Quantum Gravity

290 STRING THEORYWe have already emphasized above that the space–time metric, dilaton, andaxion play only the role of background fields. The simplest solution for them isg µν = η µν , B µν =0, Φ=const. = λ.It is usually claimed that quite generally the stationary points of S eff correspondto 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 stringtheoryvacua is also called the ‘landscape’ (Susskind 2003). A selection criterionfor the most probable wave function propagating on such a landscape backgroundis discussed in Mersini-Houghton (2005).It is clear from (9.39) that D = 26 is a necessary condition for the solutionwith constant background fields. Thus, we have recovered the old consistencycondition for the string in flat space–time. There are now, however, solutions of(9.39) with D ≠26andΦ≠ constant, which would correspond to a solutionwith 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 sinceit can be changed by a shift in the dilaton. The physical gravitational constant(in D dimensions) reads16πG D =2κ 2 0e 2λ . (9.41)Apart from α ′ -corrections, one can also consider loop corrections to (9.40). Sinceg 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,thetwo-loop approximation at order gc 2 ,andsoon. 3In Section 9.1, we saw that the graviton appears as an excitation mode forclosed strings. What is the connection to the appearance of gravity in the effectiveaction (9.40)? Such a connection is established through the ansatzg µν =ḡ µν + √ 32πGf µν(cf. (2.76)) and making a perturbation expansion in the effective action withrespect to f µν . It then turns out that the term of order f µν just yields the vertexoperator for the string graviton state (see e.g. Mohaupt 2003). Moreover, it isclaimed that exponentiating this graviton vertex operator leads to a ‘coherentstate’ of gravitons. The connection between the graviton as a string mode andgravity in the effective action thus proceeds via a comparison of scattering amplitudes.For example, the amplitude for graviton–graviton scattering from thescattering of strings at tree level coincides with the field-theoretic amplitude ofthe corresponding process at tree level as being derived from S eff . The reasonfor 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 inα ′ ∼ l 2 s . Since the string amplitude contains the parameter α ′ and the effective3 For open strings, odd orders of the coupling (g o)alsoappear.