Conformal NLED Theories

• Conformal NLED theories are nonlinear modifications of Maxwell’s framework that enforce full conformal symmetry and employ the invariant ratio u for scale-free dynamics.
• They feature constitutive relations defined solely by functions of u, allowing for both Lagrangian and non-Lagrangian formulations that respect strict invariance constraints.
• These models are pivotal in exploring gauge-gravity dualities and quantum critical phenomena, utilizing heat kernel techniques to ensure consistency, causality, and effective quantum actions.

Conformal nonlinear electrodynamics (NLED) denotes the class of classical or quantum electromagnetic theories in four-dimensional Minkowski spacetime whose equations respect the full conformal group and generalize Maxwell’s linear theory by allowing nonlinear relationships between the electromagnetic field tensor and its excitation. Conformal invariance imposes stringent algebraic and differential constraints on allowable Lagrangians and constitutive relations, ensuring the absence of intrinsic scales and preserving the tracelessness of the energy-momentum tensor. These constraints yield a rich structure of possible classical and quantum effective theories, including both Lagrangian and non-Lagrangian models, widely relevant for gauge-gravity dualities, critical phenomena, and quantum field theory.

1. Conformal Invariants and Field Content

In conformal NLED, the fundamental variables are the spacetime metric $g_{\mu\nu}$ and the electromagnetic field strength tensor $$F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$. Lorentz invariance allows the construction of two independent scalar invariants: $$I_1 = F_{\mu\nu}F^{\mu\nu}, \qquad I_2 = F_{\mu\nu}\widetilde F^{\mu\nu},$$ where $$\widetilde F^{\mu\nu} = \tfrac12 \epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$$ is the Hodge dual. Under conformal inversion, $I_1$ and $I_2$ transform as $I_{1,2}(x) \mapsto (x^2)^2 I_{1,2}(x)$ (up to sign for $I_2$), indicating that each is not invariant by itself. Instead, their ratio $u = I_2/I_1$ is invariant (up to sign) under a single inversion and fully invariant under double inversion, making $u$ the unique dimensionless conformal-invariant functional for constructing generalized NLED models (Duplij et al., 2019).

2. Constitutive Relations: Lagrangian and Non-Lagrangian Models

General NLED is specified by a nonlinear constitutive relation between $F_{\mu\nu}$ and the excitation tensor $G^{\mu\nu}$. Expressed in terms of the invariant $u$,

$$G^{\mu\nu} = N(u) F^{\mu\nu} + c\, M(u) \widetilde F^{\mu\nu},$$

where $N(u)$ and $M(u)$ are arbitrary scalar functions and $c$ is the speed of light. Maxwell electrodynamics arises in the limit $N \to 1/\mu_0,\ M \to 0$. When the model descends from a Lagrangian density $\mathcal{L}(I_1,I_2)$,

$$G^{\mu\nu} = -2 \frac{\partial \mathcal{L}}{\partial I_1} F^{\mu\nu} - \frac{\partial \mathcal{L}}{\partial I_2} \widetilde F^{\mu\nu},$$

identifications $N(u) = -2\mathcal{L}_{I_1}$, $c M(u)= -\mathcal{L}_{I_2}$ hold. For general (possibly non-Lagrangian) conformal NLED, the only requirement is that $N$ and $M$ depend exclusively on $u$ (Duplij et al., 2019).

3. Master Lagrangian and Integrability Criterion

Conformal invariance dictates that the Lagrangian take the form $$\mathcal{L}_{\mathrm{conf}}(I_1, I_2) = -\frac{1}{2\mu_0} I_1 + I_1 \int^u M(w) dw$$, or equivalently, $$\mathcal{L}_{\mathrm{conf}} = -\frac{1}{2\mu_0} I_1 + I_2 \int^u \frac{M(w)}{w} dw$$. All allowed nonlinear deformations of Maxwell’s theory—classical or quantum—that preserve conformal invariance are generated by the free choice of the function $M(u)$.

A key structural distinction arises:

• Lagrangian conformal NLED: The integrability condition,

$\frac{dN(u)}{du} - 2c u \frac{dM(u)}{du} = 0,$

must be satisfied, ensuring the constitutive relations derive from a variational principle.

• Non-Lagrangian conformal NLED: When the above is violated, the constitutive relation is non-variational but still respects conformal symmetry.

An explicit example is the quadratic conformal NLED defined by $M(u) = \kappa u$, yielding

$$\mathcal{L} = -\frac{1}{2\mu_0} I_1 + \kappa \frac{I_2^2}{2c I_1}, \qquad G^{\mu\nu} = \left(\frac{1}{\mu_0} + 2c\kappa u^2\right)F^{\mu\nu} + \kappa u \widetilde F^{\mu\nu}$$

(Duplij et al., 2019).

4. Quantum Effective Action and Heat Kernel Structure

Quantization of conformal NLED in the background field formalism produces non-minimal wave operators for fluctuations, necessitating advanced heat kernel techniques. The asymptotic expansion of the heat kernel yields DeWitt coefficients $a_n(x)$ encoding one-loop divergences and effective action corrections. For conformal NLED,

$$S[A] = \int d^4x\ L(I, J),\qquad I = \frac14 F^{ab}F_{ab},\ J = \frac14\widetilde F^{ab}F_{ab}$$

and the necessary and sufficient condition for conformal invariance is

$I L_I + J L_J = L$

(Buchbinder et al., 27 Jan 2026).

The exact, all-orders $a_0$ coefficient is

$a_0(x) = \frac{3}{L^2} + \frac{1}{L^2 + 2AL - B^2}$

where $A$ and $B^2$ are constructed from second derivatives of $L$ (via the "curvature-like" tensor $G^{abcd}$) and $L$ itself. This closed formula holds for all backgrounds and fully incorporates the (generally nonlinear) constraint structure of conformal NLED.

At one-loop, the log-divergent effective action is

$$\Gamma_{div} = -\frac{\ln\Lambda}{16\pi^2} \int d^4x\ a_2(x)$$

where, by conformality, $a_2$ contains only quartic-in-$F$ and four-derivative terms: no Maxwell-like (kinetic) counterterm is generated and no intrinsic mass scale appears. The detailed structures of $a_1$, $a_2$ at weak field are explicit expansions in powers of $F_{ab}$ and its derivatives, reflecting the allowed counterterms (Buchbinder et al., 27 Jan 2026).

5. Causality, Self-Consistency, and Wave Propagation

Causality in NLED is tightly linked with the analytic structure and convergence of the heat-kernel expansion. The following constraints ensure subluminal and hyperbolic wave propagation on arbitrary backgrounds:

• Weak-field convexity: $L_{II} \ge 0$, $L_{JJ} \ge 0$, $L_{II}L_{JJ} - (L_{IJ})^2\ge 0$
• Strong-field propagation: $$-L_I + (L_{II}-L_{JJ})I - 2L_{IJ}J - (L_{II} + L_{JJ})\sqrt{I^2 + J^2}>0$$.

For conformal NLED, these criteria are necessary and sufficient for the convergence and real-valuedness of the exact $a_0$, $a_1$, and $a_2$ coefficients. The presence of phase ambiguities or infrared divergences in one-loop quantities signals a causality violation (Buchbinder et al., 27 Jan 2026). This connection makes causality constraints both a physical and mathematical consistency requirement for conformal NLED quantized actions.

6. Exemplars: ModMax Model and Black Hole Applications

The one-parameter ModMax conformal NLED, with Lagrangian

$$L_{MM}(I, J) = -\cosh\gamma\ I + \sinh\gamma\ \sqrt{I^2 + J^2}\qquad (\gamma \in \mathbb{R})$$

is a uniquely conformal- and duality-invariant theory reducible to Maxwell ($\gamma \to 0$) (Zhang et al., 2021, Buchbinder et al., 27 Jan 2026). The ModMax model admits exact coupling to gravity and conformally coupled scalars. For example, in "conformal scalar NUT-like dyons," the full matter sector is given by

$$\mathcal{L}_{CE}(S, P) = -\frac12[S\cosh\gamma - \sqrt{S^2 + P^2}\sinh\gamma]$$

and the full action combines Einstein-Hilbert, conformal scalar, and NLED terms. Black hole solutions exhibit standard thermodynamic behavior, with modifications parametrized by $\gamma$ (nonlinearity), and the radius of ISCOs and photon shadows increase with increasing $\gamma$, reflecting enhanced NLED effects (Zhang et al., 2021).

7. Physical Implications and Theoretical Context

Conformal NLED models lack intrinsic physical scales; all nonlinear effects are functions of the ratio $u = I_2/I_1$ or $\sqrt{I^2 + J^2}/I$. This scale-free structure precludes the existence of smooth critical field cutoffs (as in Born-Infeld theory) unless the generating functions $M(u)$ or equivalents are specifically chosen to regulate the theory in a conformally invariant manner (Duplij et al., 2019). In quantum theory, counterterm structure is highly constrained: no renormalization of the Maxwell term or induced mass scales emerge at any order in perturbation. Conformal NLED can serve as effective field theories for phase transitions, critical points, or as phenomenological laboratories for renormalization-group fixed points in QED, and as tractable analogs for AdS/CFT duality and conformal field theory with interacting gauge sectors.

The theory thus provides a complete algebraic and analytic classification of NLED models—Lagrangian and non-Lagrangian—admitting full conformal symmetry in four dimensions, with quantization and causality requirements interlinked at both the classical and quantum levels (Duplij et al., 2019, Buchbinder et al., 27 Jan 2026, Zhang et al., 2021).