Are Genus Corrections in Effective Actions Invariant Under Buscher Rules?

It is well-established that the dimensional reduction of the classical effective action of string theory at any order of $\alpha'$ on a circle of arbitrary radius remains invariant under the higher-derivative extension of Buscher transformations. In this study, we extend 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 $\alpha'^3$ as derived by the S-matrix method. This result aligns with the general belief that quantum gravity has no global symmetry.