Abstract
It has been speculated in the literature that the effective actions of string theories at any order of $\alpha'$ should be invariant under the Buscher rules plus their higher covariant derivative corrections. This may be used as a constraint to find effective actions at any order of $\alpha'$, in particular, the metric, the B-field and the dilaton couplings in supergravities at order $\alpha'^3$ up to an overall factor. For the simple case of zero B-field and diagonal metric in which we have done the calculations explicitly, we have found that the constraint fixes almost all the seven independent Riemann curvature couplings. There is only one term which is not fixed, because when metric is diagonal, the reduction of two $R^4$ terms become identical. The Riemann curvature couplings that the T-duality constraint produces for both type II and Heterotic theories are fully consistent with the existing couplings in the literature which have been found by the S-matrix and by the sigma-model approaches.
Highlights
String theory is a candidate for the quantum gravity which produces the classical supergravity at low energy
The stringy signature of the quantum gravity appears in the higherderivative corrections to the supergravity
The spectrum is invariant under the transformation in which the KaluzaKelin modes and the winding modes are interchanged, and at the same time, the set of scalar fields parametrizing the tours transforms to another set of scalar fields parametrizing the dual tours
Summary
String theory is a candidate for the quantum gravity which produces the classical supergravity at low energy. We will again find that the constraint almost fixes the effective action while leaving many parameters in the T-duality transformations at orders α0, α02, α03. In this case, there are gravity couplings which result from the Green-Schwarz mechanism [25]. We show that the gravity couplings which result from the Green-Schwarz mechanism are invariant under the T-duality transformations We show that this constraint fixes the residual T-duality parameters at order α0
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