T-duality building blocks for ${\alpha }^{\prime }$ string corrections

Abstract

T-duality has been shown to constrain the higher derivative corrections of string theory. We revisit the problem of understanding the T-duality constraints imposed on the ${\alpha }^{\prime }$ corrections using the language of a torsionful connection. We find a convenient way to express the $O(d,d)$ invariants in terms of linear combinations of the metric and B-field for general $d$-dimensional torus compactifications, which we then use to revisit the heterotic and bosonic ten-dimensional string action at order ${\alpha }^{\prime }$. We also comment on the four-point functions for corrections of order ${\alpha }^{\prime 3}$ in the type II string.

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