Circular loop operators in conformal field theories

TL;DR

This paper employs conformal symmetry to classify and analyze non-local operators, such as Wilson loops, in conformal field theories, providing a framework for understanding their representations and properties.

Contribution

It introduces a method to classify non-local operators in CFTs using conformal group subgroups, specifically applying this to Wilson loops in four-dimensional theories.

Findings

Wilson loops can be characterized by fixed SL(2,R) x SO(3) representations

The conformal group approach classifies non-local operators systematically

Provides a new perspective on non-local operator representations in CFTs

Abstract

We use the conformal group to study non-local operators in conformal field theories. A plane or a sphere (of any dimension) is mapped to itself by some subgroup of the conformal group, hence operators confined to that submanifold may be classified in representations of this subgroup. For local operators this gives the usual definition of conformal dimension and spin, but some conformal field theories contain interesting nonlocal operators, like Wilson or 't Hooft loops. We apply those ideas to Wilson loops in four-dimensional CFTs and show how they can be chosen to be in fixed representations of SL(2,R) x SO(3).