nLab
Beilinson-Deligne cup-product

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)

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

where AA and BB are lattices in n\mathbb{R}^n, and m\mathbb{R}^m for some nn and mm, respectively. It is a morphism of complexes, so it induces a cup product in Deligne cohomology.

For A=B=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(//) conn\mathbf{B}^n(\mathbb{R}//\mathbb{Z})_{conn} be given by

C (,) d dR d dR Ω n() degree: 0 1 (n+1) \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

:B p(//) connB q(//) connB p+q+1(//) 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 α\alpha, β\beta as follows:

αβ:={αβ=αβ ifdeg(α)=0 αd dRβ ifdeg(α)>0anddeg(β)=q+1 0 otherwise. \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(Σ) diffU(1) S_{CS} : H^{2k+2}(\Sigma)_diff \to U(1)
C^ ΣC^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 KK-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)