Electric-Magnetic Duality of Topological Gauge Theories from Compactification

In this note, we discuss electric-magnetic duality between a pair of 4d topological field theories (TQFTs) by considering their compactifications to 2 dimensions. These TQFTs control the long-distance behavior of loop and surface operators in 4d gauge theories with gapped phases. These were recently used in work by S. Gukov and A. Kapustin in detecting phases not distinguishable by the Wilson-'t Hooft criterion and by A. Kapustin and the author to construct discrete theta-angles for lattice Yang-Mills theories. The strong-weak duality is manifested in an exchange of dynamical and background degrees of freedom in the compactified TQFTs.