Generalized ModMax Nonlinear Electrodynamics

• Generalized ModMax nonlinear electrodynamics is a framework that extends Maxwell’s theory with extra deformation parameters to control duality, conformal invariance, and causality.
• It features a multi-parameter Lagrangian that interpolates between ModMax, Maxwell, Born–Infeld, and Euler–Heisenberg models while ensuring regular field behavior and energy bounds.
• Applications span high-energy physics, gravitating systems, and early-universe cosmology, where the theory supports regular black hole solutions and non-singular cosmological evolution.

Generalized ModMax nonlinear electrodynamics (NLED) is a sophisticated framework that generalizes the unique conformal and SO(2) duality-invariant extension of Maxwell's theory—known as ModMax—by introducing additional deformation parameters. These generalizations yield broad families of nonlinear theories characterized by explicit control over conformal invariance, duality properties, and causal structure. The most extensively analyzed classes interpolate between the ModMax, Maxwell, Born–Infeld, and Euler–Heisenberg NLED, and support rich applications in high-energy theory, gravitating systems, and early-universe cosmology.

1. Foundational Principles and ModMax Theory

The starting point for Generalized ModMax theories is the unique one-parameter nonlinear Lagrangian discovered by Bandos, Lechner, Sorokin, and Townsend. In four-dimensional Minkowski space, one introduces the two electromagnetic invariants: $$S = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}, \qquad P = -\frac{1}{4} F_{\mu\nu} \widetilde{F}^{\mu\nu}$$ where $$\widetilde{F}^{\mu\nu} = \frac{1}{2} \varepsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}$$.

The ModMax Lagrangian is given by

$$\mathcal{L}_{\mathrm{ModMax}}(S,P;\gamma) = S\,\cosh\gamma + \sqrt{S^2 + P^2}\,\sinh\gamma$$

with %%%%1%%%% a real parameter. This theory satisfies:

• SO(2) electric-magnetic duality invariance
• Conformal invariance (traceless energy-momentum tensor)
• Maxwell recovery in the $\gamma\to0$ limit

Explicitly, the equations of motion are

$$\partial_\mu \left( \mathcal{L}_S F^{\mu\nu} + \mathcal{L}_P \widetilde{F}^{\mu\nu} \right) = 0$$

with the constitutive relations inheriting the duality symmetry via an SO(2) “rotation matrix” in the $(S, P)$ plane (Kosyakov, 2020, Ayón-Beato et al., 2024).

Uniqueness of ModMax is guaranteed by the combined conformal (tracelessness) and duality PDEs, whose only solution up to reparameterization is this form (Kosyakov, 2020, Ayón-Beato et al., 2024).

2. Generalized ModMax Lagrangians: Deformation Structure

Generalizations introduce further parameters controlling nonlinearity beyond the unique ModMax direction, while selectively retaining or relaxing the symmetries above.

A prominent example is the four-parameter family introduced by Kruglov: $$\mathcal{L}_{\mathrm{GMM}}(S,P;\beta, \lambda, \sigma, \gamma) = \frac{1}{\beta} \left\{ 1 - \left[ 1 - \frac{\beta}{\sigma} \mathcal{L}_{0}(S,P) - \frac{\beta\lambda}{2\sigma} P^2 \right]^\sigma \right\}$$ where

$$\mathcal{L}_0(S,P) = S \cosh \gamma + \sqrt{S^2 + P^2} \sinh \gamma$$

with parameters:

• $\beta > 0$ [(length)$^4$], controls higher-order nonlinearity,
• $\lambda \ge 0$ [(length)$^4$], proportional to $P^2$ contributions,
• $\sigma$ (dimensionless), interpolates analytic structure,
• $\gamma$ (dimensionless), the ModMax deformation (Kruglov, 2021).

Special limits reproduce standard theories:

• $$\beta \to 0,\ \lambda\to0,\, \gamma\to 0,\ \sigma\to1$$ $\to$ Maxwell,
• $\sigma=1,\,\lambda=0$ $\to$ ModMax,
• $\sigma=1/2,\,\lambda=\beta$ $\to$ Born–Infeld.

For $\sigma<1$, the electric field around a point charge is regular at $r=0$, avoiding divergences in the energy density (Kruglov, 2021, Sabido et al., 28 Jan 2026). Conformal invariance is lost unless $\sigma=1,\ \lambda=0$, but duality invariance can be maintained for $\sigma=1,\ \lambda=0$ (ModMax) and $\sigma=1/2,\,\lambda=\beta$ (BI–type).

Alternative parametrizations and root-$T\bar T$–like deformations, such as the Courant–Hilbert approach, also generate explicit duality-preserving multi-parameter families with strict convexity conditions ensuring causality and energy positivity (Babaei-Aghbolagh et al., 21 Nov 2025, Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022).

3. Duality, Conformal Symmetry, and Marginal/Irrelevant Deformations

Electric-magnetic duality is imposed via manifestly SO(2) or SL(2,$\mathbb{R}$)–covariant structures. For instance, the democratic Lagrangian approach employs pairs of electric and magnetic gauge potentials, plus auxiliary fields, with invariants constructed to guarantee duality via SO(2) invariance: $$g(\lambda_1, \lambda_2) = h\bigl( \sqrt{\lambda_1^2 + \lambda_2^2} \bigr)$$ which under further constraints yields the ModMax Lagrangian as the conformal-invariant, duality-symmetric solution (Avetisyan et al., 2021).

Higher-order or irrelevant (BI-type) deformations (parameterized by $\lambda$) can be systematically generated by root-$T\bar T$–flow equations. In Courant–Hilbert variables, duality invariance becomes the requirement that the Lagrangian be written as a function $\ell(\tau)$, with convexity (causality) reducing to

$\dot\ell(\tau)\geq1, \qquad \ddot\ell(\tau)\geq0$

Conformal invariance restricts $\ell$ to be linear (ModMax: $\ell(\tau)=e^{\gamma}\tau$), while non-linear choices interpolate between ModMax and Born–Infeld or logarithmic models (Babaei-Aghbolagh et al., 21 Nov 2025, Babaei-Aghbolagh et al., 2022).

From the point of view of solvable deformations, ModMax arises from a marginal $T\bar T$-like flow on Maxwell,

$$\frac{\partial \mathcal{L}}{\partial \gamma} = O_{T\bar T}^{(\gamma)}[\mathcal{L}]$$

while BI–type irrelevant operators deform ModMax to generalized Born–Infeld theories (Babaei-Aghbolagh et al., 2022). The interplay is clarified via the SL(2,$\mathbb{R}$)–covariant energy-momentum tensor and self-dual invariant actions (Babaei-Aghbolagh et al., 2022).

4. Hamiltonian Structure, Causality, and Energy Bounds

The Hamiltonian analysis, both in second-order and first-order Plebański variables, demonstrates that physical branches are strictly bounded below, with the effective Hamiltonian density

$$\mathcal{H}(D,B) = \sqrt{D^2 + B^2}\,\cosh\gamma - (D \cdot B)\,\sinh\gamma$$

for ModMax, and corresponding generalizations for deformed theories (Escobar et al., 2021, Kruglov, 2021). No ghost or superluminal modes arise provided convexity/positivity inequalities are satisfied,

$$\mathcal{L}_S > 0, \qquad \mathcal{L}_{SS} \ge 0, \qquad \mathcal{L}_{PP} \ge 0, \qquad \mathcal{L}_{SS}\mathcal{L}_{PP} - (\mathcal{L}_{SP})^2 \ge 0$$

(Bandos et al., 2021, Babaei-Aghbolagh et al., 21 Nov 2025).

Propagation of field discontinuities yields standard Maxwell light cones plus “extraordinary” modes on effective metrics: $$k^2 = 0,\qquad k^2 = f(D, H, \gamma)\, (F^{\mu\alpha}F^\nu_{\ \alpha}\,k_\mu k_\nu)$$ implying field-dependent birefringence and phase-velocity shifts in backgrounds (Escobar et al., 2021, Shi et al., 2024).

5. Gravitational Coupling and Black Hole Solutions

Generalized ModMax NLED coupled to general relativity yields nontrivial, often regular, black hole solutions. Examples include:

• Nonlinearly charged AdS black holes exhibiting van der Waals–type phase transitions, with singularity structure determined by the specific NLED model (Babaei-Aghbolagh et al., 21 Nov 2025).
• Accelerated AdS black holes in ModMax theory, with explicit dependence of thermodynamic quantities on the ModMax parameter $\gamma$ via redressing of electro-magnetic charges (Barrientos et al., 2022).
• Charged conformally dressed black holes (e.g., MTZ type) remain regular in core curvature invariants for suitable choices of model parameters (Ayón-Beato et al., 2024, Kruglov, 2022).

All constructions exploit the preservation of duality and conformal symmetry in the matter sector, allowing tractable generalizations of classic solutions.

6. Cosmological Applications and Field Regularization

Generalized ModMax NLEDs serve as early-universe models capable of avoiding cosmological singularities and supporting viable inflationary scenarios. In a spatially flat FRW metric, filling the universe with a purely magnetic GMM fluid ($\mathcal{G}=0$) leads to modified Friedmann equations with non-singular a(t):

• Initial energy density and pressure are finite,
• Early-time evolution exhibits de Sitter–like inflationary expansion,
• Late-time behavior matches standard radiation-dominated scaling,
• Spectral index and tensor-to-scalar ratio can be tuned to match Planck data for adjustable parameters $(\sigma, \beta, \gamma)$ (Sabido et al., 28 Jan 2026).

Regularity of Fisher–type singularities at the origin for point charges is ensured for $\sigma<1$, with finite maximum field strength and bounded energy. Corrections to Coulomb’s law and total self-energies in both electric and magnetic sectors are explicitly calculable (Kruglov, 2021).

7. Extensions: Supersymmetry and Higher-Form Generalizations

N=1 supersymmetric completions of ModMax and ModMax–BI theories have been constructed, preserving duality and (super-)conformal invariance (Bandos et al., 2021). The superspace action is manifestly duality-invariant, and higher-derivative photino terms can be eliminated via Volkov–Akulov–type superfield redefinitions.

The democratic formalism admits natural generalization to $p$-form gauge theories in $D=2p+2$ dimensions, with duality invariance imposed by requiring the Lagrangian depend only on appropriate SO(2)-invariant combinations of field strength bilinears. The on-shell equations reduce to twisted self-duality conditions for a single $(p-1)$-form (Avetisyan et al., 2021).

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Table: Limiting Cases of Generalized ModMax Lagrangians

Limiting Parameters	Theory Recovered	Principal Properties
$\sigma=1,\ \lambda=0$	ModMax	Conformal & SO(2) self-duality
$\beta\to0,\ \lambda\to0,\ \gamma\to0$	Maxwell	Linear, conformal, duality-invariant
$\sigma=1/2,\ \lambda=\beta$	Born–Infeld (BI)	SO(2) self-duality, finite energy
$\sigma < 1$	GMM, finite-energy field	Regular point-charge field, non-conformal
General $(\beta, \lambda, \sigma, \gamma)$	Generalized ModMax (GMM)	Parametrically controlled NLED

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• Democratic duality formalism (Avetisyan et al., 2021)
• Generalized ModMax constructions and applications (Kruglov, 2021, Sabido et al., 28 Jan 2026, Ayón-Beato et al., 2024, Babaei-Aghbolagh et al., 21 Nov 2025, Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022, Kruglov, 2022, Bandos et al., 2021, Escobar et al., 2021, Barrientos et al., 2022, Shi et al., 2024, Kosyakov, 2020)