nLab
Beilinson-Deligne cup-product

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)

$$\cup :A[p{]}_D^{\mathrm{\infty }}\otimes B[q{]}_D^{\mathrm{\infty }}\to (A{\otimes }_{\mathbb{Z}}B)[p+q{]}_D^{\mathrm{\infty }},$$\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 $A=B=\mathbb{Z}$, 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 $B^n(ℝ//\mathbb{Z}{)}_{\mathrm{conn}}$ be given by

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

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

$$\alpha \cup \beta :=\{\begin{array}{cc}\alpha \wedge \beta =\alpha \beta & \mathrm{if}\mathrm{deg}(\alpha )=0\\ \alpha \wedge d_{\mathrm{dR}}\beta & \mathrm{if}\mathrm{deg}(\alpha )>0\mathrm{and}\mathrm{deg}(\beta )=q+1\\ 0& \mathrm{otherwise}\end{array}.$$\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

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

$$\hat{C}\mapsto {\int }_{\Sigma }\hat{C}\cup \hat{C}.$$\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

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