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Boundary Actions and Loop Groups: A Geometric Picture of Gauge Symmetries at Null Infinity

Silvia Nagy, Javier Peraza, Giorgio Pizzolo

TL;DR

This work provides a concrete boundary action for Stueckelberg fields that realize large gauge symmetries at null infinity in Yang–Mills theory, enabling a first-principles derivation and renormalization of the associated charges. It develops a two-step Stueckelberg procedure to extend the bulk gauge symmetry to a boundary-friendly G, and recasts the construction in a rigorous fibre-bundle framework, highlighting how dressing fields implement reductions and extensions of structure groups. A geometric picture emerges in which boundary gauge transformations form a formal loop group, with the Stueckelberg field living on the boundary and transforming under this loop structure, while the boundary vector potential remains invariant. The approach yields finite, well-defined charges that reproduce subleading soft-theorem structures and provides a scalable framework for incorporating loop corrections and gravity. Overall, the paper advances a holographic-like, geometrically robust description of gauge symmetries at null infinity and lays groundwork for extensions to gravity and higher-spin algebras.

Abstract

In previous work arXiv:2407.13556, we proposed an extended phase space structure at null infinity accommodating large gauge symmetries for sub$^n$-leading soft theorems in Yang-Mills, via dressing fields arising in the Stueckelberg procedure. Here, we give an explicit boundary action controlling the dynamics of these fields. This allows for a derivation from first principles of the associated charges, together with an explicit renormalization procedure when taking the limit to null and spatial infinity, matching with charges proposed in previous work. Using the language of fibre bundles, we relate the existence of Stueckelberg fields to the notion of extension/reduction of the structure group of a principal bundle, thereby deriving their transformation rules in a natural way, thus realising them as Goldstone-like objects. Finally, this allows us to give a geometric picture of the gauge transformation structure at the boundary, via a loop group coming from formal expansions in the coordinate transversal to the boundary.

Boundary Actions and Loop Groups: A Geometric Picture of Gauge Symmetries at Null Infinity

TL;DR

This work provides a concrete boundary action for Stueckelberg fields that realize large gauge symmetries at null infinity in Yang–Mills theory, enabling a first-principles derivation and renormalization of the associated charges. It develops a two-step Stueckelberg procedure to extend the bulk gauge symmetry to a boundary-friendly G, and recasts the construction in a rigorous fibre-bundle framework, highlighting how dressing fields implement reductions and extensions of structure groups. A geometric picture emerges in which boundary gauge transformations form a formal loop group, with the Stueckelberg field living on the boundary and transforming under this loop structure, while the boundary vector potential remains invariant. The approach yields finite, well-defined charges that reproduce subleading soft-theorem structures and provides a scalable framework for incorporating loop corrections and gravity. Overall, the paper advances a holographic-like, geometrically robust description of gauge symmetries at null infinity and lays groundwork for extensions to gravity and higher-spin algebras.

Abstract

In previous work arXiv:2407.13556, we proposed an extended phase space structure at null infinity accommodating large gauge symmetries for sub-leading soft theorems in Yang-Mills, via dressing fields arising in the Stueckelberg procedure. Here, we give an explicit boundary action controlling the dynamics of these fields. This allows for a derivation from first principles of the associated charges, together with an explicit renormalization procedure when taking the limit to null and spatial infinity, matching with charges proposed in previous work. Using the language of fibre bundles, we relate the existence of Stueckelberg fields to the notion of extension/reduction of the structure group of a principal bundle, thereby deriving their transformation rules in a natural way, thus realising them as Goldstone-like objects. Finally, this allows us to give a geometric picture of the gauge transformation structure at the boundary, via a loop group coming from formal expansions in the coordinate transversal to the boundary.

Paper Structure

This paper contains 21 sections, 8 theorems, 146 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Stueckelberg procedure: review and extension
  3. Group theoretic formulation of the Stueckelberg procedure
  4. Decoupling all the symmetries via an extended Stueckelberg procedure
  5. Edge mode Lagrangian for YM
  6. Charges
  7. Renormalization
  8. Fibre bundle derivation
  9. Review of fibre bundle formulation of gauge theories
  10. Principal and associated bundles
  11. Gauge transformations as vertical automorphisms
  12. Reduction and extension of structure group
  13. The dressing field method
  14. Connection on reduced bundles
  15. Stueckelberg fields
  16. ...and 6 more sections

Key Result

Proposition 5.4

Let $\xi=(P,\pi,M)$ be a principal $G$-bundle, $F$ be a left $G$-manifold and where $L$ is the left $G$-action on $F$. Then $\mathcal{U}\xleftrightarrow{1:1}\Gamma(\xi[F])$ via where

Figures (2)

  • Figure 1: Representation of the two families of slices commonly used to define the symplectic structure at null infinity. Dashed line: $t=cnt.$ hypersurfaces. Solid line: $r=cnt.$ hypersurfaces. Light lines: light rays.
  • Figure 2: Schematic representation of the map from gauge group on $M$ and the gauge group on $\partial M$.

Theorems & Definitions (30)

  • Definition 5.1: principal bundle
  • Definition 5.2: principal bundle (iso)morphism
  • Definition 5.3: associated bundle
  • Proposition 5.4
  • Definition 5.5: adjoint bundle
  • Definition 5.6: gauge transformation
  • Proposition 5.7
  • Definition 5.8: reduction/extension
  • Lemma 5.9
  • Proposition 5.10: existence of reduction/extension
  • ...and 20 more