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

TL;DR

This paper explores the algebraic structures underlying field theories with colour-kinematics duality, demonstrating how tensor products of BV-algebras produce new theories, including a novel pure spinor supergravity action.

Contribution

It introduces a tensor product construction of BV-algebras that generalizes the double copy, with detailed mathematical framework and multiple physical examples.

Findings

Tensor product of BV-algebras yields new field theories.

Construction of a new cubic pure spinor supergravity action.

Extension of algebraic structures to various field theories.

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.