Yang-Mills theory from the worldline

TL;DR

This paper encodes the full nonlinear structure of Yang-Mills theory within a worldline model using homotopy algebra, providing new insights into gauge interactions and the kinematic algebra.

Contribution

It constructs off-shell vertex operators for the bosonic spinning particle and shows how Yang-Mills theory's nonlinear structure is captured by the commutator algebra of these operators.

Findings

Yang-Mills equations emerge as a Maurer-Cartan operator equation

The entire L-infinity algebra of Yang-Mills is realized in the worldline framework

Provides a map from nonlinear field theory to a worldline model

Abstract

We construct off-shell vertex operators for the bosonic spinning particle. Using the language of homotopy algebras, we show that the full nonlinear structure of Yang-Mills theory, including its gauge transformations, is encoded in the commutator algebra of the worldline vertex operators. To do so, we deform the worldline BRST operator by coupling it to a background gauge field and show that the coupling is consistent on a suitable truncation of the Hilbert space. On this subspace, the square of the BRST operator is proportional to the Yang-Mills field equations, which we interpret as an operator Maurer-Cartan equation for the background. This allows us to define further vertex operators in different ghost numbers, which correspond to the entire $L_\infty$ algebra of Yang-Mills theory. Besides providing a precise map of a fully nonlinear field theory into a worldline model, we expect these results will be valuable to investigate the kinematic algebra of Yang-Mills, which is central to the double copy program.