The Beilinson-Drinfeld cup product is an explicit presentation of the cup product in ordinary differential cohomology (see also at cup product in 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
Last revised on June 30, 2014 at 02:24:26. See the history of this page for a list of all contributions to it.