in [784, 785].