U(1) connection as definition

Question

My question is what is a U(1) connection? Can the U(1) connection be expanded in terms of the basis?

EDIT: i.e., is there a relation between a U(1) connection and a 1-form?

Answer 1

EDIT: i.e., is there a relation between a U(1) connection and a 1-form?

Yes. A $U(1)$-connection on a trivial $U(1)$-bundle is precisely a $1$-form. A $U(1)$-connection on a nontrivial $U(1)$-bundle is an object that looks locally but not globally like a $1$-form; in particular it has an "exterior derivative," its curvature $F$, which is a closed $2$-form that need not be exact.