ModMax Nonlinear Electrodynamics

• ModMax nonlinear electrodynamics is a one-parameter nonlinear modification of classical Maxwell theory that retains full Poincaré, conformal, and electromagnetic duality invariance.
• It introduces a deformation parameter y which employs hyperbolic functions of the field invariants S and P to regulate null configurations and ensure nonlinearity.
• The theory enables the explicit construction of deformed electromagnetic knots, offering robust insights into topological solitons in both high-energy and quantum electrodynamics.

ModMax nonlinear electrodynamics (NLED) is the unique continuous, one-parameter family of nonlinear deformations of source-free Maxwell electrodynamics in four-dimensional Minkowski vacuum that preserves the full set of Poincaré and conformal spacetime symmetries along with continuous electromagnetic duality invariance. ModMax theory therefore provides a distinguished theoretical laboratory for exploring nonlinear modifications to classical electromagnetism while retaining the exceptional symmetry properties of the original theory.

1. Structure of ModMax Theory

The ModMax theory introduces a dimensionless, positive deformation parameter $y$ (sometimes denoted $\gamma$ in the literature) such that the Lagrangian density is

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

where $S$ and $P$ are the usual Lorentz and parity invariants of the electromagnetic field: $$S = \frac{1}{2}(\mathbf{E}^2 - \mathbf{B}^2),\quad P = \mathbf{E}\cdot\mathbf{B}.$$ For $y = 0$, the theory reduces continuously to the standard Maxwell Lagrangian. The theory is constructed so that all of the following are preserved:

• Poincaré invariance,
• four-dimensional conformal invariance,
• continuous $SO(2)$ electromagnetic duality invariance.

Unlike Born–Infeld or Euler–Heisenberg nonlinear electrodynamics, ModMax maintains conformal invariance in four dimensions and electromagnetic duality as a continuous symmetry, with the latter realized manifestly in the Hamiltonian rather than the Lagrangian formulation.

2. The Deformation Parameter and Nonlinearity

The real parameter $y>0$ directly controls the nonlinearity of the theory. For $y\neq 0$, the Lagrangian contains hyperbolic functions of $y$ multiplying both the standard Maxwell invariant $S$ and the square root of $S^2 + P^2$. As $y\to 0$, $\cosh y\to 1$ and $\sinh y\to 0$, so $\mathcal{L}_0(S,P) = S$, i.e., Maxwell. For $y>0$, all null field configurations of Maxwell theory ($S=P=0$) become non-admissible as regular Lagrangian solutions, because the ModMax Lagrangian is non-analytic at $(S,P) = (0,0)$. Thus, previously null configurations must "self-adjust" under the deformation to become non-null, leading to physically regular solutions.

3. Symmetries and the Spacetime Structure

ModMax retains the entire suite of spacetime and internal symmetries present in Maxwell theory:

• Poincaré invariance: The theory is invariant under spacetime translations and Lorentz transformations.
• Conformal invariance: The theory remains invariant under conformal group actions in Minkowski space, including scale and special conformal transformations.
• Electromagnetic duality: The continuous $U(1)$ symmetry rotating the electric and magnetic fields into each other is preserved (observed most transparently in the Hamiltonian formulation).

These invariances are crucial, as many explicit constructions (e.g., via Bateman's method for null configurations, or through conformal inversions) leverage them to generate physically and topologically nontrivial solutions.

4. Null Fields and Electromagnetic Knots

In linear Maxwell theory, null field configurations ($S=P=0$) underlie field topologies known as electromagnetic knots—for example, hopfion–Rañada solutions. The Bateman construction uses pairs of self-dual complex scalar potentials to build such knotted null fields. In ModMax electrodynamics, these strictly null configurations are singular as the Lagrangian becomes non-analytic at $S=P=0$. Upon deformation ($y>0$), solutions which were strictly null "split": the field strengths must deform so that $S$ and $P$ are never exactly zero, "regularizing" the solution in the context of the Lagrangian formalism.

The topological properties—such as the knotting and linking of electric and magnetic field lines, and associated invariants like helicity—are preserved, but the local field characteristics change (in particular, the fields become non-null and the standard duality between E and B is modified via a redefinition of the Riemann–Silberstein vector).

5. Deformed Hopfion–Rañada Knots

A significant technical achievement is the explicit construction of ModMax-deformed hopfion–Rañada knots. The key result is as follows:

• When ModMax deformation is switched on ($y>0$), each standard Maxwell hopfion–Rañada knot bifurcates into two distinct deformed solutions, parameterized by $d=\pm 1$.
• As $y\to 0$, the two deformed solutions continuously coalesce to the original Maxwell hopfion–Rañada knot, recovering the null field solution in the linear limit.

These deformed knots are constructed using generalized Bateman potentials and a nontrivial redefinition of Riemann–Silberstein vectors in the nonlinear, duality-invariant context of ModMax. The duality invariance remains manifest in the Hamiltonian structure. The constitutive relation for the deformed theory in RS variables takes the general form

$$\mathcal{S}(\mathbf{R}, \mathbf{R}^*) = -i \cosh y\, \mathbf{R} + i \sinh y\, f(\mathbf{R}^2, \mathbf{R}^{*2}),$$

where $\mathbf{R}$ is the (complex) RS vector, and $f$ encodes the nonlinearity arising from ModMax.

6. Preservation and Deformation of Topology

Despite the lack of regular null Lagrangian solutions for $y>0$, the deformed knot solutions maintain the nontrivial topological structure:

• The helicity, Chern–Simons invariants, and field line linkage remain unchanged under continuous deformation.
• The Lagrangian non-analyticity at $(S,P)=0$ is circumvented by the solution's self-adjustment, so the underlying knotted topology survives in the nonlinear regime.
• The full Bateman construction can, in principle, be generalized to yield a broader class of $(p,q)$ knot solutions in nonlinear electrodynamics with the same symmetry content.

These results confirm that robust electromagnetic topological solitons persist in fully nonlinear, duality- and conformally-invariant settings, opening the possibility of studying their physical relevance in a broader range of physical theories—including those incorporating quantum or strong-field effects.

7. Implications and Research Directions

The findings provide robust evidence that ModMax nonlinear electrodynamics accommodates nontrivial field topologies akin to those permitted in Maxwell’s theory, with the deformed knots linked in a doubled, non-null fashion. This demonstrates the non-destructive effect of well-designed nonlinear deformations on electromagnetic topology. Directions for further research highlighted in the work include:

• Investigating deformations of other nontrivial field topologies (e.g., general $(p,q)$ knots) under ModMax-type nonlinearities.
• Exploring the continuous deformation of the double Hopf fibration ancillary to the hopfion–Rañada knot and the fate of their algebraic and geometric invariants.
• Studying the application of such topological nonlinear field configurations in physical scenarios where nonlinear electrodynamics is expected to be relevant—ranging from quantum vacuum birefringence to magnetohydrodynamic plasmas and possibly cosmological settings.

A plausible implication is that even when high-energy or quantum corrections introduce duality- and conformally-invariant nonlinearities, the underlying field topology remains viable and can potentially be observed or utilized in experimental and astrophysical contexts. This suggests new robustness for the paper of electromagnetic knots beyond the confines of linear theory.