Operadic Calculus for Higher Colour-Kinematics Duality

TL;DR

This paper develops a homotopy-theoretic operadic framework for algebraic structures underlying colour-kinematics duality, extending previous models to include higher-valence phenomena relevant in Yang--Mills theory.

Contribution

It introduces homotopy coexact BV-algebras via operadic Koszul duality, enabling systematic analysis of higher-level structures in colour-kinematics duality.

Findings

Provides a concrete model for homotopy coexact BV-algebras.

Enables use of homotopical tools like $$-morphisms and deformation theory.

Accommodates quartic structures in Yang--Mills theory.

Abstract

The search for algebraic foundations of colour-kinematics duality and the double-copy construction has brought into focus a generalization of Batalin--Vilkovisky algebras, referred to here as coexact BV-algebras and as $\textrm{BV}^\square$-algebras in other sources. While these structures capture the cubic sector, they fail to encode higher-valence phenomena, for which a homotopy-theoretic extension becomes necessary. This work introduces a conceptual notion of homotopy coexact BV-algebra, defined through the homotopical interplay of commutative and BV structures, and provides a concrete model in terms of generators and relations, obtained through an extension the theory of Koszul duality for operads. The resulting framework enables the systematic use of homotopical tools -- $\infty$-morphisms, homotopy transfer, rectification, and deformation theory -- and naturally accommodates the quartic-level structures recently identified in Yang--Mills theory.