Loop Equations as a Generalized Virasoro Constraints

TL;DR

This paper introduces a generalized algebra of loop equations in lattice gauge theory, extending the Virasoro algebra, with implications for understanding Wilson loop correlators and their symmetries.

Contribution

It formulates a new algebraic structure for loop equations in gauge theory that generalizes the Virasoro algebra and constructs its representations on loop space tensor fields.

Findings

The algebra of loop equations forms a closed, Virasoro-like structure.

Representations on tensor fields generalize conformal fields.

Results are valid in the continuum due to coupling-independent structure constants.

Abstract

The loop equations in the $U(N)$ lattice gauge theory are represented in the form of constraints imposed on a generating functional for the Wilson loop correlators. These constraints form a closed algebra with respect to commutation. This algebra generalizes the Virasoro one, which is known to appear in one-matrix models in the same way. The realization of this algebra in terms of the infinitesimal changes of generators of the loop space is given. The representations on the tensor fields on the loop space, generalizing the integer spin conformal fields, are constructed. The structure constants of the algebra under consideration being independent of the coupling constants, almost all the results are valid in the continuum.