ModMax: Nonlinear Extension of Maxwell Electrodynamics

• ModMax is a one-parameter nonlinear extension of Maxwell electrodynamics that preserves both conformal invariance and electric–magnetic duality using the deformation parameter γ.
• ModMax employs nonlinear constitutive relations combining electromagnetic invariants, leading to effective screening factors, exact classical solutions, and anisotropic wave propagation.
• ModMax connects to Born–Infeld theory and supersymmetry, offering insights into quantum corrections, black-hole phenomenology, and effective field theory limits.

Modified Maxwell, usually abbreviated ModMax, is a one-parameter nonlinear extension of four-dimensional Maxwell electrodynamics distinguished by the simultaneous preservation of continuous electric–magnetic duality and conformal invariance. In the conventions used in several core papers, one defines $S=-\tfrac14 F_{\mu\nu}F^{\mu\nu}$ and $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$, and writes $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$, with real deformation parameter $\gamma$ and Maxwell recovered at $\gamma=0$; across the literature, equivalent formulas appear with different sign conventions for the invariants, but the defining structure is the same: a homogeneous, duality-compatible deformation built from a linear invariant and the square root $\sqrt{S^2+P^2}$ (Bandos et al., 2021, Nastase, 2021, Lechner et al., 2022). ModMax has consequently become a focal model in nonlinear electrodynamics, with developments spanning brane-inspired constructions, exact classical solutions, propagation in external backgrounds, black-hole physics, supersymmetry, and perturbative quantization.

1. Defining structure and uniqueness

ModMax is repeatedly characterized in the literature as the unique one-parameter nonlinear deformation of Maxwell theory in four dimensions that preserves both conformal invariance and continuous electric–magnetic duality. In the standard formulation, the only parameter is the dimensionless real coupling $\gamma$, with $\gamma=0$ giving the linear Maxwell theory and increasing $\gamma$ increasing the nonlinearity. Several works also emphasize that causality and unitarity considerations restrict attention to $\gamma\ge 0$, and one quantum review states that $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$0 allows superluminal signals (Hamil, 6 Jan 2026, Martin, 2024).

The uniqueness statement is structural rather than merely phenomenological. Conformal invariance follows from the homogeneity of the Lagrangian in the electromagnetic invariants, which implies a traceless energy–momentum tensor. Duality invariance is realized at the level of the equations of motion through nonlinear constitutive relations satisfying the Gaillard–Zumino condition. In this sense, ModMax occupies a distinguished position among nonlinear electrodynamics models: it is not simply another Born–Infeld-type deformation, but the only continuous one-parameter theory preserving the two hallmark symmetries of vacuum Maxwell theory simultaneously (Bandos et al., 2021, Hamil, 6 Jan 2026, Diaz et al., 2024).

A recurrent simplification in applications is the appearance of an effective screening factor. In purely electric or purely magnetic sectors with vanishing pseudoscalar invariant, the nonlinear equations reduce to Maxwell-type equations with an overall factor $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$1, and the electrostatic field of a point charge behaves as in Maxwell theory with an effective charge reduction $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$2 (Hamil, 6 Jan 2026).

2. Field equations, constitutive relations, and conserved tensors

For a general nonlinear electrodynamics $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$3, the dynamics is expressed through the constitutive tensor $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$4, or equivalently through a tensor $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$5 defined by differentiation with respect to $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$6. The equations of motion are then the generalized Maxwell equations together with the Bianchi identity, $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$7 and $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$8, or in curved spacetime $P=-\tfrac14 F_{\mu\nu}\tilde F^{\mu\nu}$9 and $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$0 (Bandos et al., 2021, Diaz et al., 2024, Hamil, 6 Jan 2026).

For ModMax, the derivatives of the Lagrangian with respect to $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$1 and $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$2 are explicit rational functions of $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$3, $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$4, and $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$5. This gives nonlinear constitutive relations in which $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$6 and $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$7 mix. The duality condition is satisfied identically in the standard formulation, and the symmetric energy–momentum tensor is traceless for all $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$8, establishing conformal invariance at the classical level (Bandos et al., 2021, Hamil, 6 Jan 2026).

The theory is often written in the language of electrodynamics in media. Around a background field one introduces the constitutive fields $$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}$$9 and $\gamma$0, with $\gamma$1 and $\gamma$2. This recasts the nonlinear field equations into Maxwell form in an effective medium whose response tensors depend on the background invariants. That viewpoint is central to the study of wave propagation, birefringence, and optical analogues in ModMax (Neves et al., 2022, Neves et al., 13 Feb 2026).

The same constitutive structure clarifies a common point of confusion. In special sectors with $\gamma$3, ModMax indeed reduces to Maxwell with a screening factor, but in generic backgrounds the constitutive relations are anisotropic and nonlinear. The full theory therefore cannot be reduced globally to a trivial rescaling of Maxwell electrodynamics (Neves et al., 2022, Neves et al., 13 Feb 2026).

3. Born–Infeld origin, reformulations, and formal extensions

A central result in the early literature is that ModMax arises as the infinite-tension limit of a duality-invariant one-parameter generalization of Born–Infeld theory. Starting from a Hamiltonian depending on a tension parameter $\gamma$4 and performing the Legendre transform, the limit $\gamma$5 yields precisely the ModMax Lagrangian. This identifies ModMax as the low-energy limit of a Born–Infeld-type theory rather than as an isolated algebraic ansatz (Nastase, 2021).

That same work constructs a brane-like “precursor” action by replacing the field strength inside the Born–Infeld determinant with $\gamma$6, where the scalar functions $\gamma$7 and $\gamma$8 are chosen so that the determinant expansion reproduces the ModMax structure. A Dirac–Born–Infeld-like coupling to a scalar field $\gamma$9 then follows immediately by inserting $\gamma=0$0 into the determinant. This precursor theory provides a bridge between ModMax and the DBI constructions familiar from D-brane physics (Nastase, 2021).

ModMax also admits auxiliary-field reformulations. One study rewrites the square-root Lagrangian using three auxiliary scalars $\gamma=0$1 or, equivalently, a single angular variable $\gamma=0$2 obeying $\gamma=0$3. These formulations suggest an effective-field-theory relation to Maxwell theory coupled to frozen axion–dilaton-like scalars (Lechner et al., 2022).

The formal extension program is broad. An $\gamma=0$4 supersymmetrization exists for ModMax and for its Born–Infeld-like generalization, preserving duality invariance and establishing superconformal invariance through super-Weyl invariance after coupling to supergravity. Higher-derivative photino interactions can be removed by an invertible nonlinear superfield redefinition (Bandos et al., 2021). A distinct line of work develops manifestly $\gamma=0$5-invariant expressions for the ModMax energy–momentum tensor, extends them to generalized Born–Infeld theory, and identifies derivative actions with irrelevant and marginal $\gamma=0$6-like deformations (Babaei-Aghbolagh et al., 2022). A nonrelativistic analogue, formulated on a Newton–Cartan background, preserves the full Galilean Conformal Algebra and provides a Galilean cousin of the same $\gamma=0$7–$\gamma=0$8 structure (Banerjee et al., 2022).

4. Exact classical solutions and charged sources

ModMax inherits and generalizes several classical solution sectors of Maxwell and Born–Infeld theory. In the scalar-coupled DBI-like precursor, static, spherically symmetric, purely electric configurations yield the same universal integral family that appears in ordinary DBI, with solutions interpolating continuously between the catenoid, the non-BPS BIon, and the BPS Callan–Maldacena BIon. In particular, the case $\gamma=0$9 reproduces the extremal BIon interpreted as a fundamental string ending on the deformed brane (Nastase, 2021).

Knotted null solutions survive as well. Ra~nada’s exact Maxwell configurations with nontrivial Hopf index and nonzero electric or magnetic helicity satisfy $\sqrt{S^2+P^2}$0 pointwise. Because of this null structure, they remain exact solutions not only of pure ModMax but also of the finite-$\sqrt{S^2+P^2}$1 precursor theory. In the ModMax constitutive relations one finds $\sqrt{S^2+P^2}$2 and $\sqrt{S^2+P^2}$3 once $\sqrt{S^2+P^2}$4, and the overall factor does not obstruct the Maxwell evolution of the knot configuration (Nastase, 2021).

Couplings to electric and magnetic sources reveal another atypical feature. In contrast to generic nonlinear electrodynamics, Liénard–Wiechert fields generated by a moving electric charge, a magnetic monopole, or a dyon remain exact ModMax solutions. The same analysis investigates how the nonlinearity affects the Coulomb law, the Lorentz force, Dirac and Schwinger quantization conditions, and the Compton effect, while also giving an auxiliary-scalar representation of the Lagrangian (Lechner et al., 2022).

These results show that ModMax is highly nontrivial without erasing classical structures familiar from Maxwell theory. Exact retention of BIon-type, knot-like, and Liénard–Wiechert sectors is unusual among nonlinear electrodynamics models and is one reason ModMax is repeatedly treated as a particularly rigid deformation.

5. Hyperbolicity, propagation in backgrounds, and optical effects

Wave propagation in ModMax is strongly background dependent. Expanding the theory to quadratic order around constant or uniform electromagnetic backgrounds yields effective permittivity, magneto-electric, and permeability tensors, together with modified dispersion relations, group velocities, and refractive indices. For perpendicular backgrounds $\sqrt{S^2+P^2}$5, the theory displays birefringence, and in the strong-magnetic, small-$\sqrt{S^2+P^2}$6 regime one obtains $\sqrt{S^2+P^2}$7 (Neves et al., 2022).

The pure magnetic-background case admits especially compact formulas. One recent analysis finds one trivial mode with $\sqrt{S^2+P^2}$8 and a nontrivial anisotropic mode $\sqrt{S^2+P^2}$9, so that $\gamma$0 and $\gamma$1. A plane wave traveling distance $\gamma$2 therefore accumulates a phase difference $\gamma$3, which reduces in the transverse, small-$\gamma$4 limit to $\gamma$5. The same work further studies Goos–Hänchen shifts at a dielectric/ModMax interface and complex Kerr rotation for perpendicular electric and magnetic backgrounds in the regimes $\gamma$6 and $\gamma$7 (Neves et al., 13 Feb 2026).

The nonlinear propagation problem is not merely kinematic. A geometric criterion based on the intersection of characteristic cones shows that ModMax with $\gamma$8 is symmetric-hyperbolic and therefore admits a well-posed Cauchy problem. In that analysis, one polarization mode propagates on the Maxwell light cone while the second mode is subluminal for $\gamma$9 (Diaz et al., 2024).

That same study proves ModMax versions of Bekenstein-type inequalities relating energy, charge, angular momentum, and size, and reports stable numerical simulations in the highly nonlinear regime. In slab symmetry, the evolutions exhibit wave-packet splitting, shock formation, and clear birefringence as $\gamma=0$0 grows, while numerically preserving the inequality $\gamma=0$1 (Diaz et al., 2024).

6. Coupling to gravity and black-hole phenomenology

When coupled to Einstein gravity, ModMax produces a family of charged black-hole geometries that are often algebraically close to Reissner–Nordström, with the principal deformation entering through the screened combination $\gamma=0$2. In the standard static dyonic solution one has $\gamma=0$3, with horizons at $\gamma=0$4. The same screened charge combination controls the photon sphere, shadow radius, weak deflection angle, and ringdown spectrum (Pantig et al., 2022, Guzman-Herrera et al., 2023, Sucu et al., 8 Aug 2025).

The optical sector around these black holes is richer than the metric alone suggests. In an effective-metric analysis of high-frequency propagation, one polarization sees the background metric while the other experiences an effective metric with $\gamma=0$5 rescalings in the temporal-radial and angular sectors. The resulting birefringence modifies null geodesics, deflection angles, redshifts, and candidate shadow radii; the same paper notes, however, that for an asymptotic observer the two polarization shadows have the same observed angular radius, with the remaining $\gamma=0$6 dependence entering through the screened charge term in the metric functions (Guzman-Herrera et al., 2023).

The asymptotically nonflat generalizations are extensive. Topological ModMax–(A)dS black holes have $\gamma=0$7 and have been studied via quasinormal spectra for scalar, electromagnetic, and Dirac perturbations, eikonal photon-sphere formulas, and Hawking emission rates (Panah et al., 2024). In four-dimensional Einstein–Gauss–Bonnet gravity, an exact purely electric ModMax black hole displays modified horizon structure, a minimum mass, stable black-hole remnants, changes in circular orbits and the innermost stable circular orbit, and linear stability in scalar-field quasinormal-mode analyses (Hamil, 6 Jan 2026). In $\gamma=0$8 gravity, exact electrically charged solutions satisfy the first law, admit analyses of heat capacity and Helmholtz free energy, and can be studied through HPEM thermodynamic geometry (Panah, 2024). In $\gamma=0$9 dimensions, BTZ–ModMax black holes provide exact analytic solutions with screened electric sector, thermodynamic first law, and overlapping local/global stability regions only for sufficiently large entropy (Panah, 2024).

Observational and phenomenological studies use these backgrounds to compute shadow diameters, lensing in vacuum and plasma, axion–plasmon corrections, greybody bounds, and neutrino propagation. Across these models, the parameter $\gamma$0 acts as a screening or damping factor in the charge sector while leaving distinctive signatures in thermodynamic phase structure, quasinormal frequencies, and optical observables (Pantig et al., 2022, Panah et al., 2024, Sucu et al., 8 Aug 2025).

7. Supersymmetric completion and perturbative quantization

The supersymmetric development of ModMax is unusually explicit for a nonlinear electrodynamics model of this type. An $\gamma$1 superspace action can be written in terms of the chiral field-strength superfield $\gamma$2, and after eliminating the auxiliary $\gamma$3 field one exactly recovers the bosonic ModMax Lagrangian. The resulting superModMax theory satisfies the Kuzenko–Theisen duality condition and is superconformal because its coupling to old-minimal supergravity is super-Weyl invariant. Higher-derivative photino couplings, although present in the raw component expansion, can be removed by an invertible nonlinear superfield redefinition that rewrites the action in Volkov–Akulov form times the bosonic ModMax Lagrangian (Bandos et al., 2021).

The perturbative quantum theory is less rigid than the classical one. Using the background-field method and dimensional regularization, the one-loop effective action vanishes for constant field-strength backgrounds because the relevant loop integrals are scaleless. For varying backgrounds, however, the first nontrivial one-loop correction is a logarithmically divergent nonlocal operator with tensor structure not contained in the classical ModMax Lagrangian. The resulting conclusion is that ModMax is non-renormalizable and should be regarded as an effective field theory requiring an infinite tower of higher-dimension counterterms in generic backgrounds (Martin, 2024).

This quantum result places an important limit on classical uniqueness statements. ModMax remains unique at the level of local classical nonlinear electrodynamics with conformal and duality symmetry in four dimensions, but its one-loop effective action in nonuniform backgrounds does not remain within the original functional form. The contrast between classical rigidity and quantum effective non-renormalizability is therefore one of the central themes in the current literature (Martin, 2024).