The Hidden Symmetries of Yang-Mills Theory in (3+1)-dimensions

TL;DR

This work reveals hidden, integrability-like structures in classical Yang-Mills theories in $(3+1)$-D by formulating YM dynamics as loop-space integral equations with flat connections. It identifies two families of hidden symmetries: (i) global transformations generated by an infinite tower of gauge-invariant conserved charges, and (ii) symmetries of the integral YM equations realized by an infinite group of holonomies on loop space. The charges are the eigenvalues of a gauge-covariant, path-independent charge operator whose higher modes are in involution via a Sklyanin-type relation, suggesting new nonperturbative tools for YM theories. The work also introduces a novel infinite-dimensional symmetry group acting on the integral equations, offering a route to connect loop-space structures with potential mappings between vacua and nontrivial solutions. Collectively, these results point to nonperturbative avenues in YM and QCD, with possible implications for generalized global symmetries and the behavior of color-singlet charges in the hadronic spectrum.

Abstract

We show that classical, non-supersymmetric Yang-Mills theories coupled to spin-1/2 and spin-0 elementary matter fields, in (3+1)-dimensional Minkowski space-time, possess exact structures that resemble integrability, with an infinite number of conserved charges in involution. Such structures live in the space of non-abelian electric and magnetic charges, and are based on flat connections in generalized loop spaces, presenting an R-matrix, and Sklyanin relation. We present two novel symmetries of Yang-Mills theories. The first one corresponds to global transformations generated by the infinity of those conserved charges under the Poisson brackets. The gauge and matter fields, as well as Wilson lines and fluxes, have interesting transformation laws under such a global symmetry. The second one corresponds to symmetries of the integral Yang-Mills equations, which lead to the conserved charges. They generate an infinite-dimensional group, where the elements are holonomies of connections on the loop space of functions from the circle S^1 to the space-time. Our approach certainly applies to the Standard Model of the Fundamental Interactions. The conserved charges are gauge invariant, and so, in the case of QCD, they are color singlets and perhaps are not confined. Therefore, the hadrons may carry such charges. Our results open up the way for the construction of non-perturbative methods for Yang-Mills theories.