Double Copy from Tensor Products of Metric BV${}^{\color{gray} \blacksquare}$-algebras

TL;DR

The paper develops a rigorous homotopy-algebraic framework unifying colour--kinematics duality and the double copy via BV-box algebras. CK duality for field theories with cubic interactions emerges from pseudo-$\mathrm{BV}^{\square}$-algebras, whose derived brackets yield kinematic Lie algebras; taking tensor products (syngamies) of two BV-box algebras produces a new theory whose differential graded Lie structure realizes the double copy, including cases with matter through BV-box modules and Hopf-algebra enhancements. The authors provide concrete realizations across diverse theories—biadjoint scalars, Chern–Simons, self-dual YM/gravity, and especially pure spinor descriptions of YM, M2-branes, and ten-dimensional supergravity—culminating in a cubic pure spinor action for 10D supergravity. They introduce a restricted tensor product to remove doubled field content and discuss compactification as a way to manage analytical issues, thereby offering a comprehensive off-shell CK framework with explicit examples and a pathway to loop-level considerations. The work unifies action-based CK duality with the double copy in a mathematically complete setting, offers new actions (notably a cubic pure spinor SUGRA action), and broadens CK duality to gauge–matter theories via BV-box modules, with potential implications for higher-dimensional theories and beyond-cubic interactions.

Abstract

Field theories with kinematic Lie algebras, such as field theories featuring colour-kinematics duality, possess an underlying algebraic structure known as BV${}^{\color{gray} \blacksquare}$-algebra. If, additionally, matter fields are present, this structure is supplemented by a module for the BV${}^{\color{gray} \blacksquare}$-algebra. We explain this perspective, expanding on our previous work and providing many additional mathematical details. We also show how the tensor product of two metric BV${}^{\color{gray} \blacksquare}$-algebras yields the action of a new syngamy field theory, a construction which comprises the familiar double copy construction. As examples, we discuss various scalar field theories, Chern-Simons theory, self-dual Yang-Mills theory, and the pure spinor formulations of both M2-brane models and supersymmetric Yang-Mills theory. The latter leads to a new cubic pure spinor action for ten-dimensional supergravity. We also give a homotopy-algebraic perspective on colour-flavour-stripping, obtain a new restricted tensor product over a wide class of bialgebras, and we show that any field theory (even one without colour-kinematics duality) comes with a kinematic $L_\infty$-algebra.

Theorems & Definitions (87)

• definition 1
• proposition 1
• proof
• definition 2
• definition 3
• definition 4
• definition 5
• theorem 1
• proposition 2
• proof