Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories

TL;DR

This work analyzes how loop and surface operators serve as order parameters for 4d gapped gauge theories and introduces 4d TQFTs to capture their long-distance behavior. It develops abelian and nonabelian constructions, including BF-type and higher gauge theory formulations, and shows how condensates of electrically, magnetically, or dyonically charged objects determine the residual topological data and confinement properties. A key novelty is the use of nonabelian gerbes (Lie 2-groups) to model nonabelian confinement, with theta-quantization appearing in oblique phases. The results connect confinement, Higgs phenomena, and lattice formulations, illustrating a topological, duality-aware framework for classifying and understanding gapped phases in gauge theories. These TQFTs offer a structured way to compute ground-state degeneracies and nonlocal operator algebras across diverse phases, with implications for dualities and lattice realizations.

Abstract

We revisit the role of loop and surface operators as order parameters for gapped phases of four-dimensional gauge theories. We show that in some cases surface operators are confined, and that this fact can be used to distinguish phases which are not distinguished by the Wilson-'t Hooft criterion. The long-distance behavior of loop and surface operators which are neither confined nor screened is controlled by a 4d TQFT. We construct these TQFTs for phases which are characterized by the presence of electrically and/or magnetically charged condensates. Interestingly, the TQFT describing a phase with a nonabelian monopole condensate is based on the theory of nonabelian gerbes. We also show that in phases with a dyonic condensate the low-energy theta-angle is quantized.