  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.
