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

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.

💡 Research Summary

The paper tackles the long‑standing problem of constructing four‑dimensional nonlinear electrodynamics (NED) theories that are invariant under continuous electric‑magnetic SO(2) duality. The authors focus on the Gaillard‑Zumino (GZ) self‑duality condition, which can be written as a nonlinear first‑order partial differential equation (PDE) for the Lagrangian (L(S,P)) where (S=\frac14F_{\mu\nu}F^{\mu\nu}) and (P=\frac14F_{\mu\nu}\tilde F^{\mu\nu}).

Methodology – Characteristics:
By introducing the conjugate variables (p_S=\partial L/\partial S) and (p_P=\partial L/\partial P), the GZ equation becomes
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