Yang-Mills Theory and the $\mathcal{N}=2$ Spinning Path Integral

TL;DR

This work builds a worldline, BRST-based route to Yang–Mills theory by embedding the YM BV multiplet into the $ N=2$ spinning particle’s vertex-operator algebra and then pulling back worldline correlators to supermoduli space via a Poincaré dual. The quadratic action emerges after projecting to the YM Fock space, while the cubic interaction corresponds to a deformation of the BRST operator $Q(A)$; the quartic vertex arises from moduli-space boundary terms and restores gauge invariance, with higher-order terms vanishing in the zero-length limit. The construction provides an a priori bridge between the deformation problem of $Q$ and a space-time action obtained from worldline integrals, offering a pathway toward a BV/SFT-like off-shell description and suggesting extensions to more general or gravitational frameworks. The approach highlights the role of auxiliary mixed-symmetry fields and the nontrivial geometry of the supermoduli space as essential for encoding non-linear YM dynamics from a worldline perspective.

Abstract

We embed the perturbative Fock state of the Yang-Mills BV-multiplet in the vertex operator algebra of the path-integral for the $\mathcal{N}=2$ supersymmetric world line and evaluate the pull-back of the latter to an integral form on supermoduli space. Choosing a suitable Poincaré dual on the latter, we show that this integral form describes an extension of Yang-Mills theory. Upon projection back to the Fock space, we recover the Yang-Mills action from the world line. This furthermore gives an a priori justification for the construction of Yang-Mills equations of motion as emerging from deformations of the BRST differential.