Self-Dual Electrodynamics via the Characteristic Method: Relativistic and Carrollian Perspectives

TL;DR

This work develops a characteristic-method framework to solve the Gaillard-Zumino self-duality equation for nonlinear electrodynamics, recasting it as a first-order PDE in the invariants $S$ and $P$ and solving it via flows in an extended space of $(S,P,L,p_S,p_P)$. Starting from seed Lagrangians and boundary data, the authors construct a broad class of self-dual theories in both relativistic and Carrollian settings, recovering known models such as Maxwell, Born-Infeld, and ModMax, and introducing new families with closed-form Lagrangians. A notable attractor behavior is observed: different seed theories can converge to the same self-dual descendant under the characteristic flow, revealing a universal structure in the space of duality-invariant NEDs. The framework also yields Carrollian analogues of duality-preserving theories, connecting ultra-relativistic limits to their relativistic counterparts and suggesting applications to holography, flat-space physics, and higher-form generalizations.

Abstract

Electric-magnetic duality plays a pivotal role in understanding the structure of nonlinear electrodynamics (NED). The Gaillard-Zumino (GZ) criterion provides a powerful constraint for identifying self-dual theories. In this work, we systematically explore solutions to the GZ self-duality condition by applying the method of characteristics, a robust tool for solving nonlinear partial differential equations. Our approach enables the construction of new classes of Lagrangians that respect duality symmetry, both in the relativistic and Carrollian frameworks. In the relativistic setting, we not only recover well-known examples such as Born-Infeld and ModMax theories, but also identify novel models. We then generalize the GZ formalism to the Carrollian case and construct several classes of Carrollian self-dual non-linear electrodynamic models. Remarkably, we demonstrate that the characteristic flow exhibits an attractor behavior, in the sense that different seed theories that may not be self-dual can generate the same descendant self-dual Lagrangian. These findings broaden the landscape of self-dual theories and open new directions for exploring duality in ultra-relativistic regimes.

Figures (1)

• Figure 1: Schematic of characteristic flow converging to a duality-invariant surface in $(S, P)$ space. That different boundary choices can generate the same self-dual Lagrangian. For instance, all of the boundary $P=0, L_0=-e^{-\alpha}S$, the boundary $S=0, L_0= \sinh(\alpha)P$, and the boundary $P=2 \sqrt{1-S}, L_0= -e^{-\alpha}S +2 \sinh(\alpha)$ generate the ModMax.