ModMax Theory: 4D Nonlinear Electrodynamics

• ModMax theory is a four-dimensional nonlinear electrodynamics model defined by a dimensionless deformation parameter that preserves both electromagnetic duality and conformal invariance.
• It employs nonlinear constitutive relations that modify photon propagation and black hole charge screening, yielding observable optical and gravitational phenomena.
• The framework extends to supersymmetric completions, holographic transport, and non-Abelian extensions, linking DBI-like structures and T‾T-like deformations.

ModMax theory is a four-dimensional nonlinear electrodynamics defined, in one standard convention, by

$$\mathcal L_{\rm ModMax}=\cosh\gamma\,S+\sinh\gamma\,\sqrt{S^2+P^2}, \qquad S=-\frac14F_{\mu\nu}F^{\mu\nu}, \qquad P=-\frac14F_{\mu\nu}\tilde F^{\mu\nu},$$

with dimensionless deformation parameter $\gamma$. At $\gamma=0$ it reduces to Maxwell theory, while preserving the two structures repeatedly emphasized in the literature as distinctive in four dimensions: electromagnetic duality and conformal invariance (Bandos et al., 2021, Kuzenko et al., 2023). A complementary viewpoint treats ModMax as the $T\to\infty$ limit of a Born–Infeld-like precursor, placing it within a broader nonlinear and DBI-like framework (Nastase, 2021).

1. Definition, invariants, and conventions

The basic dynamical variable is the field strength

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$

The theory depends only on the two electromagnetic invariants built from $F_{\mu\nu}$ and $\tilde F_{\mu\nu}$. Across the literature, these invariants are denoted by $(S,P)$, $(\mathcal S,\mathcal P)$, or $(F,G)$, and overall sign and normalization conventions vary. One rotating Einstein–ModMax–scalar formulation, for example, uses

$\gamma$0

and explicitly notes that this differs by $\gamma$1 from another standard convention (Bixano et al., 27 Mar 2026). The Maxwell limit remains the common anchor: $\gamma$2 reproduces the linear theory.

The physically preferred branch is $\gamma$3. In the supersymmetric analysis, convexity of the bosonic Lagrangian is related to semi-classical unitarity and to the absence of superluminal propagation around constant backgrounds, and this selects the $\gamma$4 branch for ModMax (Bandos et al., 2021). In several gravitational and holographic applications the same restriction is imposed for causality and unitarity (Guzman-Herrera et al., 2023, Barrientos et al., 3 Jun 2025).

A recurring source of confusion is that many exact solutions are constructed in restricted sectors, most often purely electric ones with $\gamma$5 or $\gamma$6. In such sectors the full two-invariant nonlinear structure collapses to a much simpler effective description, often amounting to a Maxwell-like theory with $\gamma$7-dependent dressing. That simplification is exact in those sectors, but it does not exhaust the full dyonic or duality-rotating content of ModMax.

2. Symmetry content and constitutive structure

The defining structural statement about ModMax is that it is simultaneously conformal and duality invariant. In the Gaillard–Zumino formulation, one introduces

$\gamma$8

and duality acts as an $\gamma$9 rotation on the doublet $\gamma=0$0. For ModMax, the constitutive relation is nonlinear: $\gamma=0$1 so $\gamma=0$2 is no longer simply proportional to $\gamma=0$3 (Babaei-Aghbolagh et al., 2022). The corresponding stress tensor can nevertheless be reorganized into a duality-invariant form. Defining

$\gamma=0$4

the ModMax stress tensor is written as

$\gamma=0$5

and, with axion–dilaton coupling $\gamma=0$6, the same structure admits a manifestly $\gamma=0$7-invariant rewriting in terms of a scalar matrix $\gamma=0$8 and an invariant tensor $\gamma=0$9 (Babaei-Aghbolagh et al., 2022).

Conformal invariance is reflected in the tracelessness of the stress tensor. In the supersymmetric treatment this appears through the homogeneity relation

$T\to\infty$0

and for conformal electrodynamics one finds

$T\to\infty$1

In the same analysis, ModMax is presented as essentially the unique interacting one-parameter extension of Maxwell theory that preserves both conformal invariance and duality invariance in the corresponding class (Bandos et al., 2021).

This symmetry content has a direct optical consequence. In the effective-metric analysis of photon propagation near an Einstein–ModMax black hole, the birefringence indices satisfy

$T\to\infty$2

Thus one photon polarization experiences a nontrivial effective optical metric, while the other propagates on the original spacetime null cone. The paper identifies this as a concrete consequence of ModMax conformal invariance (Guzman-Herrera et al., 2023).

3. Precursor theory and deformation viewpoints

A major interpretive development is the identification of a finite-$T\to\infty$3 “ModMax precursor,” a duality-invariant Born–Infeld-like theory whose Hamiltonian is

$T\to\infty$4

with

$T\to\infty$5

The infinite-tension limit

$T\to\infty$6

gives the ModMax Hamiltonian

$T\to\infty$7

In this formulation, finite $T\to\infty$8 preserves nonlinear electromagnetic duality but breaks conformal invariance, while the $T\to\infty$9 limit removes the BI scale and restores conformal symmetry (Nastase, 2021).

The same work constructs a DBI-like determinant form for the precursor in which $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$0 is replaced by a nonlinear combination

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$1

This allows scalar couplings analogous to DBI and imports BIon, catenoid, and interpolating solutions into the precursor theory. In the strict ModMax limit, however, the nontrivial gauge–scalar interaction collapses to a free scalar kinetic term plus the ModMax gauge Lagrangian, so the richer DBI-like solitonic structure belongs most naturally to the precursor rather than to pure ModMax itself. The same paper also shows that Ra~nada’s $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$2 knotted electromagnetic configurations remain exact solutions both of ModMax and of its precursor (Nastase, 2021).

A second viewpoint interprets ModMax through stress-tensor deformations. For ModMax itself,

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$3

and this is identified as a manifestly self-dual invariant action. In the generalized Born–Infeld family, the $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$4-flow is associated with an irrelevant $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$5-like deformation, whereas the $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$6-flow is associated with a marginal $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$7-like deformation (Babaei-Aghbolagh et al., 2022). This places ModMax at the intersection of nonlinear duality-symmetric electrodynamics and deformation-theoretic constructions.

4. Supersymmetric and sigma-model formulations

ModMax admits an explicit $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$8 supersymmetric completion. A general prescription is given for supersymmetrizing any four-dimensional nonlinear electrodynamics satisfying suitable convexity conditions, and applying it to ModMax yields the superfield action

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu, \qquad \tilde F^{\mu\nu}=\frac12\epsilon^{\mu\nu\rho\lambda}F_{\rho\lambda}.$$9

Eliminating the auxiliary field reproduces the bosonic ModMax Lagrangian exactly. The same work proves that supersymmetric self-duality is preserved, that the coupling to supergravity is super-Weyl invariant, and therefore that superModMax is superconformal. It also shows that the higher-derivative photino interactions characteristic of supersymmetric nonlinear electrodynamics can be removed by an invertible nonlinear superfield redefinition (Bandos et al., 2021).

There is also a distinct supersymmetric sigma-model analogue. In the chiral–complex-linear class, every $F_{\mu\nu}$0-duality-invariant nonlinear electrodynamics determines a unique duality-invariant supersymmetric sigma model, and applying that correspondence to ModMax yields

$F_{\mu\nu}$1

Its Legendre-dual purely chiral formulation has Kähler potential

$F_{\mu\nu}$2

The target space is a Kähler cone, and the connected component of its isometry group is

$F_{\mu\nu}$3

This construction is not a vector-multiplet supersymmetrization of electrodynamics; it is a sigma-model counterpart that mirrors ModMax’s defining combination of duality invariance, conformal structure, and uniqueness (Kuzenko et al., 2023).

5. Einstein-coupled black holes and thermodynamics

A large part of the ModMax literature concerns exact black-hole solutions. In Einstein–ModMax theory, the static spherically symmetric black-hole geometry analyzed in optical studies has lapse function

$F_{\mu\nu}$4

so the spacetime is Reissner–Nordstr\"om-like with a screened charge contribution $F_{\mu\nu}$5 (Guzman-Herrera et al., 2023). That same screening pattern persists in many more elaborate settings.

In constant-curvature $F_{\mu\nu}$6-ModMax theory, a purely electric exact solution takes the form

$F_{\mu\nu}$7

The corresponding thermodynamic analysis verifies the first law, modifies the entropy through the standard $F_{\mu\nu}$8 factor $F_{\mu\nu}$9, and studies both local stability through heat capacity and global stability through Helmholtz free energy (Panah, 2024). In ModMax–dRGT-like massive gravity, the same electric-sector simplification appears as

$\tilde F_{\mu\nu}$0

and the extended-phase-space analysis verifies both the first law and a Smarr relation, with the massive couplings $\tilde F_{\mu\nu}$1 promoted to thermodynamic variables (Panah, 8 Jul 2025).

These papers also make an important limitation explicit. In the purely electric branch, $\tilde F_{\mu\nu}$2, the ModMax field equation reduces to

$\tilde F_{\mu\nu}$3

and the stress tensor becomes a Maxwell-type expression multiplied by $\tilde F_{\mu\nu}$4. This means that many exact black-hole solutions probe a highly restricted subsector of the full theory rather than the generic dyonic regime (Panah, 2024, Panah, 8 Jul 2025). The same caveat is sharpened in the three-dimensional BTZ construction, where the model is presented not as a derivation of genuine $\tilde F_{\mu\nu}$5 duality-invariant ModMax, but as a ModMax-inspired nonlinear electrodynamics chosen to mimic the four-dimensional form; in the solved electric sector it again collapses to a rescaled Maxwell theory (Panah, 2024).

6. Beyond single static black holes: multi-center, rotating, and non-Abelian generalizations

Exact many-body solutions exist in Einstein–ModMax theory. In the purely electric Weyl sector one has $\tilde F_{\mu\nu}$6, $\tilde F_{\mu\nu}$7 constant, and the traceless stress tensor, so the coupled system reduces to an Einstein–Maxwell-like one with an overall $\tilde F_{\mu\nu}$8 factor. This allows exact nonextremal multi-black-hole spacetimes analogous to multi–Reissner–Nordstr\"om, with conical struts on the axis. In the extremal limit the metric becomes isometric to Majumdar–Papapetrou, but the physical charge-to-mass ratio is modified by screening: $\tilde F_{\mu\nu}$9 The same paper uses ModMax duality invariance to generate magnetic and dyonic families, and extends the construction to positive cosmological constant in a direct analogue of Kastor–Traschen (Bokulić et al., 8 Jan 2025).

Rotation has been incorporated in a broader Einstein–ModMax–scalar framework. A generalized Ernst-type formalism with nonzero rotational function $(S,P)$0 is constructed, and exact rotating families are obtained in the frozen sector

$(S,P)$1

In that sector the nonlinear ModMax information is encoded in constant coefficients $(S,P)$2, $(S,P)$3, and $(S,P)$4. A sharp caveat accompanies this result: if the scalar is absent and the frozen sector is imposed, the theory becomes Maxwell-like after a constant linear redefinition. The paper therefore identifies scalar coupling as essential for nontrivial frozen ModMax phenomenology in the rotating stationary-axisymmetric setting (Bixano et al., 27 Mar 2026).

A separate extension replaces the Abelian gauge field by an $(S,P)$5 one. The resulting non-Abelian ModMax action is

$(S,P)$6

with

$(S,P)$7

In this theory the appropriate self-duality condition is not $(S,P)$8, which makes the ModMax square root degenerate, but

$(S,P)$9

With that substitution the theory admits BPST-like instantons, constant-curvature generalizations on $(\mathcal S,\mathcal P)$0 and $(\mathcal S,\mathcal P)$1, perturbative multi-instantons from ’t Hooft-symbol ans\"atze, and, after coupling to gravity with a conformally coupled scalar, Euclidean wormholes and smooth configurations with secondary hair (Canfora et al., 19 Nov 2025). On Euclidean AdS backgrounds the size dependence of the Chern–Pontryagin number is explicitly linked to the fact that the configuration is not a pure gauge at infinity, a property already emphasized by Callan and Wilczek in the Yang–Mills context (Canfora et al., 19 Nov 2025).

7. Lower-dimensional analogues, optical and holographic applications, and recurrent caveats

Several papers transplant the ModMax structure into settings where its original four-dimensional interpretation changes. A Galilean cousin of ModMax is built on Newton–Cartan data with two gauge fields and Galilean electromagnetic invariants, producing a nonlinear electrodynamics invariant under the Galilean Conformal Algebra. The construction preserves the square-root ModMax pattern, but the paper states explicitly that true electromagnetic duality does not survive as an intrinsic symmetry in the Galilean theory; at most a sector-exchange remnant remains (Banerjee et al., 2022). This is a useful corrective to any assumption that duality automatically survives every ModMax-like deformation.

A two-dimensional scalar analogue is obtained from a marginal $(\mathcal S,\mathcal P)$2-like deformation. For multiple scalars with target metric $(\mathcal S,\mathcal P)$3, the resulting “scalar ModMax” theory is

$(\mathcal S,\mathcal P)$4

with $(\mathcal S,\mathcal P)$5 and $(\mathcal S,\mathcal P)$6 the quadratic and quartic scalar invariants. An additional irrelevant $(\mathcal S,\mathcal P)$7 deformation yields a generalized scalar ModMax theory of Born–Infeld/Nambu–Goto type (Babaei-Aghbolagh et al., 2022). A different two-dimensional descendant, obtained by dimensional reduction of four-dimensional ModMax with a frozen electric background and an added scalar potential, shows that the static kink equation can be mapped to the canonical scalar model, while fluctuations of the second scalar obey a Sturm–Liouville problem with $(\mathcal S,\mathcal P)$8-dependent weight and can exhibit bound states absent in the canonical case (Brito et al., 24 Feb 2025).

On the phenomenological side, optical propagation near an Einstein–ModMax black hole displays polarization-dependent effective metrics, birefringence away from purely radial propagation, modified deflection angles, modified redshifts, and polarization-dependent shadow and absorption cross section. One polarization reproduces a screened Reissner–Nordstr\"om behavior, while the other yields genuinely nonlinear-optical deviations (Guzman-Herrera et al., 2023). In holography, ModMax electrodynamics coupled to Einstein–AdS gravity and linear axions produces DC magnetotransport coefficients entirely in terms of horizon data; the deformation parameter $(\mathcal S,\mathcal P)$9 changes longitudinal and Hall conductivities, suppresses the Hall angle while preserving the strange-metal scaling

$(F,G)$0

and modifies the Nernst signal in a way the paper interprets as reproducing a superconducting dome and normal phase familiar from high-$(F,G)$1 cuprates (Barrientos et al., 3 Jun 2025).

Across these applications, the most persistent caveat is methodological rather than interpretive. Whenever one restricts to $(F,G)$2, to $(F,G)$3 constant, or to sectors where the constitutive coefficients become constant, ModMax often collapses to a Maxwell-like system with a simple $(F,G)$4-dependent dressing. Those reductions are exact within their respective sectors, but they should not be mistaken for the generic theory. The full ModMax structure is most visible in settings that retain both invariants, duality-sensitive constitutive relations, or genuinely time-dependent and magnetized probes.