Color-kinematics duality from an algebra of superforms

TL;DR

This work introduces a novel algebraic framework to derive color-kinematics (CK) duality from first principles by embedding the Yang-Mills (YM) kinematic sector into the algebra of superforms on a superspace, endowed with a strict $BV^{\square}$ algebra and its BV$_{\infty}^{\square}$ generalization. By partitioning the de Rham complex into an ideal up to homotopy and taking a quotient, the color-stripped YM $C_{\infty}$ algebra emerges with a nontrivial $m_3$ that encodes YM interactions, while the quotient carries the YM kinematic structure. The bracket $b_2=[b,m_2]$ realizes a Lie-like structure up to higher homotopies, with the CK duality at tree level arising because the Jacobiator is cancelled by higher maps such as $b_3$ and $ heta_3$, rendering the obstruction exact and removable by multiparticle gauge transformations. The program outlines an algorithmic path to determine higher BV$_{\infty}^{\square}$ maps, aiming to derive gravity as the double copy from YM data and to extend the construction to more elaborate theories, including supersymmetric variants.

Abstract

Color-kinematics duality states that the kinematic numerators of the cubic tree-level Yang-Mills scattering amplitudes obey the same symmetry properties that the color factors obey due to the Jacobi identity. We present a novel strategy for deriving this duality, based on the differential forms on a superspace. This space of superforms carries a generalization of a Batalin-Vilkovisky (BV) algebra (BV$^{\square}$ algebra). We show that the homotopy algebra of color-stripped Yang-Mills theory is obtained as a quotient of this space in which a subspace, which is an ideal `up to homotopy', is modded out. This algebra is a subsector of a BV$_{\infty}^{\square}$ algebra. Deriving the latter would provide a first-principle proof of color-kinematics duality from field theory.

Figures (1)

• Figure 1: Bicomplex of superforms. The degrees $(p,q)$ in the above diagram are given by $(N,G-N)$. The spaces that descend to the Yang-Mills kinematic algebra ${\cal K}$ are displayed in green. Spaces in gray are zero, due to $c^2=0$.