Kiefer C. Quantum gravity

QUANTUM-GRAVITATIONAL ASPECTS 285
matically contained in string theory. Here we discuss other aspects which are
relevant in the context of quantum gravity.
One can generalize the Polyakov action (3.51) to the situation of a string
moving in a general D-dimensional curved space–time. It makes sense to take
into account besides gravity other massless fields that arise in string excitations—
the dilaton and the ‘axion’. One therefore formulates the generalized Polyakov
action as
S P ≡ S σ + S φ + S B
= − 1 ∫ (√
4πα ′ d 2 σ hh αβ ∂ α X µ ∂ β X ν g µν (X)
−α ′√ )
h (2) RΦ(X)+ɛ αβ ∂ α X µ ∂ β X ν B µν (X)
. (9.26)
The fields g µν (D-dimensional metric of the embedding space), Φ (dilaton), and
B µν (antisymmetric tensor field) are background fields, that is, they will not be
integrated over in the path integral. The fields X µ define again the embedding of
the worldsheet into the D-dimensional space which is also called ‘target space’.
An action in which the coefficients of the kinetic term depend on the fields themselves
(here, g µν depends on X) is for historic reasons called a non-linear sigma
model. This is why the first part on the right-hand side of (9.26) is abbreviated
as S σ . The second part S φ is, in fact, independent of the string parameter α ′ ,
since in natural units (where = 1), the dilaton is dimensionless. We emphasize
that (9.26) describes a quantum field theory on the worldsheet, not the target
space. For the latter, one uses an effective action (see below).
In string theory, it has been proven fruitful to employ a path-integral approach
in which the worldsheet is taken to be Euclidean instead of Minkowskian;
cf. Section 2.2. This has the advantage that the integral over the metric is better
defined. Polchinski (1998a, p. 82) presents a formal argument why the resulting
theory is equivalent to the original Minkowskian version.
In the Euclidean formulation, where σ 1 = σ and σ 2 =iτ, the starting point
would thus be
∫
Z = DXDh e −SP , (9.27)
where X and h are a shorthand for the embedding variables and the worldsheet
metric, respectively. Only these variables are to be integrated over. In order to
get a sensible expression, one must employ the gauge-fixing procedure outlined
in Section 2.2.3. The invariances on the worldsheet involve two local diffeomorphisms
and one Weyl transformation. Since h ab (σ 1 ,σ 2 ) has three independent
parameters, one can fix it to a given ‘fiducial’ form ˜h ab , for example, ˜h ab = δ ab
(‘flat gauge’) or ˜h ab =exp[2ω(σ 1 ,σ 2 )]δ ab (‘conformal gauge’). As discussed in
Section 2.2.3, the Faddeev–Popov determinant can be written as a path integral
over (anticommuting) ghost fields. The action in (9.27) has then to be replaced
by the full action S P + S ghost + S gf , that is, augmented by ghost and gauge-fixing
action.