Kosmann derivative and momentum maps from a duality covariant framework

A covariant implementation of diffeomorphisms in the presence of local symmetries is a nontrivial aspect of gravitational theories. In Double Field Theory, this is achieved through the so-called generalized Kosmann derivative. In this work, we show that the generalized Kosmann derivative admits a natural formulation entirely in terms of generalized fluxes through the inclusion of a compensating term that plays the role of a generalized momentum map, yielding a fully determined and covariant operator that provides a covariant realization of generalized diffeomorphisms. When parameterized in terms of the field content of heterotic supergravity, the resulting symmetry transformations give rise to momentum maps at the supergravity level, offering a duality-covariant interpretation of these objects. This framework provides a natural setting for the construction of conserved currents and Noether charges in doubled geometry with internal symmetries, with direct implications for black hole thermodynamics and its higher-derivative corrections in a duality-covariant setting.

💡 Research Summary

The paper addresses the long‑standing problem of implementing diffeomorphisms covariantly in theories that also possess local internal symmetries, focusing on Double Field Theory (DFT) where T‑duality is made manifest. In ordinary gravity the Kosmann derivative supplies a covariant extension of the Lie derivative by adding a compensating Lorentz rotation built from the spin connection; its action vanishes on Killing vectors and thus preserves both the metric and the orthonormal frame. The authors ask how this construction generalises to the doubled geometry of DFT, where generalized diffeomorphisms mix conventional coordinate transformations with gauge transformations of the Kalb–Ramond two‑form, and where an additional double‑Lorentz symmetry acts on the doubled tangent space.

After a concise review of DFT (generalized coordinates (X^{M}=(x^{\mu},\tilde x_{\mu})), O(D,D) invariant metric (\eta_{MN}), generalized metric (\mathcal H_{MN}), and the strong constraint), the authors move to the flux formulation. In this picture the fundamental object is the generalized frame (E_{M}{}^{A}) and its associated spin connection (\omega_{M}{}^{AB}). Compatibility of the covariant derivative with the frame and vanishing torsion allow one to express the dynamics entirely in terms of the fluxes \