Gravity and Strings

438 The string effective action and T duality
we call the KK scalar and vector field k and A and, in the other, we call them k−1 and B.
These two dual compactifications give the same d-dimensional string background. This is
abetter way to think about T-duality transformations because it is the one that generalizes
to type-II effective actions.
In both cases, it is easy to relate the d-dimensional ˆ
metric, KR 2-form, and dilaton of the
two dual string backgrounds (primed and unprimed): 6
ˆg ′ zz
= 1/ ˆgzz,
ˆB
ˆg ′ µz = ˆBµz/ ˆgzz,
′ µz =ˆgµz/ ˆgzz,
ˆB ′ µν = ˆBµν + 2 ˆg[µ|z| ˆBν]z/ ˆgzz,
ˆg ′ µν =ˆgµν − ( ˆgµz ˆgνz − ˆBµz ˆBνz)/ ˆgzz, ˆφ ′ = ˆφ − 1
ln |ˆgzz|.
2
(15.28)
These relations are known as Buscher’s rules [198–200] and relate two backgrounds with
one isometry that are completely equivalent7 from the string-theory point of view and, in
particular, are classical solutions of the string effective-action Eq. (15.1). If we set ℓz = ℓs,
we immediately obtain the relations Eqs. (14.61) and (14.62) between the moduli of the
two dual theories.
The rules were originally derived using the string σ -model, as we are going to do in
the next section (although at the classical level), but the effective-action method [121, 125,
130, 675] turns out to give the correct rules in a much simpler way. In Section 15.3 we will
study some simple examples of string solutions and T dualization using Buscher’s rules,
although string solutions and their duality relations are the main theme of Part III and we
will see many more examples in later chapters.
Buscher’s rules refer only to solutions with an isometry. 8 However, from the string point
of view, it seems that it should be possible to define T duality whenever strings can be
wrapped around non-contractible cycles. However, the only (partial) realization of this
more general duality has been achieved in [476].
To end this discussion on Buscher’s T-duality rules, let us make some important remarks.
1. These rules are valid only to lowest order in α ′ .
2. T duality does not commute with gauge transformations (reparametrizations or gauge
transformations of the KR 2-form).
3. In the presence of fermions, Buscher’s rules have to be formulated in terms of the
Vielbein instead of the metric. We have used the Scherk–Schwarz recipe, which employs
the Vielbein formalism, to derive the rules and one could draw the conclusion
6 These rules are valid only for the heterotic-string background fields (all in the NSNS sector) at lowest order
in α ′ .Athigher orders in α ′ one has to take into account the Yang–Mills fields and also corrections to these
rules [34, 126].
7 If the isometric direction is not compact or corresponds to an isometry with fixed points (a rotation instead
of a translation) so that strings cannot wrap around it, the stringy equivalence between the two solutions
related by Buscher’s rules need not be true. Still, the new configuration solves the string equations of motion
and it is another string solution [805].
8 It is clear, though, that they can be extended to the case of several mutually commuting symmetries (toroidal
compactifications). The rules follow from the results of Section 16.5.