Soft Charges and Electric-Magnetic Duality

TL;DR

This paper investigates magnetic soft charges in four-dimensional Maxwell theory, revealing their algebraic structure and duality properties, with implications for boundary symmetries and gauge transformations.

Contribution

It introduces a detailed analysis of magnetic soft charges, their algebra, and duality symmetry in Maxwell theory, extending previous electric charge studies.

Findings

Electric and magnetic soft charges form a complex $U(1)$ current algebra.

The algebra of charges includes two $U(1)$ Kac-Moody algebras.

Electric-magnetic duality relates soft charges to infinite $iso(2)$ algebra.

Abstract

The main focus of this work is to study magnetic soft charges of the four dimensional Maxwell theory. Imposing appropriate asymptotic falloff conditions, we compute the electric and magnetic soft charges and their algebra both at spatial and at null infinity. While the commutator of two electric or two magnetic soft charges vanish, the electric and magnetic soft charges satisfy a complex $U(1)$ current algebra. This current algebra through Sugawara construction yields two $U(1)$ Kac-Moody algebras. We repeat the charge analysis in the electric-magnetic duality-symmetric Maxwell theory and construct the duality-symmetric phase space where the electric and magnetic soft charges generate the respective boundary gauge transformations. We show that the generator of the electric-magnetic duality and the electric and magnetic soft charges form infinite copies of $iso(2)$ algebra. Moreover, we study the algebra of charges associated with the global Poincar\'e symmetry of the background Minkowski spacetime and the soft charges. We discuss physical meaning and implication of our charges and their algebra.