# nLab Beilinson-Deligne cup-product

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

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)

$\cup :A\left[p{\right]}_{D}^{\infty }\otimes B\left[q{\right]}_{D}^{\infty }\to \left(A{\otimes }_{ℤ}B\right)\left[p+q{\right]}_{D}^{\infty },$\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 ${ℝ}^{n}$, and ${ℝ}^{m}$ for some $n$ and $m$, respectively. It is a morphism of complexes, so it induces a cup product in Deligne cohomology.

For $A=B=ℤ$, 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 ${B}^{n}\left(ℝ//ℤ{\right)}_{\mathrm{conn}}$ be given by

$\begin{array}{cccccccc}& ℤ& ↪& {C}^{\infty }\left(-,ℝ\right)& \stackrel{{d}_{\mathrm{dR}}}{\to }& \cdots & \stackrel{{d}_{\mathrm{dR}}}{\to }& {\Omega }^{n}\left(-\right)\\ \\ \mathrm{degree}:& 0& & 1& & \cdots & & \left(n+1\right)\end{array}$\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

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

$\alpha \cup \beta :=\left\{\begin{array}{cc}\alpha \wedge \beta =\alpha \beta & \mathrm{if}\phantom{\rule{thinmathspace}{0ex}}\mathrm{deg}\left(\alpha \right)=0\\ \alpha \wedge {d}_{\mathrm{dR}}\beta & \mathrm{if}\phantom{\rule{thinmathspace}{0ex}}\mathrm{deg}\left(\alpha \right)>0\phantom{\rule{thinmathspace}{0ex}}\mathrm{and}\phantom{\rule{thinmathspace}{0ex}}\mathrm{deg}\left(\beta \right)=q+1\\ 0& \mathrm{otherwise}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

${S}_{\mathrm{CS}}:{H}^{2k+2}\left(\Sigma {\right)}_{\mathrm{diff}}\to U\left(1\right)$S_{CS} : H^{2k+2}(\Sigma)_diff \to U(1)
$\stackrel{^}{C}↦{\int }_{\Sigma }\stackrel{^}{C}\cup \stackrel{^}{C}\phantom{\rule{thinmathspace}{0ex}}.$\hat C \mapsto \int_\Sigma \hat C \cup \hat C \,.

For more on this see higher dimensional Chern-Simons 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

Revised on September 27, 2012 02:22:46 by Urs Schreiber (82.169.65.155)