Gravity and Strings

15.2 T duality and background fields: Buscher’s rules 439
that our results automatically imply a transformation rule for the Vielbein. However,
the rules involve only world tensors and they determine the transformation rules for
the Vielbeins up to (z-independent) local Lorentz transformations, and only by considering
T duality with fermions is the indeterminacy eliminated and one finds just
two possible transformation rules for the Vielbein [121]:
ê â ′ z =∓ê â z/ ˆgzz, ê â ′ µ =ê â µ − ( ˆgµz ± ˆBµz)ê â z/ ˆgzz. (15.29)
Both signs lead to the same Buscher rules for world tensors Eqs. (15.28). Now, if we
start with the standard gauge choice for the Vielbein Eq. (11.33), the two possible
T-dual Vielbeins are given by
â
ê
′
ˆµ =
eµ a ±k−1Bµ 0 ±k−1
,
êâ
ˆµ ′
=
ea µ
−Ba
. (15.30)
0 ±k
We will see in Section 17.4 that T duality in type-II theories requires the use of the
lower (“non-standard”) sign for it to work in the fermionic sector.
15.2.2 T duality in the bosonic-string worldsheet action
We can also gain some insight by studying T duality from the point of view of the twodimensional
σ -model that describes the motion of a string in a d-dimensional ˆ
spacetime
with a metric ˆg ˆµˆν and a KR 2-form ˆB ˆµˆν:
ˆS =− T
d
2
2 ξ |γ |γ ij ˆg ˆX)∂i ˆµˆν( ˆX ˆµ ∂ j ˆX ˆν + T
d
2
2 ξɛ ij ˆB ˆX)∂i ˆµˆν( ˆX ˆµ ∂ j ˆX ˆν . (15.31)
We do not include the dilaton term Eq. (15.8) since, in our purely classical approach, it
is not going to play any role at all. 9 As in the effective action, we assume that the spacetime
fields are independent of z = x ˆd−1 and, thus, the embedding coordinate Z appears
only through its derivatives. We then decompose the d-dimensional ˆ
fields into ( dˆ − 1)dimensional
fields using Eqs. (11.28), (15.20), and (15.22) and, on substituting into the
above, we obtain
ˆS =− T
2
d 2 ξ |γ | γ ij gij − k 2 F 2 + T
2
d 2 ξɛ ij
Bij + Ai B j − 2Fi B j , (15.32)
where gij, Bij, Ai, and Bi are the pull-backs of the d-dimensional metric, KR 2-form, KK
vector, and winding vector and where
is the field strength of Z, which is invariant under the shifts
Fi = ∂i Z + Ai, (15.33)
δZ =−(X), δ Aµ = ∂µ. (15.34)
9 The T-duality transformation rule of the dilaton is a quantum effect. We found it in the string effective action
because this action contains information about the quantum theory.