Topological Disorder Operators in Three-Dimensional Conformal Field Theory

TL;DR

This work constructs and analyzes monopole operators in a non-supersymmetric 3d QED with $N_f$ flavors, treating the gauge field as a classical background in the large $N_f$ limit. Monopole operators are defined as local topological disorder operators that create Abrikosov-Nielsen-Olesen vortices, with their properties extracted via radial quantization on $S^2\times R$ in the presence of magnetic flux. The dimensions of these operators scale with $N_f$ and depend on the Chern-Simons coupling $k$, and their flavor quantum numbers arise from fermionic zero modes, yielding explicit representations of $SU(N_f)$ (and dependenices on $k$). The results provide a constructive handle on topological disorder in a 3d CFT and have potential implications for 3d dualities and mirror symmetry, with possible extensions to SUSY theories and higher dimensions.

Abstract

Many abelian gauge theories in three dimensions flow to interacting conformal field theories in the infrared. We define a new class of local operators in these conformal field theories which are not polynomial in the fundamental fields and create topological disorder. They can be regarded as higher-dimensional analogues of twist and winding-state operators in free 2d CFTs. We call them monopole operators for reasons explained in the text. The importance of monopole operators is that in the Higgs phase, they create Abrikosov-Nielsen-Olesen vortices. We study properties of these operators in three-dimensional QED using large N_f expansion. In particular, we show that monopole operators belong to representations of the conformal group whose primaries have dimension of order N_f. We also show that monopole operators transform non-trivially under the flavor symmetry group, with the precise representation depending on the value of the Chern-Simons coupling.