The Beilinson-Drinfeld cup product is an explicit presentation of the cup product in ordinary differential cohomology for the case that the latter is modeled by the Cech-Deligne cohomology. It sends (see cup product in abelian Cech cohomology)
where $A$ and $B$ are lattices in $\mathbb{R}^n$, and $\mathbb{R}^m$ for some $n$ and $m$, respectively. It is a morphism of complexes, so it induces a cup product in Deligne cohomology.
For $A=B=\mathbb{Z}$, the Beilinson-Deligne cup product is associative and commutative up to homotopy, so it induces an associative and commutatvive cup product on differential cohomology
Let the Deligne complex $\mathbf{B}^n(\mathbb{R}//\mathbb{Z})_{conn}$ be given by
where we refer to degrees as indicated in the bottom row.
The Beilinson-Deligne product is the morphism of chain complexes of sheaves
given on homogeneous elements $\alpha$, $\beta$ as follows:
The action functional of abelian higher dimensional Chern-Simons theory is given by the fiber integration in ordinary differential cohomology over the BD cup product of differential cocycles
For more on this see higher dimensional Chern-Simons theory.
The original references are
Pierre Deligne, Théorie de Hodge II , IHES Pub. Math. (1971), no. 40, 5–57.
Alexander Beilinson, Higher regulators and values of L-functions , J. Soviet Math. 30 (1985), 2036—2070
Alexander Beilinson, Notes on absolute Hodge cohomology , Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II, Contemp. Math., 55, Amer. Math. Soc., Providence, RI, 1986.
A survey is for instance around prop. 1.5.8 of
and in section 3 of
For the cup product of Cheeger-Simons differential characters see also