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Duality, asymptotic charges and higher form symmetries in $p$-form gauge theories

Federico Manzoni

TL;DR

The work analyzes surface charges for $p$-form gauge fields in the Bondi patch of $D$-dimensional Minkowski space and shows that under Hodge duality, electric-like charges map to magnetic-like charges of the dual $(D-p-2)$-form, with the complexified charge transforming via a Möbius map. It then develops a geometrized CCFT picture in $D=4$ as a $\mathrm{PGL}(2,\mathbb{C})$-equivariant bundle over the celestial sphere, and proves an existence and uniqueness theorem for the duality map using de Rham cohomology, revealing the duality as a topological relation between charges. The paper also connects asymptotic charges to higher-form symmetries, showing how smeared charges reproduce standard higher-form charges when residual gauge parameters are chosen appropriately, thereby linking infrared structures to generalized global symmetries. Collectively, these results offer a unified, topologically grounded view of dualities, asymptotic charges, and celestial holography for abelian $p$-form theories, with clear avenues for extending to non-abelian sectors and richer holographic structures.

Abstract

The surface charges associated with $p$-form gauge fields in the Bondi patch of $D$-dimensional Minkowski spacetime are computed. We show that, under the Hodge duality between the field strengths of the dual formulations, electric-like charges for $p$-forms are mapped to magnetic-like charges for the dual $q$-forms, with $q=D-p-2$. We observe that the complex combination of electric-like and magnetic-like charges transforms under duality according to a specific Möbius transformation. This leads to a possible construction of CCFT in $D=4$ as a Möbius-principal equivariant bundle, together with its associated bundles, in order to recover celestial operators. We prove an existence and uniqueness theorem for the duality map relating the asymptotic electric-like charges of the dual descriptions, and we provide an algebraic-topological interpretation of this map. As a result, the duality map has a topological nature and ensures that the charge of one formulation contains information about the dual formulation, leading to a deeper understanding of gauge theories, the non-trivial charges associated with them, and the duality of their observables. Moreover, we propose a link between higher-form symmetry charges, naturally associated with a $p$-form gauge theory, and their asymptotic charges. The higher-form charges are reproduced by choosing the gauge parameter to be constant and supported only on an appropriate codimension submanifold. This could partially answer an open question in the celestial holography program.

Duality, asymptotic charges and higher form symmetries in $p$-form gauge theories

TL;DR

The work analyzes surface charges for -form gauge fields in the Bondi patch of -dimensional Minkowski space and shows that under Hodge duality, electric-like charges map to magnetic-like charges of the dual -form, with the complexified charge transforming via a Möbius map. It then develops a geometrized CCFT picture in as a -equivariant bundle over the celestial sphere, and proves an existence and uniqueness theorem for the duality map using de Rham cohomology, revealing the duality as a topological relation between charges. The paper also connects asymptotic charges to higher-form symmetries, showing how smeared charges reproduce standard higher-form charges when residual gauge parameters are chosen appropriately, thereby linking infrared structures to generalized global symmetries. Collectively, these results offer a unified, topologically grounded view of dualities, asymptotic charges, and celestial holography for abelian -form theories, with clear avenues for extending to non-abelian sectors and richer holographic structures.

Abstract

The surface charges associated with -form gauge fields in the Bondi patch of -dimensional Minkowski spacetime are computed. We show that, under the Hodge duality between the field strengths of the dual formulations, electric-like charges for -forms are mapped to magnetic-like charges for the dual -forms, with . We observe that the complex combination of electric-like and magnetic-like charges transforms under duality according to a specific Möbius transformation. This leads to a possible construction of CCFT in as a Möbius-principal equivariant bundle, together with its associated bundles, in order to recover celestial operators. We prove an existence and uniqueness theorem for the duality map relating the asymptotic electric-like charges of the dual descriptions, and we provide an algebraic-topological interpretation of this map. As a result, the duality map has a topological nature and ensures that the charge of one formulation contains information about the dual formulation, leading to a deeper understanding of gauge theories, the non-trivial charges associated with them, and the duality of their observables. Moreover, we propose a link between higher-form symmetry charges, naturally associated with a -form gauge theory, and their asymptotic charges. The higher-form charges are reproduced by choosing the gauge parameter to be constant and supported only on an appropriate codimension submanifold. This could partially answer an open question in the celestial holography program.

Paper Structure

This paper contains 13 sections, 1 theorem, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. p-form gauge theories, their duality and their asymptotic charges
  3. p-form gauge theories gauge invariant dynamics
  4. Duality in $p$-form gauge theories
  5. Lorenz-like gauge fixing, residual gauge and asymptotic charges
  6. Magnetic-like charges for $p$-form gauge theories
  7. Geometrization of CCFT
  8. An existence and uniqueness theorem for the duality map
  9. Duality map for well-defined charges
  10. Physical interpretation
  11. Comments on the duality for power law divergent/vanishing charges
  12. Asymptotic charges and $p$-form symmetries
  13. Conclusions and outlook

Key Result

Theorem 3.1

Let $(M_D,\boldsymbol{\eta})$ be the $D$-dimensional Minkowski spacetime where the de Rham cohomology groups in positive degree $n>0$ vanish, $H^n=0$. Let $p\in [1, \frac{D-2}{2}]$ and $q:=D-p-2$; let $H$ and $\tilde{H}$ be the field strengths of a $p$-form gauge field $B$ and of a $q$-form gauge fi

Figures (1)

  • Figure 1: Schematic picture of the action of the duality on the complex electromagnetic-like asymptotic charge $\mathcal{Q}_{q,D}^{(\mathrm{em})}$. A similar scheme can be drawn for $\mathcal{Q}_{p,D}^{(\mathrm{em})}$.

Theorems & Definitions (4)

  • Theorem 3.1: Existence and uniqueness of the duality map for well-defined charges
  • proof
  • Definition 3.1: Power law weak fall-offs
  • Conjecture 3.1: Link between trivial gauge transformations and power law weak fall-offs