Gravity and Strings

436 The string effective action and T duality
15.2.1 T duality in the bosonic-string effective action
T duality relates closed d-dimensional ˆ
string theories compactified on circles of relatively
dual radii. The effective-field theories will be dˆ − 1 = d-dimensional field theories for the
massless modes and the KK formalism that was developed in Chapter 11 is perfectly suited
to obtaining them from the effective actions of the uncompactified d-dimensional ˆ
effective
theories. 5
Our starting point is the action Eq. (15.1) with hats on every object, following the notation
of Chapter 11. We denote the compact coordinate by x ˆd−1 ≡ z ∈ [0, 2πℓs], and assume
that all fields are independent of it. We can use the standard KK Ansatz Eq. (11.33)
and the results concerning the spin connection, Eqs. (11.36) and (11.35), and volume element,
Eq. (11.37). Before substituting in the Einstein–Hilbert part of the action, we use the
d-dimensional ˆ
Palatini identity Eq. (D.4) with K = e−2 ˆφ and immediately obtain
d ˆ
d −2 ˆφ ˆx |ˆg| e ˆR = dz d ˆd−1 √
−2 ˆφ x |g| e k − ωb baωc c a − ωa bcωbc a
+ 2ωb ba∂a ln (e−2 ˆφ k) − 2∂a ln k ∂a ln e−2 ˆφ 1
− 4k2 F 2 (15.15)
(A) .
It is evident that the combination e −2 ˆφ k now plays the role of a d-dimensional dilaton,
and thus we define
The kinetic term for the dilaton gives
d ˆ
d −2 ˆφ ˆx |ˆg| e −4(∂ ˆφ) 2
=
φ ≡ ˆφ − 1
ln k. (15.16)
2
dz
d ˆd−1 √
−2 ˆφ x |g| e k −4(∂ ˆφ) 2
. (15.17)
On combining these two terms and using now the d-dimensional Palatini identity with
K = e−2φ ,weobtain, straightforwardly,
d ˆ
d −2 ˆφ
ˆx |ˆg| e [ ˆR − 4(∂ ˆφ) 2
] = dz d ˆd−1
−2φ
x |g| e R − 4(∂φ) 2
+ (∂ ln k) 2 − 1
4 k2 F 2 (A) .
(15.18)
The reduction of the KR 2-form is a bit trickier. First, we reduce the field strength by
identifying, in tangent-space indices,
where
ˆHabc ≡ Habc, ˆHabz = k −1 Fab(B), (15.19)
F(B) = 2∂ B, Bµ ≡ ˆBµz. (15.20)
5 Observe that the KK formalism may describe the massive KK modes of the string but not the massive
winding modes. The massless modes have zero momentum and winding number except at the self-dual
radius, at which there are additional massless modes with non-trivial KK momentum and winding number.
The effective action that we are going to write cannot describe the enhancement of symmetry that takes
place at the self-dual radius. A direct calculation of the string effective action for that radius is necessary.