Vertex operators for the kinematic algebra of Yang-Mills theory

TL;DR

This paper explores the algebraic structures underlying Yang-Mills theory, introducing vertex operators to represent the $C_{ olinebreak\infty}$ algebra within a first-quantized framework, and investigates hidden algebraic structures related to color-kinematics duality.

Contribution

It presents a novel representation of the $C_{ olinebreak\infty}$ algebra using vertex operators in a first-quantized setting, connecting homotopy algebras with quantum operators.

Findings

Vertex operators realize the $C_{ olinebreak\infty}$ algebra on the Hilbert space.

Introduction of $A_{ olinebreak\infty}$ morphisms to define vertex operators.

Initial steps towards representing hidden algebraic structures.

Abstract

The kinematic algebra of Yang-Mills theory can be understood in the framework of homotopy algebras: the $L_{\infty}$ algebra of Yang-Mills theory is the tensor product of the color Lie algebra and a kinematic space that carries a $C_{\infty}$ algebra. There are also hidden structures that generalize Batalin-Vilkovisky algebras, which explain color-kinematics duality and the double copy but are only partially understood. We show that there is a representation of the $C_{\infty}$ algebra, in terms of vertex operators, on the Hilbert space of a first-quantized worldline theory. To this end we introduce $A_{\infty}$ morphisms, which define the vertex operators and which inject the $C_{\infty}$ algebra into the strictly associative algebra of operators on the Hilbert space. We also take first steps to represent the hidden structures on the same space.