nLab Beilinson–Deligne cup product

Idea

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)

$$\cup: A[p]^\infty_D\otimes B[q]^\infty_D\to (A\otimes_{\mathbb{Z}} B)[p+q]^\infty_D,$$

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

Definition

Let the Deligne complex $\mathbf{B}^n(\mathbb{R}//\mathbb{Z})_{conn}$ be given by

$$\array{ & \mathbb{Z} &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d_{dR}}{\to}& \cdots &\stackrel{d_{dR}}{\to}& \Omega^{n}(-) \\ \\ degree: & 0 && 1 && \cdots && (n+1) }$$

where we refer to degrees as indicated in the bottom row.

Definition

The Beilinson-Deligne product is the morphism of chain complexes of sheaves

$$\cup : \mathbf{B}^p (\mathbb{R}//\mathbb{Z})_{conn} \otimes \mathbf{B}^q (\mathbb{R}//\mathbb{Z})_{conn} \to \mathbf{B}^{p+q+1} (\mathbb{R}//\mathbb{Z})_{conn}$$

given on homogeneous elements $\alpha$, $\beta$ as follows:

$$\alpha \cup \beta := \left\{ \array{ \alpha \wedge \beta = \alpha \beta & if\,deg(\alpha) = 0 \\ \alpha \wedge d_{dR}\beta & if\,deg(\alpha) \gt 0\,and\,deg(\beta) = q+1 \\ 0 & otherwise } \right. \,.$$

Applications

In higher Chern-Simons theory

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

$$S_{CS} : H^{2k+2}(\Sigma)_diff \to U(1)$$

$$\hat C \mapsto \int_\Sigma \hat C \cup \hat C \,.$$

For more on this see higher dimensional Chern-Simons theory.

Related concepts

• Deligne line bundle

• intersection pairing

• self-dual higher gauge theory

References

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

• Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization Birkhäuser (1993)

and in section 3 of

• Helene Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)

For the cup product of Cheeger-Simons differential characters see also

• Mike Hopkins and Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory (arXiv)