Non-abelian uni-vector deformations in gauged supergravities and non-abelian Einstein-Maxwell theories

Abstract: We construct non-abelian uni-vector deformations of solutions in non-abelian Einstein--Maxwell theories and gauged supergravities, obtained as Scherk--Schwarz reductions of general relativity and double field theory, respectively. We provide examples of deformed backgrounds for both cases. We show that non-abelian deformations in Einstein--Maxwell theories can be presented as coordinate transformations in the parent theory, general relativity, extending the result of arXiv:2508.09637 for abelian uni-vector deformations.

• The paper develops explicit non-abelian uni-vector deformation rules for solutions of non-abelian Einstein-Maxwell theories and DFT-derived gauged supergravities, with examples including deformed AdS and flat backgrounds.
• The paper proves that, in the Einstein-Maxwell case, the deformation is a path-ordered coordinate transformation in the extended parent space, with path independence ensured by matching the Killing-vector and twist algebras.
• The paper identifies open directions including the DFT diffeomorphism interpretation, the condition that contracted Killing vectors vanish, the physical meaning of singularities, and possible holographic duals.

This paper extends the framework of uni-vector deformations of supergravity and Einstein--Maxwell solutions to the non-abelian setting. The construction applies to two parent setups: non-abelian Einstein--Maxwell theories with a dilaton (EM), obtained as Scherk--Schwarz reductions of general relativity in $\mathfrak{d}+\mathfrak{n}$ dimensions, and gauged supergravities obtained as Scherk--Schwarz reductions of O($\mathfrak{d}+\mathfrak{n}$,$\mathfrak{d}+\mathfrak{n}$) double field theory (DFT). Two central results are established: an explicit deformation rule that maps solutions of these theories to new solutions, and a proof that, in the EM case, the non-abelian deformation is a coordinate transformation (a diffeomorphism) in the extended space of the parent gravitational theory. The latter generalizes the abelian result of (Gubarev et al., 13 Aug 2025).

Setup: Scherk--Schwarz reductions of general relativity

The starting point is the Einstein--Hilbert action in $\mathfrak{d}+\mathfrak{n}$ dimensions, written in terms of a generalized vielbein $\check{E}^{A}{}_{M}$ and a flat metric $S_{AB}$, with cosmological constant $\Lambda$. The Scherk--Schwarz ansatz $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$, together with the requirement that the generalized fluxes $\check{F}_{AB}{}^{C}$ be independent of the internal coordinates $y$, constrains the twist matrices $\mathfrak{d}+\mathfrak{n}$0 so that only the internal block $\mathfrak{d}+\mathfrak{n}$1 is nontrivial and satisfies

$\mathfrak{d}+\mathfrak{n}$2

with $\mathfrak{d}+\mathfrak{n}$3 constant structure constants. The reduced theory is a non-abelian EM model with fields $\mathfrak{d}+\mathfrak{n}$4, a non-abelian field strength $\mathfrak{d}+\mathfrak{n}$5, and a covariant derivative $\mathfrak{d}+\mathfrak{n}$6; the action contains the characteristic flux-induced scalar potential terms built from $\mathfrak{d}+\mathfrak{n}$7 and the cosmological term.

Non-abelian uni-vector deformation in EM theories

The deformation requires a set of Killing vectors $\mathfrak{d}+\mathfrak{n}$8 of all background fields that realize the same algebra as the Scherk--Schwarz structure constants,

$\mathfrak{d}+\mathfrak{n}$9

The deformed metric $\mathfrak{d}+\mathfrak{n}$0 is generated by a triangular deformation matrix $\mathfrak{d}+\mathfrak{n}$1 with lower-left block $\mathfrak{d}+\mathfrak{n}$2. The resulting transformation rules for $\mathfrak{d}+\mathfrak{n}$3, $\mathfrak{d}+\mathfrak{n}$4, and $\mathfrak{d}+\mathfrak{n}$5 are explicit and polynomial in the Killing vectors, gauge fields, and internal metric.

Two worked examples demonstrate that the transformation is genuinely nontrivial. For Euclidean $\mathfrak{d}+\mathfrak{n}$6 with $\mathfrak{d}+\mathfrak{n}$7 and $\mathfrak{d}+\mathfrak{n}$8, the deformed metric acquires coordinate-dependent components mixing the $\mathfrak{d}+\mathfrak{n}$9 directions with denominators $\mathfrak{d}+\mathfrak{n}$0; the gauge fields become nonzero and the internal metric $\mathfrak{d}+\mathfrak{n}$1 develops a nontrivial profile. For $\mathfrak{d}+\mathfrak{n}$2 with $\mathfrak{d}+\mathfrak{n}$3, the deformed background carries nontrivial scalar invariants,

$\mathfrak{d}+\mathfrak{n}$4

confirming that the deformation does not merely relabel coordinates of the original solution. The authors note that the denominator $\mathfrak{d}+\mathfrak{n}$5 resembles the geometry of an expanding spherical brane with a horizon at $\mathfrak{d}+\mathfrak{n}$6, while explicitly flagging that a full brane interpretation requires further checks.

Deformations as diffeomorphisms in the parent theory

The most substantive result is the proof that the non-abelian deformation is a coordinate transformation in the extended $\mathfrak{d}+\mathfrak{n}$7 space of the parent $\mathfrak{d}+\mathfrak{n}$8-dimensional theory. A naive interpretation of the deformed line element as arising from $\mathfrak{d}+\mathfrak{n}$9 fails for non-abelian algebras: the integrability condition $\check{E}^{A}{}_{M}$0 holds only for abelian Killing algebras.

The resolution introduces the twist matrices $\check{E}^{A}{}_{M}$1 into the shifted differentials, $\check{E}^{A}{}_{M}$2. The integrability condition then becomes

$\check{E}^{A}{}_{M}$3

which holds precisely because the Killing vectors obey the non-abelian algebra. The coordinate transformation itself is given by a path-ordered exponential,

$\check{E}^{A}{}_{M}$4

which is well-defined (path-independent) only when the twist matches the Killing algebra, so that $\check{E}^{A}{}_{M}$5 acts as a flat connection. The proof that this transformation generates the deformation proceeds order by order in the twist expansion: defining the recursive quantities $\check{E}^{A}{}_{M}$6 and $\check{E}^{A}{}_{M}$7, an induction argument using the Killing equation $\check{E}^{A}{}_{M}$8 shows that all contributions at each order in $\check{E}^{A}{}_{M}$9 cancel, establishing $S_{AB}$0. In the abelian limit $S_{AB}$1, the construction reduces to the known abelian uni-vector deformation. An important structural point is that the deformation is only meaningful for metrics depending on internal coordinates through the Scherk--Schwarz form $S_{AB}$2.

Gauged supergravities from double field theory

The second half of the paper constructs the analogous deformation for gauged supergravities arising from Scherk--Schwarz reductions of O($S_{AB}$3,$S_{AB}$4) DFT. The reduced action is a standard gauged supergravity with fields $S_{AB}$5, an $S_{AB}$6-flux combining the $S_{AB}$7-field and Chern--Simons terms of the non-abelian gauge fields, and a scalar potential built from the structure constants $S_{AB}$8 of the O($S_{AB}$9,$\Lambda$0) twist.

The deformation acts on the generalized metric via a matrix $\Lambda$1 built from Killing vectors $\Lambda$2 satisfying $\Lambda$3, yielding explicit transformation rules for $\Lambda$4, $\Lambda$5, $\Lambda$6, and $\Lambda$7. Examples are given for Euclidean and Minkowski "flat" backgrounds (trivial metric and scalar sector but a linear dilaton $\Lambda$8), with $\Lambda$9. The Minkowski case is notable for exhibiting a genuine physical singularity at $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$0:

$$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$1

A striking empirical regularity is reported: in every successful deformation, the combination $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$2 vanishes identically, and every attempt to deform a background where it does not vanish—including heterotic supergravity—failed to produce a solution. The authors conjecture that this condition is necessary for a geometric (diffeomorphism) interpretation in DFT, but this remains unproven.

Limitations and open questions

Several caveats are stated explicitly in the paper. The brane-like interpretation of the $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$3 deformed background rests on a resemblance of the denominator $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$4 to an expanding brane worldvolume and is not substantiated by an embedding computation. The diffeomorphism interpretation is proven only for the EM case; its DFT counterpart—whether the generalized-metric deformation can be reformulated as a diffeomorphism in the doubled space—is left open, with the vanishing of $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$5 as a candidate necessary condition requiring clarification. The deformations are discrete rather than continuous: because the Killing vectors must realize the fixed structure constants $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$6 of the reduced theory, they cannot be rescaled to generate a continuous family of solutions, in contrast to the abelian case; this aligns the construction with T-, S-, and U-duality-type discrete transformations but obscures a direct link to the brane sedimentation mechanism of (Gubarev et al., 13 Aug 2025) and related work. Finally, the holographic interpretation of the deformed backgrounds—what operators or vacuum expectation values they correspond to on the dual QFT side—is not identified.

Sigma-model structure

As a step toward the integrability and holographic questions, the paper analyzes the particle sigma model on the deformed EM background. After introducing momenta $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$7 conjugate to the internal coordinates, the equations of motion for $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$8 decouple from $$\check{E}_{A}{}^{M}(x,y) = U_{N}{}^{M}(y)\, E_{A}{}^{N}(x)$$9 entirely: the twist matrices $\check{F}_{AB}{}^{C}$0 enter only through the structure constants. The result is a system describing a particle in the deformed background with a time-dependent effective charge $\check{F}_{AB}{}^{C}$1 obeying a non-abelian parallel-transport-like equation. The authors argue that this structure indicates the deformation preserves the integrable properties of the original sigma models and can be regarded as a non-abelian generalization of duality transformations, though this claim is supported by the decoupling structure rather than by an explicit Lax pair or spectral-curve analysis.

Conclusion

The paper delivers a consistent non-abelian extension of uni-vector deformations for both non-abelian EM theories and DFT-reduced gauged supergravities, with explicit deformed backgrounds for $\check{F}_{AB}{}^{C}$2, Euclidean $\check{F}_{AB}{}^{C}$3, and Euclidean/Minkowski "flat" spaces. Its principal contribution is the proof that the non-abelian EM deformation is a path-ordered-exponential coordinate transformation in the extended space of the parent gravitational theory, with path-independence guaranteed precisely by the matching of the twist to the Killing algebra. The open problems—the DFT geometric interpretation, the role of the condition $\check{F}_{AB}{}^{C}$4, the brane interpretation of the singular deformed backgrounds, and the holographic duals—define a concrete research agenda that follows directly from the results established here. All computations were verified with Mathematica and Cadabra, with the notebooks made publicly available.