O(D,D) and the string ${\alpha }^{\prime }$ expansion: An obstruction

Abstract

Double Field Theory (DFT) is an attempt to make the O(d,d) T-duality symmetry of string theory manifest, already before reducing on a d-torus. It is known that supergravity can be formulated in an O(D,D) covariant way, and remarkably this remains true to the first order in ${\alpha }^{\prime }$. We set up a systematic way to analyze O(D,D) invariants, working order by order in fields, which we carry out up to order ${\alpha }^{\prime 3}$. At order ${\alpha }^{\prime }$ we recover the known Riemann squared invariant, while at order ${\alpha }^{\prime 2}$ we find no independent invariant. This is compatible with the ${\alpha }^{\prime }$ expansion in string theory. However, at order ${\alpha }^{\prime 3}$ we show that there is again no O(D,D) invariant, in contradiction to the fact that all string theories have a quartic Riemann invariant with coefficient proportional to $\zeta (3)$. We conclude that DFT and similar frameworks cannot capture the full ${\alpha }^{\prime }$ expansion in string theory.

20 pages; v2: Clarifications and references added. Published version

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• Cosmological ${\alpha }^{\prime }$-corrections from the functional renormalization group
• Higher Derivative Supergravities in Diverse Dimensions
• Cosmology at the top of the ${\alpha }^{\prime }$ tower
• Supersymmetry, T-duality and Heterotic ${\alpha }^{\prime }$-corrections
• ${\alpha }^{\prime }$-corrections and their double formulation
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• Completing $R^4$ using $O(d,d)$
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