ModMax Electrodynamics

• ModMax electrodynamics is a one-parameter nonlinear extension of Maxwell theory that preserves four-dimensional conformal symmetry and SO(2) duality invariance.
• It is exactly solvable via a marginal T̄T-like deformation and has applications in gravitation, supersymmetry, and black hole physics.
• The theory ensures causal and birefringence-free propagation in homogeneous fields while modifying classical and quantum electromagnetic phenomena.

Modification Maxwell (ModMax)

Modification Maxwell (ModMax) theory is the unique one-parameter, conformal and duality-invariant nonlinear extension of standard Maxwell electrodynamics in four spacetime dimensions. Proposed initially by Bandos–Lechner–Sorokin–Townsend, ModMax modifies the Maxwell Lagrangian by introducing a dimensionless deformation parameter that simultaneously preserves the full SO(2) electric–magnetic duality group and four-dimensional conformal symmetry. The resulting theory is notable for its exact solvability via marginal $T\bar{T}$-like deformations, admits transparent couplings to gravitation and supersymmetry, and has nontrivial implications for classical and quantum field propagation, black hole physics, strong-coupling transport, and effective field theory constructions.

1. Definition, Lagrangian Structure, and Symmetries

The Lorentz-invariant building blocks of Maxwell’s field strength %%%%1%%%% are the two quadratic invariants: $$S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}, \qquad P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu},$$ where $$\widetilde{F}^{\mu\nu} = \tfrac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$ is the Hodge dual.

The ModMax Lagrangian density is

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

where $\gamma\ge 0$ is the ModMax parameter. For $\gamma=0$, this reduces to the standard Maxwell Lagrangian $\mathcal{L}=S$.

Key properties:

• Conformal Invariance: The theory is scale and conformally invariant in four dimensions; the stress-energy tensor is traceless, $T^\mu{}_\mu = 0$ (Lechner et al., 2022, Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022).
• SO(2) Duality Invariance: The equations of motion and constitutive relations are exactly invariant under electric–magnetic duality rotations:

$$\begin{pmatrix} F \ \widetilde{F} \end{pmatrix} \to \begin{pmatrix} \cos\alpha & \sin\alpha \ -\sin\alpha & \cos\alpha \end{pmatrix} \begin{pmatrix} F \ \widetilde{F} \end{pmatrix}.$$

(Bandos et al., 2021, Babaei-Aghbolagh et al., 2022).

• Causality and Unitarity: The Hessian of $\mathcal{L}_{\rm ModMax}(S,P)$ with respect to the electric field is positive-definite for all $\gamma \ge 0$, ensuring no ghosts/tachyons and causal propagation for small fluctuations (Bandos et al., 2021).

2. Marginal $T\bar{T}$-Like Deformation and Construction

The ModMax theory can be realized as the result of an exactly solvable marginal $T\bar{T}$-like deformation—specifically a flow generated by the composite bilinear operator $T^{\mu}{}_{\nu}T^{\nu}{}_{\mu}$ on Maxwell theory (Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022).

• Flow Equation:

$$\frac{\partial \mathcal{L}(y)}{\partial y} = \frac{1}{2}T^{\mu}{}_{\nu}T^{\nu}{}_{\mu},\quad \mathcal{L}(0) = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu},$$

with dimensionless flow parameter $y$ (identified with $\gamma$).

• Closed-Form Solution: The Lagrangian along the flow resums to

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

showing that ModMax is the unique one-parameter family of conformal, SO(2) duality-invariant interacting gauge theories arising from such a marginal deformation (Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022).

• Parallel for Scalars in 2d: Analogous flows in 2d (on free conformal scalar theories) yield strict analogs:

$$\mathcal{L}_{\rm SMM}(y) = \cosh y\,P_1 + \sinh y\,\sqrt{P_1^2 - P_2},$$

where $P_1, P_2$ are scalar kinetic invariants (Babaei-Aghbolagh et al., 2022).

• Interpretation: The marginality ensures that the deformation preserves scale/conformal invariance, and the cosh/sinh structure encodes exactly solvable nonlinear interactions that interpolate between the free and strongly interacting regimes.

3. Field Equations, Constitutive Structure, and Duality

The Euler-Lagrange equations for ModMax are nonlinear generalizations of Maxwell's equations: $$\partial_\mu G^{\mu\nu} = 0, \qquad \partial_\mu \widetilde{F}^{\mu\nu} = 0,$$ where the "excitation" tensor is

$$G^{\mu\nu} = -2\,\frac{\partial \mathcal{L}}{\partial F_{\mu\nu}} = \mathcal{L}_S F^{\mu\nu} + \mathcal{L}_P \widetilde{F}^{\mu\nu},$$

with

$$\mathcal{L}_S = \cosh\gamma + \sinh\gamma\,\frac{S}{\sqrt{S^2 + P^2}}, \qquad \mathcal{L}_P = \sinh\gamma\,\frac{P}{\sqrt{S^2 + P^2}}.$$

(Lechner et al., 2022, Bandos et al., 2021, Banerjee et al., 2022)

Constitutive relations for the macroscopic fields $(\mathbf{D},\mathbf{H})$ are nonlinear but preserve duality: $$\mathbf{D} = \mathcal{L}_F\,\mathbf{E} + \mathcal{L}_G\,\mathbf{B}, \qquad \mathbf{H} = \mathcal{L}_F\,\mathbf{B} - \mathcal{L}_G\,\mathbf{E}.$$ This ensures no birefringence for $\gamma>0$ in parallel field backgrounds, unlike in generic NLED models (Lechner et al., 2022).

• Energy-Momentum Tensor: The stress tensor can be written in a manifestly SL(2,ℝ)-invariant form, with

$$T_{\mu\nu} = g_{\mu\nu}\mathcal{L} + F_{\mu\alpha}G_{\nu}^{\;\;\alpha},$$

and is traceless for any $\gamma$ (Babaei-Aghbolagh et al., 2022).

4. Couplings to Gravity, Black Holes, and Extensions

ModMax nonlinear electrodynamics couples naturally to general relativity and supergravity, yielding a wide variety of black hole and cosmological solutions. The structure of the coupling is of the form

$$I = \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\,\left[ R - 4\mathcal{L}_{\rm ModMax}(S,P) \right],$$

with $S$ and $P$ promoted to their covariant analogs (Panah, 2024, Lechner et al., 2022, Panah et al., 27 Dec 2025).

• Spherically Symmetric Black Holes: The metric function for dyonic (electric and magnetic charge) ModMax black holes is

$$f(r) = 1 - \frac{2M}{r} + \frac{Q_e^2 + Q_m^2}{r^2} e^{-\gamma},$$

where $\gamma$ appears as a "screening" parameter exponentially suppressing the charge interaction at large $\gamma$ (Pantig et al., 2022, Sucu et al., 8 Aug 2025, Panah et al., 27 Dec 2025).

• Thermodynamics: ModMax black holes in AdS spacetimes exhibit well-defined first-law thermodynamics, heat engine interpretation, extended phase space (with $P \sim -\Lambda$), and modified heat capacity and phase structure depending on $\gamma$ (Panah et al., 27 Dec 2025, Sucu et al., 8 Aug 2025, Panah, 2024, Panah, 8 Jul 2025, Panah, 2024, Kruglov, 2022). Large $\gamma$ suppresses electromagnetic backreaction, restoring Schwarzschild-type behavior.
• Supersymmetry: N=1 supersymmetric generalizations (superModMax) preserve both superconformal and duality invariance. Theories are constructed in superspace, with higher-derivative photino terms removable via non-linear superfield redefinitions (Volkov–Akulov structure) (Bandos et al., 2021).
• Generalizations and Limits: ModMax arises as the $T\to\infty$ ("infinite tension") limit of a two-parameter, Born–Infeld-like family combining duality invariance and nonlinearity. The generalized theories admit alternative, determinant-based DBI-like formulations and (partially) string-theoretic analogs. Scalar field couplings can be introduced via DBI-like actions (Nastase, 2021). The Galilean (non-relativistic) analog preserves invariance under the Galilean Conformal Algebra (GCA) (Banerjee et al., 2022).

5. Quantum Properties and Non-Renormalizability

At the quantum level, ModMax displays particular features distinct from those of Maxwell and generic nonlinear electrodynamics:

• Perturbative Quantization: For constant backgrounds, all one-loop quantum corrections vanish. For inhomogeneous backgrounds, nontrivial divergences arise that cannot be absorbed into a renormalization of the original Lagrangian (Martin, 2024).
• Non-Renormalizability: New divergent operators at one-loop require an infinite tower of counterterms, reflecting the non-renormalizable nature of ModMax (and its 2d analogs). However, the theory remains a consistent low-energy effective field theory.
• 2d Analogs: Quantum properties for the 2d ModMax-like scalar theory mirror the 4d case, with vanishing quantum corrections in uniform backgrounds and logarithmic divergences in varying ones.

6. Physical Implications: Propagation, Observables, and Holography

ModMax’s nonlinearity and symmetry content have direct observational and theoretical implications:

• Electrodynamics: Despite its nonlinearity, ModMax admits standard Liénard–Wiechert solutions for point charge, monopole, and dyon configurations (exact), and preserves the standard Dirac–Schwinger quantization conditions (Lechner et al., 2022). The Coulomb potential is modified by $e^{-\gamma}$.
• Photon Propagation: ModMax is birefringence-free for homogeneous backgrounds but exhibits birefringence for orthogonal electric/magnetic backgrounds, with explicit expressions for refractive indices and their dependence on $\gamma$, background fields, and polarization (Neves et al., 2022).
• Axion Coupling and Confinement: Coupling to an axion induces non-local photon interactions, leading to static potentials interpolating between exponential screening and linear confinement for static charges. The confining part depends non-trivially on $\gamma$, axion mass, coupling constant, and external magnetic field (Neves et al., 2022, Sucu et al., 8 Aug 2025).
• Black Hole Shadows and Gravitational Lensing: The parameter $\gamma$ modifies photon sphere location, shadow radius (and constraints from EHT), Einstein ring formation, quasinormal spectra, and lensing in both strong and weak fields. The exponential damping governs the electromagnetic backreaction and stability regions (Pantig et al., 2022, Sucu et al., 8 Aug 2025, Panah, 8 Jul 2025).
• Thermal and Quantum Stability: ModMax–charged black holes exhibit rich phase structure, with shifts in heat capacity critical points, global/local stability regimes, and quantum-corrected remnant formation controlled by $\gamma$ (Panah et al., 27 Dec 2025, Panah, 2024).
• Holographic Transport: In AdS/CFT, ModMax models with momentum relaxation yield analytic DC conductivities displaying nontrivial Hall and Nernst response. For large $\gamma$, a transition to an insulating, quasiparticle-dominated regime is found, while the Nernst signal and superconducting dome are tunable by $\gamma$ (Barrientos et al., 3 Jun 2025).

7. Comparative Structure and Broader Extensions

ModMax theory sits in a broader classification of electromagnetic effective field theories:

• Comparison to Maxwell and Born–Infeld: Maxwell theory is the unique linear, conformal, duality-invariant model ($\gamma=0$); Born–Infeld is a non-conformal but duality-invariant nonlinear model. ModMax uniquely saturates both conformal and duality invariance constraints and can be seen as the infinite tension limit of Born–Infeld (Babaei-Aghbolagh et al., 2022, Nastase, 2021).
• Deformation Hierarchy: ModMax is generated by a marginal ($T\bar{T}$-type) operator, while Born–Infeld is associated with an irrelevant $T\bar{T}$-type deformation. The two-parameter (GBI) theories interpolate between both and admit generalized SL(2,ℝ)-invariant stress tensors (Babaei-Aghbolagh et al., 2022).
• Generalizations: Extensions include generalized ModMax models with additional parameters controlling field regularization and criticality, higher-derivative corrections, and auxiliary-scalar (axion-dilaton) representations suggestive of an effective field theory or D-brane origin (Kruglov, 2022, Lechner et al., 2022).

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In summary, ModMax electrodynamics introduces minimal, symmetry-preserving nonlinearity into Maxwell theory through a uniquely defined marginal deformation, yielding a rich landscape of classical, gravitational, quantum, and holographic phenomena while retaining exact integrability and duality/conformal invariance at every stage (Babaei-Aghbolagh et al., 2022, Bandos et al., 2021, Babaei-Aghbolagh et al., 2022, Martin, 2024, Panah et al., 27 Dec 2025).