Potential Space Symmetries in Ernst-like Formulations of Einstein-Maxwell/ModMax-Scalar field Theories

Abstract: We complete the visible, hidden, sectorial, and discrete symmetries of Ernst-like potential spaces in stationary, axisymmetric Einstein-Maxwell-Scalar Field (EMSF) and Einstein-ModMax-Scalar Field (EMMSF) theories. In the real potential space ((f,ε,ψ,χ,κ)), we determine the exact visible symmetries and their solvable Lie algebra. We characterize the hidden symmetries on invariant subspaces: Ehlers acts in the gravito-rotational sector, while electric and magnetic Harrison transformations act in static electromagnetic sectors. In the frozen EMMSF regime, (v=v_0,\ w=w_0), we show how EMSF sectorial transformations are deformed in ModMax theory. We also show that coexistence of electric and magnetic sectorial Harrison transformations imposes (d w=0) and (d[(v^2+w^2)/w]=0), selecting precisely the frozen ModMax sector. We study the Hamiltonian formulation, Noether charges, and Casimir invariants of the sectorial algebras. In harmonic branches of the (A,B,C) one-forms, the affine geodesic energy is constant, so the quadrature for (k) is controlled by the affine-geodesic Hamiltonian. The functions (ω) and (A_\varphi) follow from Noether charges along the Killing directions of (ε) and (χ), and are written using dual harmonic functions. We examine the electric and magnetic Lewis-Weyl-Papapetrou frames and their discrete map, which sends (κ\mapstoκ^{-1}), (ψ\mapstoχ), (χ\mapsto-ψ), and (ε\mapstoε-ψχ). Finally, we apply the sectorial transformations to harmonic scalar--acuum Weyl seeds with independent gravitational and scalar harmonics. Frozen-ModMax Harrison maps generate charged branches, while Ehlers generates the gravito-rotational branch. For these solutions we give the final quadratures for (k), (ω), and (A_\varphi).

• The paper provides a comprehensive symmetry analysis, revealing both manifest and hidden (sectorial) transformations within Einstein–Maxwell/ModMax-scalar field theories.
• It constructs explicit solution-generating maps using electric, magnetic, and Ehlers transformations, demonstrating their role in integrating field equations via Hamiltonian invariants.
• The work unifies potential space geometry with scalar and nonlinear electromagnetic sectors, detailing how symmetry obstructions and duality maps constrain exact solution constructions.

Symmetry Analysis of Potential Space in Einstein-Maxwell/ModMax-Scalar Field Sigma Models

Introduction

The paper systematically investigates the symmetries—both manifest and hidden—of the real potential space in stationary, axisymmetric Einstein–Maxwell–scalar field (EMSF) theories and their nonlinear electromagnetic extensions characterized by ModMax electrodynamics (EMMSF). The analysis extends the Ernst potential formalism, incorporating scalar and nonlinear electromagnetic sectors and reconstructs the associated solution-generating transformations, with an explicit focus on the algebraic structure of their target-space metrics, associated Hamiltonians, Noether charges, and sectorial Casimir invariants. This framework allows a unified treatment of solution generation and provides a granular understanding of symmetry obstructions imposed by scalar couplings and nonlinearities.

Target-Space Geometry and Manifest Symmetries

The real potential space is parametrized by $(f, \epsilon, \psi, \chi, \kappa)$, where $f$ is the norm of the stationary Killing field, $\epsilon$ and $\chi$ are twist and magnetic potentials, $\psi$ is the electric potential, and $\kappa$ encodes the scalar field. The effective target space exhibits a nontrivial geometry, with metric couplings enforcing the constraint structure already familiar from the classical Ernst sigma model. The inclusion of a scalar induces a split of electric and magnetic sectors due to different scalar weights.

The explicit identification of manifest (visible) symmetries yields a solvable algebra, specifically a semidirect product $$\mathfrak{h}_3 \rtimes (\mathbb{R} \oplus \mathbb{R})$$ where the Heisenberg part is generated by the translations in $(\epsilon, \chi, \psi)$—the twist, magnetic, and electric sectors respectively—together with gravitational and scalar-electromagnetic dilatations. All visible symmetries are realized as explicit isometries of the target-space metric, provided certain target parameters $(v, w)$ (appearing in the ModMax deformation) are constant.

Sectorial (Hidden) Symmetries: Ehlers and Harrison Transformations

A central outcome is the exhaustive characterization of sectorial (hidden) symmetries, which do not act globally but are preserved on invariant subspaces of the potential space. The gravito-rotational (Ehlers) symmetry persists when the electromagnetic potentials are trivial ($\psi = 0, \chi = 0$). The electric Harrison transformation operates in the $f$0 sector ($f$1), and the magnetic Harrison transformation in the $f$2 sector ($f$3). The symmetry algebra in these subspaces is identified as $f$4 in the generic dilatonic case, contracting to the Heisenberg algebra $f$5 at special scalar coupling values.

A key technical result is the explicit construction and integration of these sectorial transformations, producing finite maps generalizing Harrison’s classical solution-generating formulas, including their deformation by ModMax parameters. The explicit commutator relations and finite maps facilitate the construction of families of (rotating or charged) solutions from vacuum seeds.

Crucially, the simultaneous existence of both electric and magnetic sectorial Harrison maps is possible only in the “frozen” ModMax regime, with $f$6 constant. The algebraic compatibility equations $f$7, $f$8 are shown to enforce this sector.

Obstructions and Enlargement of Full Hidden Symmetry Groups

The analysis of the root-structure and Cartan weights assigned to each potential demonstrates that, in the generic scalar-coupled theory, global extension of these sectorial transformations is obstructed. Only for special values (e.g., the Kaluza–Klein case with $f$9) does the system’s root structure close to an $\epsilon$0 (i.e., $\epsilon$1) pattern, restoring the full non-Abelian hidden symmetry group. In generic EMSF and EMMSF one obtains only a maximal solvable subalgebra. This analysis recovers and generalizes previous results on global symmetry enhancement in special cases, clarifying the geometric origin of the obstruction via the Cartan metric and root algebra.

Hamiltonian Structure, Noether Charges, and Casimirs

The target-space geodesic Lagrangian is formulated, with canonical momenta derived for all potentials. The associated geodesic Hamiltonian provides a natural quadratic invariant, shown to control the “energy” associated with affine geodesics in potential space. Every explicit target-space Killing vector produces a constant of motion—a Noether charge—providing a direct algebraic route to identifying invariants which control the physical reconstruction of the spacetime metric and gauge field.

For each sectorial algebra (Ehlers, electric Harrison, magnetic Harrison), the quadratic Casimir is explicitly constructed in canonical variables, manifesting as a commuting invariant with respect to all algebra generators. The mapping between these Casimirs and the physical metric functions (notably the Weyl function $\epsilon$2) is demonstrated by explicit calculation.

Harmonic Branch Reduction: Solution Generating and Quadrature Reconstruction

A significant technical result is the demonstration that, upon further reduction where all potentials become functions of a single harmonic function, the full system reduces to an affine geodesic in target space. This allows all field equations to be integrated to quadratures, with the affinely parametrized geodesic energy directly dictating the integration coefficient of the $\epsilon$3 function (responsible for reconstructing the conformal factor of the metric). The visible Noether charges determine the rotational potential $\epsilon$4 and the azimuthal electromagnetic potential $\epsilon$5, both via dual-harmonic functions.

Application to more general seeds—including those with two independent harmonic functions corresponding to fully decoupled gravitational and scalar sectors—shows that electric and magnetic Harrison transformations act with distinct weights, preserving the superposition property and allowing for broad generality in solution generation.

Lewis-Weyl-Papapetrou Frames and Electric-Magnetic Discrete Duality

The analysis is extended to both electric and magnetic Lewis-Weyl-Papapetrou frames, with the discrete duality map between these frames constructed explicitly in terms of potential-space variables. This map is shown to act nontrivially on the scalar field $\epsilon$6 in scalar-coupled theories, effectively mapping $\epsilon$7, $\epsilon$8 up to sign, and shifting $\epsilon$9. This duality is a canonical transformation, preserving the Hamiltonian and the Casimir structure, and its concrete utilization allows for the consistent construction and identification of dual solution orbits.

Explicit Sectorial Solution-Generating Maps

Sectorial solution-generating maps, parametrized by finite transformation parameters, are applied to scalar-vacuum Weyl seeds with two independent harmonic functions. The explicit construction of the three sectorial orbits—electric, magnetic, Ehlers-twisted—shows that each produces an exact, closed-form solution, with explicit expressions provided for the metric functions, electromagnetic and scalar potentials, and the Weyl function. It is demonstrated that the coefficients entering the $\chi$0-quadrature are unchanged by these transformations, highlighting the geometric invariance of the Hamiltonian interpretation.

Conclusion

The paper provides a comprehensive and explicit algebraic and geometric framework for analyzing the symmetries of real potential space in stationary axisymmetric Einstein–Maxwell–scalar and Einstein–ModMax–scalar field theories. The reconstruction of manifest and sectorial symmetries, detailed algebraic structure, Casimir invariants, and their implications for solution generation mark a significant advance in the systematic study of the associated gravitational sigma models. The explicit demonstration of Hamiltonian control in solution reconstruction, the identification of global symmetry obstructions, and the construction of canonical dualities offer robust tools for the generation and classification of solutions—in particular, in the context of scalar-coupled and nonlinear electrodynamics extensions. These developments are expected to be instrumental in further investigations of exact solutions, dualities, and integrability properties of extended gravitational theories.

Cited reference: "Potential Space Symmetries in Ernst-like Formulations of Einstein-Maxwell/ModMax-Scalar field Theories" (2605.17843).