ModMax: Nonlinear Conformal Electrodynamics

• ModMax theory is a one-parameter nonlinear conformal extension of Maxwell electrodynamics characterized by the deformation parameter γ, recovering Maxwell theory at γ=0.
• It preserves electromagnetic duality (SO(2) rotations) and scale invariance while ensuring convexity, causality, and absence of ghosts for γ ≥ 0.
• The theory supports rich extensions including Born–Infeld generalizations, supersymmetric versions, and applications in black hole thermodynamics and stress-tensor flows.

Modification Maxwell (ModMax) Theory is the unique one-parameter, nonlinear conformal extension of Maxwell electrodynamics in four spacetime dimensions. It is characterized by precise invariance under continuous electromagnetic duality (SO(2) rotations in the electric and magnetic fields) and exact scale/conformal symmetry. The theory is parameterized by a real, dimensionless deformation parameter γ and admits generalizations, including Born-Infeld-type deformations, scalar couplings, supersymmetric extensions, and nonrelativistic/axion couplings. ModMax theory has recently been the focus of extensive research, particularly for its role in the classification of nonlinear electrodynamics, integrable stress-tensor flows, and black hole thermodynamics.

1. Mathematical Structure and Defining Properties

ModMax theory is defined by the Lagrangian density

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

where $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ and $P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$ are the standard Lorentz and dual invariants of the electromagnetic field with $$\widetilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$. The parameter γ controls the strength of the nonlinearity; the Maxwell theory is recovered at γ = 0.

Key features:

• Non-analytic Structure: The presence of $\sqrt{S^2+P^2}$ makes the theory non-analytic at $F_{\mu\nu}=0$, in contrast to analytic NLEDs such as Born-Infeld.
• Convexity and Causality: The Lagrangian is strictly convex in the electric field for γ ≥ 0—ensuring absence of ghosts/tachyons—while γ > 0 enforces subluminality of wavefronts (causality) (Bandos et al., 2021).
• Duality and Conformal Invariance: It is the unique one-parameter deformation of Maxwell theory that preserves both SO(2) electric-magnetic duality and classical conformal symmetry (traceless stress tensor) in D=4 (Babaei-Aghbolagh et al., 2022).

The table below summarizes these conditions:

Property	Condition on γ	Manifestation
Duality invariance	all γ	Lagrangian form (SO(2) rotational symmetry)
Conformal invariance	all γ	Traceless stress-energy tensor
Strict convexity	γ ≥ 0	No ghosts, positive energy
Full causality	γ > 0	No superluminal small fluctuations

2. Symmetry Content and Uniqueness

ModMax electrodynamics is distinguished by its symmetry algebra:

• Electromagnetic Duality: Under SO(2) rotations,

$$\begin{pmatrix} F \ \widetilde{F} \end{pmatrix} \mapsto \begin{pmatrix} \cos\theta & \sin\theta \ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} F \ \widetilde{F} \end{pmatrix},$$

the Lagrangian transforms linearly in (S, P) but the “square root” term is invariant. This ensures the equations of motion are duality-covariant and the theory maps physical electric/magnetic fields into each other without distinguishing a preferred direction (Bandos et al., 2021, Babaei-Aghbolagh et al., 2022).

• Conformal Invariance: Homogeneity of degree one in (S, P) guarantees scale invariance. Explicitly, the Euler relation

$$S\,\partial_S \mathcal{L} + P\,\partial_P \mathcal{L} = \mathcal{L}$$

ensures the tracelessness of the stress-energy tensor (Bandos et al., 2021).

The theory emerges as the endpoint of current-squared or $T\bar{T}$-like flows on the stress tensor. The flow and uniqueness only close (in the sense of producing another duality/conformal-invariant action) in four dimensions—analogous constructions do not exist in other spacetime dimensions due to the algebraic structure of the stress tensor and Lorentz group (Ferko et al., 2022, Babaei-Aghbolagh et al., 2022, Babaei-Aghbolagh et al., 2022).

3. Field Equations, Constitutive Relations, and Physical Observables

Variation of the action yields generalized Maxwell equations: $$\partial_\mu G^{\mu\nu} = 0, \qquad G^{\mu\nu} = -2\frac{\partial\mathcal{L}}{\partial F_{\mu\nu}}$$ with

$S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$0

The constitutive relations for the displacement and magnetic fields (in 3+1 split): $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$1

$S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$2

These reduce to linear Maxwell relations at γ=0 (Hamil, 6 Jan 2026, Panah, 2024). In the presence of sources (electric, magnetic, dyonic), the analysis is unchanged modulo the usual quantization conditions arising from Dirac-Schwinger constraints (Lechner et al., 2022).

The theory predicts:

• Absence of Birefringence: Both polarizations propagate on the same light-cone, with the Gibbons-Rasheed criterion saturated (Bandos et al., 2021, Neves et al., 2022).
• Vacuum Birefringence in Background Fields: For nontrivial backgrounds, ModMax exhibits anisotropic permittivity and permeability tensors, leading to direction-dependent refractive indices and birefringence Δn proportional to the deformation parameter γ, with explicit analytic formulas in the backgrounds of orthogonal fields (Neves et al., 2022).

4. Deformations, Born–Infeld-Like Generalization, and Flows

ModMax admits a Born–Infeld-type deformation: $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$3 which in the limit $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$4 returns the ModMax Lagrangian, and for γ = 0 yields ordinary Born–Infeld. This extension maintains duality invariance for all T, and conformal invariance only in the strict $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$5 limit. The entire family is constructed via a current-squared flow (irrelevant $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$6-type deformation), with √TT̄-like (marginal) and true TT̄-like (irrelevant) structure corresponding to derivatives of the Lagrangian with respect to γ and T, respectively (Ferko et al., 2022, Babaei-Aghbolagh et al., 2022, Bandos et al., 2021, Nastase, 2021).

The table below summarizes limiting cases:

Parameter Regime	Lagrangian Reduces to	Symmetries
γ = 0, T → ∞	Maxwell	Duality, Conformal
γ ≠ 0, T → ∞	ModMax	Duality, Conformal
γ = 0, finite T	Born–Infeld	Duality
γ ≠ 0, finite T	BI–ModMax	Duality

The structure extends to $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$7 superfield flows, where both ModMax and its BI-extension can be supersymmetrized under explicit convexity constraints, maintaining superconformal invariance and SO(2) duality (Bandos et al., 2021, Ferko et al., 2022).

5. Coupling to Gravity, Black Holes, and Thermodynamics

ModMax electrodynamics admits minimal and non-minimal couplings to various gravity sectors, including Einstein, $S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$8, Gauss-Bonnet, dRGT-like massive gravity, dilatonic gravity, and Kaluza-Klein/entanglement-relativity scenarios (Hamil, 6 Jan 2026, Panah, 8 Jul 2025, Panah, 2024, Bixano et al., 27 Mar 2026). In all cases, ModMax modifies the electromagnetic contribution to black-hole metric functions and energy-momentum:

$S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$9

for Einstein gravity. The effective charge is exponentially damped for large γ ($P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$0), softening field strengths near singularities but preserving the existence of horizons and critical points (Sucu et al., 8 Aug 2025).

In black hole thermodynamics:

• The Hawking temperature, entropy, and classical first law receive γ-dependent corrections.
• Heat capacity exhibits second-order phase transitions, with critical radii and stability boundaries shifted as functions of γ.
• For large γ, only single, thermodynamically stable black holes persist; multiple horizons and instability regions shrink (Panah, 8 Jul 2025, Panah, 2024).
• ModMax-based black holes saturate the isoperimetric ratio, and evade some of the pathological behaviors typical of strong-field Einstein-Maxwell solutions (Panah, 8 Jul 2025).

ModMax thus interpolates between classical Einstein-Maxwell and nearly Schwarzschild-like, weakly charged geometries, depending on the nonlinearity parameter.

6. Quantum Effects, Flows, and Non-Renormalizability

The perturbative quantization of ModMax via the background field method and dimensional regularization reveals several distinctive features:

• In constant (homogeneous) field backgrounds, all one-loop quantum corrections vanish—no effective field-theory induced nonlinearities appear at this order.
• In general, non-constant backgrounds, divergent one-loop structures arise whose tensorial form lies outside the original ModMax family; this indicates the theory is not renormalizable in the usual field-theoretic sense but remains a consistent low-energy effective theory.
• The non-analytic nature of the ModMax Lagrangian prevents a weak-field expansion about $P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$1; all perturbative quantization must be performed about finite-field backgrounds (Martin, 2024).
• The current-squared (O_{T^2}) flow construction for ModMax and its BI generalization is strictly limited to $P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$2. In lower or higher dimensions the analogous operators do not close onto duality- and conformal-invariant deformations (Ferko et al., 2022).

7. Generalizations and Physical Extensions

A range of extensions and analogues of ModMax theory have been developed:

• Generalized ModMax (GenModMax): Four-parameter generalizations include extra scales, exponents, and explicit $P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$3-type deformations. These typically break at least one of duality or conformal invariance, except for the pure ModMax case (Kruglov, 2022).
• Axion and Scalar Couplings: ModMax can be coupled to axions, producing photon-axion mixing and confining potentials for static charges in external magnetic fields; the effect depends on γ, the axion mass, and the coupling (Neves et al., 2022, Bixano et al., 27 Mar 2026).
• Supersymmetrization: For Lagrangians satisfying the convexity criterion, ModMax admits $P = -\frac{1}{4}F_{\mu\nu}\widetilde{F}^{\mu\nu}$4 supersymmetric extensions preserving duality and superconformal invariance via explicit superfield constructions (Bandos et al., 2021).
• Nonrelativistic (Galilean) Cousins: A Galilean ModMax version exists, replacing Lorentz invariants with Newton–Cartan analogues and maintaining Galilean Conformal Algebra invariance (Banerjee et al., 2022).
• Brane-type and String-Theoretic Precursors: The ModMax Lagrangian emerges as the infinite-tension limit of a generalized Born–Infeld/DBI action, suggesting deep connections to brane dynamics in string theory. All standard brane solitons and topological solutions (BIons, catenoids, Rańada “knotted” fields) remain valid solutions of the ModMax precursor (Nastase, 2021).

These generalizations establish ModMax theory as a universal organizing principle for nonlinear electrodynamics with maximal allowable symmetry in four spacetime dimensions.

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References:

• (Bandos et al., 2021): Foundational exposition of ModMax theory and its symmetries, unitarity, convexity, causality, and Born–Infeld extension.
• (Lechner et al., 2022): Charged particle interactions, field equations, “axion–dilaton” representations, and explicitly solvable sectors.
• (Sucu et al., 8 Aug 2025): Einstein–Dyonic–ModMax black holes, black-hole thermodynamics, lensing, and GUP corrections.
• (Bixano et al., 27 Mar 2026): Exact rotating solutions in Einstein–ModMax–scalar field theories.
• (Martin, 2024): Perturbative quantization and non-renormalizability results.
• (Babaei-Aghbolagh et al., 2022): SL(2,ℝ) duality symmetries, TT̄-like deformations, and invariant stress tensors.
• (Ferko et al., 2022, Babaei-Aghbolagh et al., 2022): TT̄/T² flow constructions and their uniqueness in D=4.
• (Nastase, 2021): Precursor brane actions, scalar couplings, and solitonic/knotted solutions.
• (Banerjee et al., 2022): Galilean-ModMax theory and nonrelativistic limits.
• (Neves et al., 2022): Linearized ModMax in external backgrounds, birefringence, and axion coupling.
• (Hamil, 6 Jan 2026, Panah, 2024, Panah, 8 Jul 2025, Panah, 2024): ModMax black holes in various gravitational frameworks and their thermodynamic properties.