Higher-order $p$-form asymptotic symmetries in $D = p + 2$

We investigate higher-order asymptotic symmetries for a $p$-form gauge field in $(p + 2)$-dimensional Minkowski spacetime, where Hodge duality with a scalar holds. Employing symplectic renormalization, we identify $N + 1$ independent asymptotic charges, with each charge being parametrised by an arbitrary function of the angular variables. By means of the Hodge decomposition, these charges share the same formal structure independently from p and are manifestly dual to a scalar charge. We work in Lorenz gauge, therefore the gauge parameters require a radial expansion involving logarithmic (subleading) terms to ensure nontrivial angular dependence at leading order. At the same time we assume a power expansion for the field strength, allowing logarithms in the gauge field expansions within pure gauge sectors.

💡 Research Summary

This paper investigates the structure of higher‑order asymptotic symmetries for a p‑form gauge field in flat Minkowski space of dimension D = p + 2, a setting where the field is Hodge‑dual to a scalar. Working in Lorenz gauge (∇·B = 0), the authors show that the residual gauge parameters ε must admit a radial expansion that includes subleading logarithmic terms. These logarithmic pieces are essential: they allow the angular dependence of ε to be non‑trivial at leading order, thereby generating non‑vanishing asymptotic charges.

The analysis begins with a thorough review of the necessary mathematical tools: Bondi‑type retarded coordinates (u, r, xⁱ), the de Rham complex, Hodge star, co‑differential d†, and the Hodge decomposition on the p‑sphere Sᵖ. Because Sᵖ has non‑trivial cohomology only in degrees 0 and p, any (p − 1)‑form on Sᵖ can be written uniquely as an exact piece dα plus a co‑exact piece d†β, with no harmonic contribution. This decomposition is applied both to the angular part of the gauge parameter ε and to the angular components of the field strength, yielding a universal expression for the asymptotic charge that is independent of the form degree p.

The field strength F = dB is assumed to have a pure power‑law expansion in r, while logarithms are permitted only in the pure‑gauge sector of the potential B. Solving the equations of motion together with the Lorenz condition determines the allowed fall‑off of the physical components and the necessary form of the gauge parameters. The authors introduce the notion of “order” of an asymptotic symmetry: an O(rᴺ) symmetry produces an O(rᴺ) charge before renormalization. This mirrors the familiar hierarchy in electromagnetism where supertranslations are O(1) and super‑rotations are O(r).

A central technical achievement is the implementation of symplectic renormalization. Starting from the Lagrangian variation δL = EOM·δΦ + dθ, the presymplectic potential θ possesses ambiguities of the type θ → θ + δΞ + dΥ. By expanding θ in both the large‑t (time) and large‑u (retarded time) regimes, the authors isolate divergent pieces that scale as powers of r and as r ln r. Appropriate counterterms Υ are constructed to cancel these divergences while leaving the finite part untouched. The resulting renormalized charge Q₍ren₎ is finite, integrable, and depends on N + 1 independent functions on the sphere (the coefficients of α and β in the Hodge decomposition).

The charge algebra is then computed using the covariant phase‑space formalism. For two residual gauge transformations ε₁ and ε₂, the Poisson bracket of their charges yields another charge associated with the commutator