Soft Charges and Electric-Magnetic Duality

TL;DR

This work analyzes electric and magnetic soft charges in four-dimensional Maxwell theory and in its duality-symmetric formulation, revealing that while electric and magnetic soft charges commute within their own type, they do not commute with each other, forming a complex current algebra $\{Q_f^E,Q_g^B\}=\oint df\wedge dg$. The authors construct the duality-symmetric phase space with boundary edge modes, introduce the duality generator $Q_\theta$ (optical helicity), and show that the electric/magnetic soft charges together with $Q_\theta$ generate infinite copies of $iso(2)$ algebras. They analyze the charges at spatial and null infinity, derive their Poisson brackets, and relate the algebra to physical observables such as memory effects and Aharonov-Bohm phases, while connecting to Poincaré charges and angular momentum. The Sugawara construction yields two $U(1)$ Kac-Moody algebras, and the discussion extends to quantization, Virasoro structures, and potential generalization to higher-form theories, highlighting the infrared structure and boundary dynamics of gauge theories.

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 Poincaré symmetry of the background Minkowski spacetime and the soft charges. We discuss physical meaning and implication of our charges and their algebra.

Figures (4)

• Figure 1: Penrose diagram of 4d flat space. ${\cal I}^+_\pm$ denote the past and future boundaries of the null infinity ${\cal I}^+$. These boundaries are essentially future time like infinity $i^+$ and the spatial infinity $i^0$.
• Figure 2: Different ways of turning $\mathcal{I}^+$ into a Cauchy surface. Either a) consider a constant time $t=T\to \infty$ as in the left panel, or b) cutting the null infinity at large $u$ and attaching to it a spacelike surface which extends to $r=0$ as in the right panel.
• Figure 3: The phase space of soft charges and a depiction of \ref{['algebra-spatial']}. The horizontal planes depict configurations of given constant magnetic charge. ${P}^E$ moves us between these horizontal planes while $Q^E_n$, $Q^B_n$ and ${P}^B$ move us on each constant $Q_0^B$ plane. Vertical arrows show flows generated by ${P}^E$ and corresponds to addition of a Dirac string piercing the celestial sphere and appears as a surface magnetic charge for the local boundary observers. One could have drawn a similar figure using any other conjugate pairs of charges instead of ${P}^E$ and $Q^B_0$, e.g. ${P}^B$ and $Q^E_0$. This figure may be contrasted with the "just electric" residual gauge symmetry phase space which is usually considered in the Maxwell theory. In the absence of magnetic soft charges, the electric soft algebra is Abelian and hence action of residual electric gauge transformations does not create a flow on the phase space.
• Figure 4: A depiction of the pure gauge transformation which at the celestial sphere becomes $\ln z$. This gauge transformation may be viewed as a Dirac string connecting the south and north poles of the celestial sphere with some bulk extension. It produces a "magnetic charge" at the north pole and an "anti-magnetic charge" at the south pole. The observer at the north pole who uses complex coordinates $z,\bar{z}$ however, does not have access to the charge in the south pole and hence only sees a net magnetic charge. Similar picture may also be drawn for other higher pole charges. Note that in $2+1$ spacetime (boundary observers), there are three residual gauge symmetry components, two for the electric one-form (or vector) and one for the magnetic two-form (or scalar). Therefore, for the boundary observer the net magnetic monopole is computed as the integral of two-form $B$ on the region confined by the contour $c$, or equivalently, $\oint_c A$.