Are genus corrections in effective actions invariant under Buscher rules?

Abstract It is well-established that the dimensional reduction of the classical effective action of string theory at any order of $$\alpha '$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>α</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> on a circle of arbitrary radius remains invariant under the higher-derivative extension of Buscher transformations. In this study, we investigate the extension of this symmetry to higher-genus levels. By leveraging the validity of Buscher rules for any genus of the world-sheet, we find that the measure of the effective action remains invariant only when reduced on a self-dual circle. Our findings indicate that the invariance of the Lagrangian density under the higher-derivative and higher-genus extension of the corresponding restricted Buscher rules does not yield the one-loop effective action at order $$\alpha '^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mo>′</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> as derived by the S-matrix method. This result aligns with the general belief that quantum gravity has no global symmetry.