Obstruction to a Double Field Theory Coordinate Interpretation

Determine whether the non-vanishing contraction \(\gamma^m{}_{\bar{C}}\gamma^{n\bar{C}}\) obstructs interpreting non-abelian uni-vector deformations of gauged supergravities as coordinate transformations in double field theory.

Background

The paper constructs non-abelian uni-vector deformations of gauged supergravity solutions obtained through Scherk–Schwarz reductions of double field theory. The deformation is implemented by a matrix acting on the generalized metric, with γmCˉ\gamma^m{}_{\bar C} identified as Killing vectors of the undeformed background and constrained by the relevant gauge-algebra structure constants.

In every successful example considered, the contraction γmCˉγnCˉ\gamma^m{}_{\bar C}\gamma^{n\bar C} vanishes. The authors report that attempts involving a non-vanishing contraction, including heterotic supergravity, failed to produce valid solutions, and explicitly conjecture that this term may obstruct a coordinate-transformation interpretation in double field theory. Establishing whether this conjectured obstruction is genuine would clarify the geometric status and applicability of the deformation.

References

We also note that, in all the successful deformations presented below, the quantity $ \gammam{}_{\bar{C} \gamma{ n \bar{C}$ vanishes. Conversely, every attempt to deform a background for which this quantity is non-vanishing failed to yield a valid solution, including the case of heterotic supergravity. We conjecture that the presence of this term may obstruct the interpretation of the deformation as a coordinate transformation in double field theory. A detailed investigation of this possibility is left for future work.

— Non-abelian uni-vector deformations in gauged supergravities and non-abelian Einstein-Maxwell theories  (2608.18845 - Gubarev et al., 19 Aug 2026) in Section 3.2, “Non-abelian uni-vector deformation” (\ref{gaugedsugradef})

Also, in all the consistent examples we have constructed, the combination $\gammam{}_{\bar{C}\gamma{n\bar{C}$ vanishes identically; it is therefore plausible that this condition is necessary for the deformation to be well-defined and to allow a geometric interpretation in double field theory. Clarification of this point remains an open question.

— Non-abelian uni-vector deformations in gauged supergravities and non-abelian Einstein-Maxwell theories  (2608.18845 - Gubarev et al., 19 Aug 2026) in Conclusion and discussion, Section 4 (\ref{Concl})