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Asymptotic Symmetries of Yang-Mills Theory

Andrew Strominger

TL;DR

The paper analyzes asymptotic symmetries at future null infinity I^+ for gauge theories with massless charges, arguing for an infinite-dimensional symmetry generated by large gauge transformations that act holomorphically on patches of the conformal S^2. It constructs a boundary Kac-Moody current J_z from the asymptotic gauge field and shows its Ward identities reproduce Weinberg's soft-photon and soft-gluon theorems, with the non-Abelian case yielding a G-valued current algebra. The analysis is performed in a semiclassical setting using radiation gauge, detailing the boundary data, conformal properties, and Green functions, and it discusses the implications for a possible nonzero Kac-Moody level. The work also speculates on a string-theoretic realization, suggesting a spacetime current might arise from a lift of worldsheet current algebra, hinting at deeper connections between asymptotic symmetries and holographic-like structures.

Abstract

Asymptotic symmetries at future null infinity (I+) of Minkowski space for electrodynamics with massless charged fields, as well as non-Abelian gauge theories with gauge group G, are considered at the semiclassical level. The possibility of charge/color flux through I+ suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a G Kac-Moody symmetry whose generators are "large" gauge transformations which approach locally holomorphic functions on the conformal two-sphere at I+ and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of I+. The current Ward identities include Weinberg's soft photon theorem and its colored extension.

Asymptotic Symmetries of Yang-Mills Theory

TL;DR

The paper analyzes asymptotic symmetries at future null infinity I^+ for gauge theories with massless charges, arguing for an infinite-dimensional symmetry generated by large gauge transformations that act holomorphically on patches of the conformal S^2. It constructs a boundary Kac-Moody current J_z from the asymptotic gauge field and shows its Ward identities reproduce Weinberg's soft-photon and soft-gluon theorems, with the non-Abelian case yielding a G-valued current algebra. The analysis is performed in a semiclassical setting using radiation gauge, detailing the boundary data, conformal properties, and Green functions, and it discusses the implications for a possible nonzero Kac-Moody level. The work also speculates on a string-theoretic realization, suggesting a spacetime current might arise from a lift of worldsheet current algebra, hinting at deeper connections between asymptotic symmetries and holographic-like structures.

Abstract

Asymptotic symmetries at future null infinity (I+) of Minkowski space for electrodynamics with massless charged fields, as well as non-Abelian gauge theories with gauge group G, are considered at the semiclassical level. The possibility of charge/color flux through I+ suggests the symmetry group is infinite-dimensional. It is conjectured that the symmetries include a G Kac-Moody symmetry whose generators are "large" gauge transformations which approach locally holomorphic functions on the conformal two-sphere at I+ and are invariant under null translations. The Kac-Moody currents are constructed from the gauge field at the future boundary of I+. The current Ward identities include Weinberg's soft photon theorem and its colored extension.

Paper Structure

This paper contains 10 sections, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Electrodynamics
  3. The boundary data problem
  4. Conformal transformations
  5. The boundary current
  6. Ward identities and Weinberg's theorem
  7. Green functions
  8. Kac-Moody level?
  9. The non-Abelian case
  10. Stringy realization?

Figures (1)

  • Figure 1: Penrose diagram for Minkowski scattering. The figure depicts incoming photons (blue) from ${\cal I}^-$ ($v=-\infty$) creating a matter charge current $j^M$ (green) outgoing at ${\cal I}^+$ ($r=\infty$). The total charge $Q^E(v)$ is measured on a conformal sphere at ${\cal I}^+$ (parameterized by $(z,{\bar{z}})$), and depends on the retarded time $v$. The (red) cones are null surfaces of constant $v$.