Gravity and Strings

434 The string effective action and T duality
action is given by
S =
1
16πG (d)
N
d d x
|gE| RE + 4
d − 2 (∂φ)2 + 1 −8
e d−2
2 · 3! φ H 2
− (d − 2)e 4
d−2 φ
. (15.13)
The solutions to these equations describe backgrounds (vacua) for bosonic-string theory
in which strings can be consistently quantized, to lowest order in α ′ and the string coupling
constant. The simplest is evidently ten-dimensional Minkowski spacetime, which should
remain a good vacuum to all orders 2 because all fields are trivial. Other vacua can be argued
to be exact and not to receive higher α ′ corrections due to their unbroken supersymmetry
and/or the vanishing of their curvature invariants as is the case with pp-wave solutions
[41, 129, 493, 558–60, 910] and the four-dimensional solutions of [250] which are based
on the classification of the four-dimensional metrics that have all the curvature invariants
vanishing [789]. The next step is to try to quantize string theory on these vacua (for instance
in the KG10 solution [694, 696]).
It is amusing to see that quantizing string theory in non-trivial backgrounds amounts to
finding the generalization of Pythagoras’ law for vibrating strings (arguably the first law
in the history of physics) and that the generalization is possible only for backgrounds that
satisfy the above generalization of the Einstein equations.
The effective action has been obtained perturbatively in both α ′ and the string coupling
constant g and, furthermore, in the low-energy (long-distance) approximation. As a general
rule, the results obtained working with it can be trusted as long as e φ