Gravity and Strings

15.1 Effective actions and background fields 431
gauge group it must be the effective action of the heterotic and type-I strings (but already
here the couplings of the 2-form are different in the two theories).
The fields of these effective theories are given in Table 14.3. The NSNS fields are sometimes
called the common sector since they occur in all of them, including the bosonic
(oriented)-string theories. The fields in the common sector are the metric gµν, associated
with the graviton, the KR 2-form Bµν, andthe dilaton φ whose vacuum expectation value
φ0 gives the string coupling constant g = e φ0 (see Eqs. (15.8) and (14.14)). The action for
the common sector in the string frame to be defined below is given (in d dimensions with
d = 10 and 26) by
S =
g2 16πG (d)
N
dd x √ |g| e−2φ
R − 4(∂φ) 2 + 1
2 H , (15.1)
2 · 3!
where, in our notation in which indices not shown are all antisymmetrized,
H = 3∂ B (15.2)
is the KR field strength, which is invariant under the gauge transformations necessary for
the consistent quantization of a massless 2-form field,
δB = 2∂, (15.3)
where µ is an arbitrary vector field. The overall factor e−2φ is associated with the genus-0
(tree-level) origin of these terms and the normalization is conventional. In particular, the
factor g2 (g is defined in Eq. (14.57)) compensates the factor e−2φ0 that appears in the
weak field expansion of the action around the vacuum gµν = ηµν,φ= φ0, soG (d)
N can be
interpreted as the d-dimensional Newton constant. Its value Eq. (19.26) can be determined
by using duality arguments that relate it to the string coupling constant and the string length
just as we determined the Newton constant in terms of the compactification radius of KK
theory in Eq. (11.97), as we will see in Section 19.3.
Observe that, up to normalization, the above action is also invariant under shifts of the
dilaton field that change its vacuum expectation value and thus g, which is a free parameter.
A potential for the dilaton could give it a mass and fix g, solving two problems simultaneously:
the determination of g and the existence of a massless scalar that couples
as the Jordan–Brans–Dicke field to all kinds of matter, inducing violations of the equivalence
principle [287, 288]. The massless KR 2-form that also couples to matter can also
be a source of violations of the equivalence principle. The KR 3-form field strength can
be understood as a completely antisymmetric dynamical torsion field, as we discussed on
page 132 and, in the same spirit, the dilaton can be understood as part of a non-metricity
tensor of the type considered by Weyl [834]. The above string effective action can then be
written as an Einstein–Hilbert action for a torsionful and non-metric-compatible connection
using Eq. (1.55) (see also Eq. (1.58)).
The RR fields are differential forms C (n) µ1···µn of even (odd) rank in the N = 2B (A)
theory and appear in the respective actions Eqs. (17.4) and Eq. (16.38) with no couplings