Generalized Path Dependent Representations for Gauge Theories

TL;DR

This paper introduces a set of differential operators acting on path-dependent functionals in gauge theories, providing a geometric framework that satisfies key identities and generalizes existing operators.

Contribution

It develops a unified geometric formalism for path-dependent operators in gauge theories, including covariant derivatives and Taylor expansions, extending previous approaches.

Findings

Path derivative satisfies Ricci and Bianchi identities.

Provides a geometric derivation of covariant Taylor expansions.

Includes special cases like endpoint and area derivatives.

Abstract

A set of differential operators acting by continuous deformations on path dependent functionals of open and closed curves is introduced. Geometrically, these path operators are interpreted as infinitesimal generators of curves in the base manifold of the gauge theory. They furnish a representation with the action of the group of loops having a fundamental role. We show that the path derivative, which is covariant by construction, satisfies the Ricci and Bianchi identities. Also, we provide a geometrical derivation of covariant Taylor expansions based on particular deformations of open curves. The formalism includes, as special cases, other path dependent operators such as end point derivatives and area derivatives.