S-duality in higher-derivative corrections of heterotic supergravity

This study examines the consistency of heterotic supergravity under T-duality when the $B$-field gauge transformation is rendered anomalous by the Green-Schwarz mechanism. We demonstrate that T-duality invariance mandates an infinite tower of higher-derivative couplings, all scaling as $e^{-2\mathrm{\Phi }}$. Within this tower, the couplings at orders ${\alpha }^{\mathrm{\prime }}$ and ${\alpha }^{\mathrm{\prime }2}$ are protected from quantum corrections, making them exact and therefore amenable to analysis under the S-duality between heterotic and type I string theory. Our results confirm that the standard S-duality map itself does not receive higher-derivative modifications. Leveraging this exact correspondence, we derive the explicit form of the type I effective action at order ${\alpha }^{\mathrm{\prime }}$ in a scheme that omits dilaton derivatives.