In theoretical physics, -form electrodynamics is a generalization of Maxwell's theory of electromagnetism.

Ordinary (via. one-form) Abelian electrodynamics

, a gauge symmetry

is any arbitrary fixed 0-form is the exterior derivative, and a gauge-invariant vector current with density 1 satisfying the continuity equation

is the Hodge star operator.

as a closed -form, but we do not consider that case here.

is a gauge-invariant 2-form.

satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

is the spacetime manifold.

p-form Abelian electrodynamics

We have a , a gauge symmetry

is any arbitrary fixed is the exterior derivative, and a gauge-invariant p-vector with density 1 satisfying the continuity equation

is the Hodge star operator.

as a closed -form.

is a gauge-invariant .

satisfies the equation of motion

(this equation obviously implies the continuity equation).

This can be derived from the action

where is the spacetime manifold.

Other sign conventions do exist.

The Kalb–Ramond field is an example with in string theory; the Ramond–Ramond fields whose charged sources are D-branes are examples for all values of . In eleven-dimensional supergravity or M-theory, we have a 3-form electrodynamics.

Non-abelian generalization

Just as we have non-abelian generalizations of electrodynamics, leading to Yang–Mills theories, we also have nonabelian generalizations of -form electrodynamics. They typically require the use of gerbes.

Asymptotic symmetries

The study of asymptotic symmetries began in the early 1960s with the work of Bondi, van der Burg, Metzner, and Sachs on asymptotically flat gravity. The central result was that the symmetry group at zero infinity is not simply the Poincaré group, but the BMS group, which contains an infinite number of transformations called supertranslations.

Recently, thanks in particular to the work of Strominger and collaborators, it has emerged that asymptotic symmetries are deeply connected to soft theorems and memory effects. This connection is often called the infrared triangle.

Physical relevance

, the critical dimension. was studied.-forms and the connection with on-shell duality has been studied in its generalities since 2024.

Technical points

the opposite occurs. This distinction is crucial because it determines whether certain charges are finite, divergent, or subleading.

. These terms can produce slight violations of the potential falloff, but in many cases they remain confined to the pure gauge sectors and do not modify the physical components of the force field or they do not modify observable quantities. The presence of logarithms is important because without them, some asymptotic charges would be trivialized by the equation of motion or gauge fixing.

Electric and magnetic charges

-form, the asymptotic electric charge schematically has the form

, describe the same on-shell degrees of freedom but have different gauge fields and gauge parameters. Asymptotically, Hodge duality maps electric charges of a description to magnetic charges of the dual description:

For radiative data, this map connects finite and non-zero charges in both formulations. For Coulomb data, however, the situation is more subtle: a divergent charge in one description may correspond to a subleading or vanishing charge in the dual description. This suggests that the complete structure of asymptotic symmetries should be attributed not only to a single formulation of the theory but also to the set of its dual descriptions. Moreover, under topological condition such as the vanishing of suitable de Rham cohomology groups, there exists a unique duality map such that

and viceversa.

Highe-form symmetries

-forms suggests that asymptotic charges can be viewed as a localized, or refined, version of higher-form symmetry charges at null infinity. In other words, asymptotic symmetries provide an angular resolution of global topological charges.-forms can be viewed as a localization at null infinity of the higher-form symmetries, with possible interpretation in terms of celestial currents and extended topological operators.

This connection is also important for celestial holography, where celestial conformal theory operators would encode infrared scattering charges in flat space, since it provides a possible solution to one of today's open questions of the flat holography program concerning the connection between asymptotic charges and higher-form symmetries.

Higher-order asymptotic symmetries and renormalization

, up to numerical factors.However, higher-order charges are often divergent before renormalization. To make them finite, symplectic renormalization is used, which exploits the ambiguities of the presymplectic potential. By adding appropriate counterterms to the boundary, the divergences are canceled while maintaining the physically significant finite part. This procedure is similar, in spirit, to holographic renormalization. The charge algebra thus obtained is abelian in the purely electric sector but central extensions or anomalous terms may appear in the mixed electric-magnetic sectors, or due to the residual ambiguities of symplectic renormalization.

References