Abstract
Kaluza-Klein reductions of low energy string effective actions possess a continuous O(d, d) symmetry. The non-geometric elements of this group, parameterized by a bi-vector β, are not inherited from the symmetries of the higher-dimensional theory, but constitute instead a symmetry enhancement produced by the isometries of the background. The realization of this enhancement in the parent theory was recently defined as β symmetry, a powerful tool that allows to avoid the field reparameterizations of the Kaluza-Klein procedure. In this paper we further explore this symmetry and its impact on the first order α′-corrections. We derive the β transformation rules from the frame formulation of Double Field Theory (DFT), and connect them to the corresponding rules in the Metsaev-Tseytlin and Bergshoeff-de Roo supergravity schemes. It follows from our results that β symmetry is a necessary condition for the uplift of string α′-expansions to DFT.
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Acknowledgments
We warmly thank C. Hull and D. Waldram for hospitality and discussions at Imperial College, where part of this project was developed. We also thank O. Hohm for pointing out reference [17]. Support by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Agencia Nacional de Promoción Científica y Técnica (ANPCyT), Universidad de La Plata (UNLP) and Universidad de Buenos Aires (UBA) is also gratefully acknowledged.
Note added 1. Our results confirm those of our previous paper [2] and are in direct contradiction with supposed obstructions pointed out in [23]-v5 and [24]-v2. Contrary to what is stated in [24]-v2, we confirm here, through explicit computations in subsection 5.1, that the \( \mathcal{O} \)(α′) bulk Lagrangian of the heterotic string is exactly invariant under appropriate deformations of the β transformations. We also disagree with statements in [23]-v5, where it is asserted that there is a unique β transformation at \( \mathcal{O} \)(α′) that leaves the bulk bosonic action invariant and that it does not form a closed symmetry algebra. The transformation found in equation (31) of [23]-v5, corresponds to the particular choice a = b = −α′, \( \gamma =-\frac{\alpha^{\prime }}{2} \) and χ = 0 in our equations (5.13a)–(5.13b). Actually, β transformations at \( \mathcal{O} \)(α′) turn out not to be unique, but there is a two-parameter family of deformations in the bosonic string that lead to on-shell closure of the symmetry algebra.
Note added 2. After our manuscript appeared in arXiv:2307.02537 [hep-th], the author of [24] found an error in his calculations, and corrected the results, producing the revised outcome [25].
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Baron, W.H., Marqués, D. & Núñez, C.A. Exploring the β symmetry of supergravity. J. High Energ. Phys. 2023, 6 (2023). https://doi.org/10.1007/JHEP12(2023)006
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DOI: https://doi.org/10.1007/JHEP12(2023)006