U(1) connection ($A,F=dA$)

A connection on a principal circle bundle has local real one-form representatives $A$ transforming by $A\mapsto A+d\chi$. Its curvature is the globally defined two-form $F=dA$. In unit-charge normalization its holonomy around a closed loop is $e^{i\oint A}$, and curvature periods are integer multiples of $2\pi$ on closed surfaces. A circulation representative is defined modulo $2\pi$ under large gauge transformations.