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

TL;DR

This paper introduces a new integral formulation of (1+1)-dimensional Yang-Mills theory, revealing an infinite hierarchy of gauge-invariant conserved charges through loop-space holonomies and analyzing their algebraic properties.

Contribution

It develops a novel integral and symplectic formalism for Yang-Mills in 1+1 dimensions, uncovering hidden symmetries and conserved charges with potential implications for quantum gauge theories.

Findings

Holonomy eigenvalues are path-independent, leading to conserved charges.

Conserved charges generate symmetry transformations preserving the Hamiltonian.

Charges are in involution if boundary constants are central in the gauge group.

Abstract

We present an integral formulation of classical Yang-Mills theory coupled to fermionic and scalar matter fields in (1+1)-dimensional Minkowski spacetime. By reformulating the local dynamics in terms of loop-space holonomies, we demonstrate that the path independence of the holonomy eigenvalues constitutes a conservation law, yielding an infinite hierarchy of gauge-invariant, dynamically conserved charges. While a zero-curvature equation is associated with a necessary condition for this path invariance, we note that it is not strictly sufficient on its own. Employing a first-order symplectic formalism, we show that these non-abelian charges generate global symmetry transformations on the fundamental phase-space variables. We rigorously prove that these transformations preserve the physical dynamics, leaving the total Hamiltonian invariant up to first-class constraints. Furthermore, an analysis of the Poisson algebra reveals that these conserved charges are in involution, provided the boundary integration constant lies within the center of the gauge group. This exact, lower-dimensional framework provides a highly tractable setting to investigate the algebraic structures of these hidden symmetries and the meaning of the conserved charges as physical observables, establishing a classical foundation for exploring their role in the quantum regime, such as in strongly coupled lattice gauge theories.