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Scalar Asymptotic Charges and Dual Large Gauge Transformations

Miguel Campiglia, Laurent Freidel, Florian Hopfmüller, Ronak M Soni

TL;DR

The paper shows that scalar soft theorems in four dimensions can be understood as Ward identities for large gauge transformations by recasting the massless scalar as a Hodge-dual two-form gauge field. It derives the corresponding asymptotic charges from the two-form perspective, demonstrating that these charges generate large gauge transformations and, crucially, act trivially on the scalar data, placing them in the center of the algebra. It also identifies a missing charge associated with a constant-on-the-sphere parameter, interpreting it as boundary centre degrees of freedom supported by a Kramers-Wannier-type edge-mode structure, with a lattice duality providing the analogous intuition. The work thus extends the gauge-symmetry/soft-theorem correspondence to scalar theories and highlights subtle edge-mode phenomena at infinity. A lattice analogue reinforces the interpretation and points to interesting future directions in quantum and boundary dynamics of scalar theories.

Abstract

In recent years soft factorization theorems in scattering amplitudes have been reinterpreted as conservation laws of asymptotic charges. In gauge, gravity, and higher spin theories the asymptotic charges can be understood as canonical generators of large gauge symmetries. Such a symmetry interpretation has been so far missing for scalar soft theorems. We remedy this situation by treating the massless scalar field in terms of a dual two-form gauge field. We show that the asymptotic charges associated to the scalar soft theorem can be understood as generators of large gauge transformations of the dual two-form field. The dual picture introduces two new puzzles: the charges have very unexpected Poisson brackets with the fields, and the monopole term does not always have a dual gauge transformation interpretation. We find analogs of these two properties in the Kramers-Wannier duality on a finite lattice, indicating that the free scalar theory has new edge modes at infinity that canonically commute with all the bulk degrees of freedom.

Scalar Asymptotic Charges and Dual Large Gauge Transformations

TL;DR

The paper shows that scalar soft theorems in four dimensions can be understood as Ward identities for large gauge transformations by recasting the massless scalar as a Hodge-dual two-form gauge field. It derives the corresponding asymptotic charges from the two-form perspective, demonstrating that these charges generate large gauge transformations and, crucially, act trivially on the scalar data, placing them in the center of the algebra. It also identifies a missing charge associated with a constant-on-the-sphere parameter, interpreting it as boundary centre degrees of freedom supported by a Kramers-Wannier-type edge-mode structure, with a lattice duality providing the analogous intuition. The work thus extends the gauge-symmetry/soft-theorem correspondence to scalar theories and highlights subtle edge-mode phenomena at infinity. A lattice analogue reinforces the interpretation and points to interesting future directions in quantum and boundary dynamics of scalar theories.

Abstract

In recent years soft factorization theorems in scattering amplitudes have been reinterpreted as conservation laws of asymptotic charges. In gauge, gravity, and higher spin theories the asymptotic charges can be understood as canonical generators of large gauge symmetries. Such a symmetry interpretation has been so far missing for scalar soft theorems. We remedy this situation by treating the massless scalar field in terms of a dual two-form gauge field. We show that the asymptotic charges associated to the scalar soft theorem can be understood as generators of large gauge transformations of the dual two-form field. The dual picture introduces two new puzzles: the charges have very unexpected Poisson brackets with the fields, and the monopole term does not always have a dual gauge transformation interpretation. We find analogs of these two properties in the Kramers-Wannier duality on a finite lattice, indicating that the free scalar theory has new edge modes at infinity that canonically commute with all the bulk degrees of freedom.

Paper Structure

This paper contains 11 sections, 63 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scalar Asymptotic Charges
  3. Scalar/Two-Form Duality in Four Dimensions
  4. Two-Form Asymptotic Symmetries
  5. Fall-offs
  6. Charge
  7. Understanding the Symmetry Action
  8. The Missing Charge
  9. Analogous Properties from Kramers-Wannier duality
  10. Conclusions
  11. Conventions

Figures (2)

  • Figure 1: The local generator of the gauge transformations. The $E$s are added up on all four plaquettes adjoining a link with orientations given by the arrows.
  • Figure 2: An illustration of Kramers-Wannier duality. The two-form fields live on the faces of the box, and the scalar field lives on the vertices of the dotted lattice. The dual vertices outside the original box, whose $\phi$ variables generate large gauge transformation, don't have a $\pi$ variable.