Circular loop operators in conformal field theories

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)× SO(3). Conformal field theories (CFTs) play an important role in physics. They arise naturally at the fixed points of the renormalization group flow and describe critical phenomena in a wide class of systems. From the theorists perspective CFTs offer an enlargement of the space-time symmetry group that puts constraints on the theory and makes it easier to study than a general quantum field theory. Many field theories contain non-local observables, like Wilson loop operators, or topological defects, like Nielsen-Olesen vortices, or ’t Hooft loops. Such objects may appear also in theories that have a conformal symmetry, for example Wilson loops in N = 4 supersymmetric Yang-Mills theory. In this note we propose some tools to study non-local operators in conformal field theories. Let us recall the construction of local conformal operators. The conformal group in d-dimensional Euclidean space is SO(d+ 1, 1). The subgroup that will keep a fixed point (the origin) invariant is SO(d)× R, comprising of rotations and the dilatation. Hence local operators may be classified by representations of this subgroup, the spin and conformal dimension. To generalize this construction for non-local operators consider an n-dimensional sphere in R. The subgroup of the conformal group that maps the sphere to itself is SO(d − n) × SO(n + 1, 1). A simple way to see this symmetry is to map the sphere to a plane by a stereographic projection. SO(n + 1, 1) is the conformal symmetry in this plane and SO(d − n) are the rotations around the plane. In the specific case of n = 1 and d = 4 on which we concentrate later we call these operators “circular loop operators” and the symmetry group is SL(2,R)× SO(3) [1]. Our main proposition concerns any non-local operator localized on a sphere. The claim is Operators localized on S in a CFT can be classified by representations of SO(n + 1, 1) × SO(d − n) in much the same way that local operators are classified by spin and conformal dimension. This statement follows immediately from the preceding discussion. In the remainder of the note we will develop some tools for analyzing loop operators in this setting and demonstrate it in a few examples.