# nLab Deligne cohomology

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

Deligne cohomology – or Deligne-Beilinson cohomology is an abelian sheaf cohomology that models ordinary differential cohomology.

The standard Deligne complex (of abelian sheaves) is under the Dold-Kan correspondence the sheaf of n-groupoids of smooth n-functors from the path n-groupoid to the $n$-fold delooping ${B}^{n}U\left(1\right)$:

$ℤ\left(n+1{\right)}_{D}^{\infty }\simeq {\overline{B}}^{n}U\left(1\right)\stackrel{N}{{\to }^{\simeq }}\left[{P}_{n}\left(-\right),{B}^{n}U\left(1\right)\right]\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{Z}(n+1)_D^\infty \simeq \bar \mathbf{B}^n U(1) \stackrel{N}{\to^\simeq} [P_n(-), \mathbf{B}^n U(1)] \,.

Smooth Deligne cohomology in degree $n$, of a smooth space $X$ is cohomology with coefficients in ${\overline{B}}^{n}U\left(1\right)$.

$\mathrm{Deligne}\mathrm{cohomology}=H\left(X,{\overline{B}}^{n}U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$Deligne cohomology = H(X, \bar \mathbf{B}^n U(1)) \,.

Here the notation on the right is as at the end of motivation for sheaves, cohomology and higher stacks.

This is a realization of the differential refinement (or smooth extension) ${\overline{H}}^{n}\left(X,ℤ\right)$ of the integral cohomology ${H}^{n}\left(X,ℤ\right)$ of $X$ in terms of abelian sheaf cohomology.

Recall that analgous to how ${H}^{n}\left(X,ℤ\right)$ classifies line $\left(n-1\right)$-bundles and equivalently line $\left(n-2\right)$-gerbes on $X$, ${\overline{H}}^{n}\left(X,ℤ\right)$ classifies line $\left(n-2\right)$-gerbes with connection.

Accordingly, the Deligne complex of sheaves $ℤ\left(n{\right)}_{D}^{\infty }$ is a complex of sheaves of differential forms.

## Definition

###### Definition

For $k\in ℕ$ write ${\Omega }^{k}\left(-\right):U↦{\Omega }^{k}\left(U\right)$ for the [[sheaf] of smooth differential $k$-forms on $X$ and ${C}^{\infty }\left(-,V\right)$ for the sheaf of smooth $V$-valued functions on $X$.

The degree $\left(n+1\right)$ Deligne complex is the complex of sheaves

$ℤ\left(n+1{\right)}_{D}^{\infty }\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}\left(\cdots \to 0\to {C}^{\infty }\left(-,ℤ\right)↪{C}^{\infty }\left(-,ℝ\right)\stackrel{d}{\to }{\Omega }^{1}\left(-\right)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^{n}\left(-\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{Z}(n+1)_D^\infty \; := \; \left( \cdots \to 0 \to C^\infty(-,\mathbb{Z}) \hookrightarrow C^\infty(-,\mathbb{R}) \stackrel{d }{\to} \Omega^1(-) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^n(-) \right) \,.

Often it is useful to consider the quasi-isomorphic complex

${\overline{B}}^{n}U\left(1\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:=\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(\cdots 0\to {C}^{\infty }\left(-,U\left(1\right)\right)\stackrel{d\mathrm{log}}{\to }{\Omega }^{1}\left(-\right)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^{n}\left(-\right)\right)$\bar \mathbf{B}^n U(1) \;\; := \;\; \left( \cdots 0 \to C^\infty(-,U(1)) \stackrel{d log}{\to} \Omega^1(-) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^n(-) \right)

Here ${C}^{\infty }\left(-,U\left(1\right)\right)\stackrel{d\mathrm{log}}{\to }{\Omega }^{1}\left(-\right)$ is the morphism of sheaves induced by regarding a $U\left(1\right)\simeq ℝ/ℤ$-valued function locally as a $ℝ$-valued function and applying the deRham differential $d$ to that.

The obvious morphism of complexes

$\begin{array}{ccccccccc}{C}^{\infty }\left(-,ℤ\right)& ↪& {C}^{\infty }\left(-,ℝ\right)& \stackrel{d\mathrm{log}}{\to }& {\Omega }^{1}\left(-\right)& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^{n}\left(-\right)\\ ↓& & {↓}^{\left(-\right)/ℤ}& & {↓}^{\mathrm{Id}}& & & & {↓}^{\mathrm{Id}}\\ 0& \to & {C}^{\infty }\left(-,U\left(1\right)\right)& \stackrel{d\mathrm{log}}{\to }& {\Omega }^{1}\left(-\right)& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^{n}\left(-\right)\end{array}$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d log}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow && \downarrow^{(-)/\mathbb{Z}} && \downarrow^{Id} && && \downarrow^{Id} \\ 0 &\to& C^\infty(-,U(1)) &\stackrel{d log}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) }

clearly induces isomorphism on homology groups: the homology in degree $n$ is locally constant $ℝ$-valued functions modulo locally constant $ℤ$-valued functions in the first case and constant $U\left(1\right)$-valued functions in the second case, which is the same.

###### Definition

Deligne cohomology in degree $n+1$ of $X$ is the cohomology (which is abelian sheaf cohomology in this case) with coefficients in ${\overline{B}}^{n}U\left(1\right)$.

$H\left(X,ℤ\left(n+1{\right)}_{D}^{\infty }\right)\simeq H\left(X,{\overline{B}}^{n}U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$H(X, \mathbb{Z}(n+1)_D^\infty) \simeq H(X, \bar \mathbf{B}^n U(1)) \,.

Here the notation on the right is motivated from the discussion at the end of motivation for sheaves, cohomology and higher stacks.

## Properties

### Characteristic classes of Deligne cocycles

There are two natural morphisms of abelian cohomology groups out of Deligne cohomology:

$\mathrm{cl}:H\left(X,{\overline{B}}^{n}U\left(1\right)\right)\to H\left(X,{B}^{n}U\left(1\right)\right)\simeq H\left(X,{B}^{n+1}ℤ\right)\simeq {H}^{n+1}\left(X,ℤ\right)$cl : H(X,\bar \mathbf{B}^n U(1)) \to H(X,\mathbf{B}^n U(1)) \simeq H(X, \mathbf{B}^{n+1} \mathbb{Z}) \simeq H^{n+1}(X,\mathbb{Z})
• the map to the curvature characteristic class
$\left[F\right]:H\left(X,{\overline{B}}^{n}U\left(1\right)\right)\to {H}_{\mathrm{dR}}^{n+1}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$[F] : H(X,\bar \mathbf{B}^n U(1)) \to H_{dR}^{n+1}(X) \,.

These are induced from the canonical morphisms of coefficient objects

${\overline{B}}^{n}U\left(1\right)\simeq ℤ\left(n+1{\right)}_{D}^{\infty }\to {B}^{n+1}ℤ$\bar \mathbf{B}^n U(1) \simeq \mathbb{Z}(n+1)_D^\infty \to \mathbf{B}^{n+1} \mathbb{Z}

given by

$\begin{array}{ccccccccc}{C}^{\infty }\left(-,ℤ\right)& ↪& {C}^{\infty }\left(-,ℝ\right)& \stackrel{d}{\to }& {\Omega }^{1}\left(-\right)& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^{n}\left(-\right)\\ {↓}^{\mathrm{Id}}& & {↓}^{0}& & {↓}^{0}& & & & {↓}^{0}\\ {C}^{\infty }\left(-,ℤ\right)& \to & 0& \to & 0& \to & \cdots & \to & 0\end{array}$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d }{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow^{Id} && \downarrow^{0} && \downarrow^{0} && && \downarrow^{0} \\ C^\infty(-, \mathbb{Z}) &\to& 0 &\to& 0 &\to& \cdots &\to& 0 }

and

${\overline{B}}^{n}U\left(1\right)\simeq ℤ\left(n+1{\right)}_{D}^{\infty }\to \left({\Omega }^{0}\left(-\right)\stackrel{d}{\to }{\Omega }^{1}\left(-\right)\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\Omega }^{n+1}\left(-\right)\right)$\bar \mathbf{B}^n U(1) \simeq \mathbb{Z}(n+1)_D^\infty \to (\Omega^0(-) \stackrel{d}{\to} \Omega^1(-) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^{n+1}(-) )

given by

$\begin{array}{ccccccccc}{C}^{\infty }\left(-,ℤ\right)& ↪& {C}^{\infty }\left(-,ℝ\right)& \stackrel{d}{\to }& {\Omega }^{1}\left(-\right)& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^{n}\left(-\right)\\ {↓}^{0}& & {↓}^{d}& & {↓}^{d}& & & & {↓}^{d}\\ {C}^{\infty }\left(-,ℝ\right)& \stackrel{d}{\to }& {\Omega }^{1}\left(-\right)& \stackrel{d}{\to }& {\Omega }^{2}\left(-\right)& \stackrel{d}{\to }& \cdots & \stackrel{d}{\to }& {\Omega }^{n+1}\left(-\right)\end{array}$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d }{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow^{0} && \downarrow^d && \downarrow^d && && \downarrow^d \\ C^\infty(-,\mathbb{R}) &\stackrel{d}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \Omega^2(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^{n+1}(-) }
###### Theorem

These two morphisms exhibit Deligne cohomology as a refinement in differential cohomology of ordinary (i.e. integral Eilenberg-MacLane) cohomology, in that the diagram

$\begin{array}{ccc}H\left(X,{\overline{B}}^{•}U\left(1\right)\right)& \stackrel{\left[F\right]}{\to }& {H}_{\mathrm{dR}}^{•+1}\left(X\right)\\ {↓}^{\mathrm{cl}}& & ↓\\ {H}^{•+1}\left(X,ℤ\right)& \to & {H}^{•+1}\left(X,ℝ\right)\end{array}$\array{ H(X,\bar \mathbf{B}^\bullet U(1)) &\stackrel{[F]}{\to}& H^{\bullet+1}_{dR}(X) \\ \downarrow^{cl} && \downarrow \\ H^{\bullet+1}(X,\mathbb{Z}) &\to& H^{\bullet + 1}(X,\mathbb{R}) }

is the cohomology of a homotopy pullback diagram, i.e. satisfies the axioms described at differential cohomology.

### Interpretation in terms of higher parallel transport

There is a natural way to understand the Deligne complex of sheaves as a sheaf which assigns to each patch the Lie $n$-groupoid of smooth higher parallel transport n-functors. This perspective is helpful for understanding how Deligne cohomology relates to the bigger picture of differential cohomology.

We start by discussing this in low degree.

There is path groupoid ${P}_{1}\left(X\right)$ whose smooth space of objects is $X$ and whose smooth space of morphisms is a space of classes of smooth paths in $X$. Every smooth 1-form $A\in {\Omega }^{1}\left(X\right)$ induces a smooth functor ${\mathrm{tra}}_{A}:{P}_{1}\left(X\right)\to BU\left(1\right)$ from ${P}_{1}\left(X\right)$ to to the smooth groupoid $BU\left(1\right)$ with one object and $U\left(1\right)$ as its smooth space of morphisms by sending each path $\gamma :\left[0,1\right]\to X$ to $\mathrm{exp}\left(2\pi i{\int }_{0}^{1}{\gamma }^{*}A\right)$. This map from 1-forms to smooth functors turns out to be bijective: every smooth functor of this form uniquely arises this way. Similarly, one finds that smooth natural transformation ${\eta }_{f}:{\mathrm{tra}}_{A}\to {\mathrm{tra}}_{A\prime }$ between two such functors is in components precisely a smooth function $f:X\to U\left(1\right)$ such that $A\prime =A+d\mathrm{log}f$.

Since the analogous statements are true for every open subset $U\subset X$ this defines a sheaf of Lie groupoids

${\mathrm{Funct}}^{\infty }\left({P}_{1}\left(-\right),BU\left(1\right)\right):\mathrm{Op}\left(X{\right)}^{\mathrm{op}}\to \mathrm{LieGrpd}\phantom{\rule{thinmathspace}{0ex}}.$Funct^\infty(P_1(-), \mathbf{B}U(1)) : Op(X)^{op} \to LieGrpd \,.

By the Dold-Kan correspondence this sheaf of groupoids corresponds to a sheaf of complexes of groups. This complex of sheaves is nothing but the degree 2 Deligne complex

${\mathrm{Funct}}^{\infty }\left({\Pi }_{1}\left(-\right),BU\left(1\right)\right)\simeq ℤ\left(2{\right)}_{D}^{\infty }\phantom{\rule{thinmathspace}{0ex}}.$Funct^\infty(\Pi_1(-), \mathbf{B}U(1)) \simeq \mathbb{Z}(2)^\infty_D \,.

This way Deligne cohomology is realized as computing the stackification of the pre-stack ${\mathrm{Funct}}^{\infty }\left({P}_{1}\left(-\right),B\left(1\right)\right)$ of smooth $U\left(1\right)$-valued parallel transport functors.

The identification generalizes: for all $n$ there is a path n-groupoid ${P}_{n}\left(X\right)$ whose $k$-morphisms are $k$-dimensional smooth paths in $X$. Smooth $n$-functors ${\mathrm{tra}}_{C}{:}_{n}\left(X\right)\to {B}^{n}U\left(1\right)$ are canonically identified with smooth $n$-forms $C\in {\Omega }^{n}\left(X\right)$ and under the Dold-Kan correspondence the Deligne-complex in degree $n+1$ is identified with the sheaf of $n$-groupoids of such smooth $n$-functors

$n{\mathrm{Funct}}^{\infty }\left({P}_{n}\left(-\right),{B}^{n}\right)\simeq ℤ\left(n+1{\right)}_{D}^{\infty }\phantom{\rule{thinmathspace}{0ex}}.$n Funct^\infty(P_n(-), \mathbf{B}^n) \simeq \mathbb{Z}(n+1)^\infty_D \,.

See

• John Baez, Urs Schreiber, Higher Gauge Theory (arXiv)

The full proof for $n=1$ this is in

• Urs Schreiber, Konrad Waldorf, Parallel transport and functors (arXiv);

for $n=2$ in

• Urs Schreiber, Konrad Waldorf, Smooth functors versus differential forms (arXiv)

For more on this see infinity-Chern-Weil theory introduction.

For higher $n$ there is as yet no detailed proof in the literature, but the low dimensional proofs have obvious generalizations.

## Examples

As described in some detail at electromagnetic field in abelian higher gauge theories the background field naturally arises as a Čech–Deligne cocycle, i.e. a Čech cocycle representative with values in the Deligne complex.

• Degree 2 Deligne cohomology classifies $U\left(1\right)$-principal bundles with connection. The Deligne complex $\overline{B}U\left(1\right)$ in this case coincides with the groupoid of Lie-algebra valued forms for the Lie algebra of $U\left(1\right)$.

• In physics the electromagnetic field is modeled by a degree 2 Deligne cocycle. See there for a derivation of Čech–Deligne cohomology from physical input.
• Degree 3 Deligne cohomology classifies bundle gerbes with connection.

• Degree 4 Deligne cohomology classifies bundle 2-gerbes with connection. In particular Chern-Simons bundle 2-gerbes whose degree 4 curvature characteristic class is a multiple of the Pontryagin 4-form on some $\mathrm{SO}\left(n\right)$-principal bundle.

## References

Deligne cohomology was introduced in

• Pierre Deligne, Théorie de Hodge II , IHES Pub. Math. (1971), no. 40, 5–57.

Surveys are for instance in section 5 of

• Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and geometric Quantization, Birkhaeuser

• Hélène 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)

A concise review is for instance section 2 of

• Kiyonori Gomi, Projective unitary representations of smooth Deligne cohomology groups (arXiv)

Revised on August 30, 2011 14:16:28 by Urs Schreiber (131.211.239.161)