Completing R4 using O(d, d)

A bstract The tree-level string effective action is known to contain quartic Riemann terms with coefficient ζ (3) α ′ 3 . In the case of the type II string this is the first α ′ correction. We use the requirement that the action reduced on a d -torus should have an O ( d, d ) symmetry to find the B-field couplings up to fifth order in fields. The answer turns out to have a surprisingly intricate structure.