Completing $R^4$ using $O(d,d)$

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.

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