String Dualities at Order $\alpha'^{\,3}$

We compute the cosmological reduction of the fourth powers of the Riemann tensor claimed to arise in string theory at order $\alpha'^{\,3}$, with overall coefficient proportional to $\zeta(3)$, and show that it is compatible with an $O(9,9)$ symmetry. This confirms the general result in string theory, due to Sen, that classical string theory with $d$-dimensional translation invariance admits an $O(d,d)$ symmetry to all orders in $\alpha'$.