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Homotopy kinematic algebras at null infinity

Felipe Díaz-Jaramillo, Silvia Nagy, Giorgio Pizzolo

TL;DR

The paper addresses how to encode asymptotic data near null infinity within a homotopy-algebraic framework for gauge theories, focusing on self-dual Yang–Mills in Bondi coordinates. It introduces a $L_\infty$-based formulation compatible with a $1/r$ expansion, then derives a master system that yields physically correct fall-offs and a spectrum of slice-dependent kinematic algebras. By constructing projections and cochain maps, the authors produce an infinite family of sliced $L_\infty$ algebras and, through refinements, obtain several strict kinematic Lie algebras on refined slices; they also establish a quasi-isomorphism to the boundary theory. The results deepen the algebraic underpinning of color–kinematics duality and the double copy in the asymptotic regime and point toward extensions to gravity and BMS-like symmetries with potential implications for celestial holography and infrared structure.

Abstract

We present the first formulation of a homotopy algebra adapted to a $1/r$ expansion near future null infinity ($\mathcal{I^+}$). Focusing on self-dual Yang-Mills theory in Bondi coordinates, we demonstrate that imposing the homotopy algebra relations naturally yields the physically consistent fall-off behavior of the fields near $\mathcal{I^+}$. Furthermore, we employ this framework to systematically construct kinematic algebras, uncovering novel infinite families of such algebras that satisfy the Jacobi identity on slices near $\mathcal{I^+}$.

Homotopy kinematic algebras at null infinity

TL;DR

The paper addresses how to encode asymptotic data near null infinity within a homotopy-algebraic framework for gauge theories, focusing on self-dual Yang–Mills in Bondi coordinates. It introduces a -based formulation compatible with a expansion, then derives a master system that yields physically correct fall-offs and a spectrum of slice-dependent kinematic algebras. By constructing projections and cochain maps, the authors produce an infinite family of sliced algebras and, through refinements, obtain several strict kinematic Lie algebras on refined slices; they also establish a quasi-isomorphism to the boundary theory. The results deepen the algebraic underpinning of color–kinematics duality and the double copy in the asymptotic regime and point toward extensions to gravity and BMS-like symmetries with potential implications for celestial holography and infrared structure.

Abstract

We present the first formulation of a homotopy algebra adapted to a expansion near future null infinity (). Focusing on self-dual Yang-Mills theory in Bondi coordinates, we demonstrate that imposing the homotopy algebra relations naturally yields the physically consistent fall-off behavior of the fields near . Furthermore, we employ this framework to systematically construct kinematic algebras, uncovering novel infinite families of such algebras that satisfy the Jacobi identity on slices near .

Paper Structure

This paper contains 43 sections, 8 theorems, 258 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. Color-kinematics duality and the kinematic algebra
  4. Review of physical fall-off at null infinity
  5. Homotopy algebras
  6. L infty algebras
  7. C infty algebras
  8. Kinematic algebras and BVBV infty box algebras
  9. Summary and notation
  10. Homotopy algebra formulation of SDYM in Bondi coordinates
  11. L infty algebra
  12. Color stripping and kinematic algebra
  13. Physical fall-offs from $L_{\infty}$ algebras
  14. Requirements on the L infty algebra near I
  15. Solutions of the master system
  16. ...and 28 more sections

Key Result

Lemma 7.2

$\bar{B}_2=B_2|_{\bar{\mathcal{X}}\otimes\bar{\mathcal{X}}}$

Figures (3)

  • Figure 1: The drawing illustrates the slices $\mathfrak{S}_k$ (where $k\in\mathbb{Z}^{\leq0}$) obtained by projecting the initial graded vector space $\mathcal{X}$ via the maps $\Pi_k$ in \ref{['res:master system solution']}, that are the solutions of the master system \ref{['eq:master_system']}. Each slice is represented in a different shade of green. The lighter slices always include the darker ones, in other words $\mathfrak{S}_{k-1}\subset\mathfrak{S}_k$ for all $k$. Moreover, all slices share the same maxima and therefore the same fall-offs.
  • Figure 2: The fall-offs follow directly from the general properties of the master system: each condition in the system restricts specific powers in the fall-off, and the figure illustrates how these conditions determine the excluded powers, which are represented by red crosses.
  • Figure 3: Visual representation of the discrete intervals $N_I\in\mathcal{N}^\mathrm{test}$, see \ref{['pf:sets N_E_munu']} and \ref{['pf:trial solution N_A_mu with equalities']}. Each square represents an integer. The red crosses indicate the numbers forbidden by \ref{['eq:fall off from first two conditions']}. The square with the exclamation mark corresponds to the element $-1\in N_{A_r}$, that is the hypothesis of the proof by contradiction. The use of colours help illustrate how the intervals $N_{A_\mu}$ and $N_\Lambda$ depend on $N_{E_{\mu\nu}}$ through equations \ref{['pf:inclusions for N_A_mu']} and \ref{['pf:inclusions for N_Lambda']}, respectively.

Theorems & Definitions (26)

  • proof
  • Lemma 7.2
  • proof
  • Lemma A.1: Properties of projection
  • proof
  • Lemma A.2: Cochain map condition
  • proof
  • Lemma A.3: Leibniz rule condition
  • proof
  • Lemma A.4: Fall-off
  • ...and 16 more