Invariance of interaction terms in new representation of self-dual electrodynamics

TL;DR

This paper introduces a new representation for 4D nonlinear electrodynamics using auxiliary bispinor fields, simplifying the characterization of self-duality and duality symmetries in the theory.

Contribution

It presents a novel formulation with auxiliary fields that makes duality symmetries more transparent and can be extended to multiple gauge fields with U(n) invariance.

Findings

Interaction Lagrangian E(V) depends on auxiliary fields and exhibits invariance properties.

Continuous SO(2) duality symmetry corresponds to E being a function of |V|^4.

Discrete self-duality corresponds to E(V)= E(iV).

Abstract

A new representation of Lagrangians of 4D nonlinear electrodynamics is considered. In this new formulation, in parallel with the standard Maxwell field strength F, an auxiliary bispinor (tensor) field V is introduced. The gauge field strength appears only in bilinear terms of the full Lagrangian, while the interaction Lagrangian E depends on the auxiliary fields, E = E(V). Two types of self-duality inherent in the nonlinear electrodynamics models admit a simple characterization in terms of the function E. The continuous SO(2) duality symmetry between nonlinear equations of motion and Bianchi identities amounts to requiring E to be a function of the SO(2) invariant quartic combination |V|^4. The discrete self-duality (or self-duality under Legendre transformation) amounts to a weaker condition E(V)= E(iV). This approach can be generalized to a system of n Abelian gauge fields exhibiting U(n) duality. The corresponding interaction Lagrangian should be U(n) invariant function of n bispinor auxiliary fields.