Kiefer C. Quantum gravity

290 STRING THEORY
We have already emphasized above that the space–time metric, dilaton, and
axion play only the role of background fields. The simplest solution for them is
g µν = η µν , B µν =0, Φ=const. = λ.
It is usually claimed that quite generally the stationary points of S eff correspond
to possible ground states (‘vacua’) of the theory. String theory may, in fact,
predict a huge number of such vacua; cf. Douglas (2003). The space of all stringtheory
vacua is also called the ‘landscape’ (Susskind 2003). A selection criterion
for the most probable wave function propagating on such a landscape background
is discussed in Mersini-Houghton (2005).
It is clear from (9.39) that D = 26 is a necessary condition for the solution
with constant background fields. Thus, we have recovered the old consistency
condition for the string in flat space–time. There are now, however, solutions of
(9.39) with D ≠26andΦ≠ constant, which would correspond to a solution
with a large cosmological constant ∝ (D − 26)/6α ′ , in conflict with observation.
The parameter κ 0 in (9.40) does not have a physical significance by itself since
it can be changed by a shift in the dilaton. The physical gravitational constant
(in D dimensions) reads
16πG D =2κ 2 0e 2λ . (9.41)
Apart from α ′ -corrections, one can also consider loop corrections to (9.40). Since
g c is determined by the value of the dilaton, see (9.36), the tree-level action
(9.40) is of order gc
−2 . The one-loop approximation is obtained at order gc 0,the
two-loop approximation at order gc 2 ,andsoon. 3
In Section 9.1, we saw that the graviton appears as an excitation mode for
closed strings. What is the connection to the appearance of gravity in the effective
action (9.40)? Such a connection is established through the ansatz
g µν =ḡ µν + √ 32πGf µν
(cf. (2.76)) and making a perturbation expansion in the effective action with
respect to f µν . It then turns out that the term of order f µν just yields the vertex
operator for the string graviton state (see e.g. Mohaupt 2003). Moreover, it is
claimed that exponentiating this graviton vertex operator leads to a ‘coherent
state’ of gravitons. The connection between the graviton as a string mode and
gravity in the effective action thus proceeds via a comparison of scattering amplitudes.
For example, the amplitude for graviton–graviton scattering from the
scattering of strings at tree level coincides with the field-theoretic amplitude of
the corresponding process at tree level as being derived from S eff . The reason
for this coincidence is the vanishing of the Weyl anomaly for the worldsheet.
The coincidence continues to hold at higher loop order and at higher orders in
α ′ ∼ l 2 s . Since the string amplitude contains the parameter α ′ and the effective
3 For open strings, odd orders of the coupling (g o)alsoappear.