# 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 $\mathbf{B}^n U(1)$:

$\mathbb{Z}(n+1)_D^\infty \simeq \bar \mathbf{B}^n U(1) \stackrel{N}{\to^\simeq} [P_n(-), \mathbf{B}^n U(1)] \,.$

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

$Deligne cohomology = H(X, \bar \mathbf{B}^n U(1)) \,.$

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) $\bar H^n(X,\mathbb{Z})$ of the integral cohomology $H^n(X, \mathbb{Z})$ of $X$ in terms of abelian sheaf cohomology.

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

Accordingly, the Deligne complex of sheaves $\mathbb{Z}(n)^\infty_D$ is a complex of sheaves of differential forms.

## Definition

###### Definition

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

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

$\mathbb{Z}(n+1)_D^\infty \; := \; \left( \cdots \to 0 \to C^\infty(-,\mathbb{Z}) \hookrightarrow C^\infty(-,\mathbb{R}) \stackrel{d }{\to} \Omega^1(-) \stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^n(-) \right) \,.$

Often it is useful to consider the quasi-isomorphic complex

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

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

The obvious morphism of complexes

$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d log}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow && \downarrow^{(-)/\mathbb{Z}} && \downarrow^{Id} && && \downarrow^{Id} \\ 0 &\to& C^\infty(-,U(1)) &\stackrel{d log}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) }$

clearly induces isomorphism on homology groups: the homology in degree $n$ is locally constant $\mathbb{R}$-valued functions modulo locally constant $\mathbb{Z}$-valued functions in the first case and constant $U(1)$-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 $\bar \mathbf{B}^n U(1)$.

$H(X, \mathbb{Z}(n+1)_D^\infty) \simeq H(X, \bar \mathbf{B}^n U(1)) \,.$

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:

$cl : H(X,\bar \mathbf{B}^n U(1)) \to H(X,\mathbf{B}^n U(1)) \simeq H(X, \mathbf{B}^{n+1} \mathbb{Z}) \simeq H^{n+1}(X,\mathbb{Z})$
• the map to the curvature characteristic class
$[F] : H(X,\bar \mathbf{B}^n U(1)) \to H_{dR}^{n+1}(X) \,.$

These are induced from the canonical morphisms of coefficient objects

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

given by

$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d }{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow^{Id} && \downarrow^{0} && \downarrow^{0} && && \downarrow^{0} \\ C^\infty(-, \mathbb{Z}) &\to& 0 &\to& 0 &\to& \cdots &\to& 0 }$

and

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

given by

$\array{ C^\infty(-,\mathbb{Z}) &\hookrightarrow& C^\infty(-,\mathbb{R}) &\stackrel{d }{\to}& \Omega^1(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^n(-) \\ \downarrow^{0} && \downarrow^d && \downarrow^d && && \downarrow^d \\ C^\infty(-,\mathbb{R}) &\stackrel{d}{\to}& \Omega^1(-) &\stackrel{d}{\to}& \Omega^2(-) &\stackrel{d}{\to}& \cdots &\stackrel{d}{\to}& \Omega^{n+1}(-) }$
###### 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

$\array{ H(X,\bar \mathbf{B}^\bullet U(1)) &\stackrel{[F]}{\to}& H^{\bullet+1}_{dR}(X) \\ \downarrow^{cl} && \downarrow \\ H^{\bullet+1}(X,\mathbb{Z}) &\to& H^{\bullet + 1}(X,\mathbb{R}) }$

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 $P_1(X)$ 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 $A \in \Omega^1(X)$ induces a smooth functor $tra_A : P_1(X) \to \mathbf{B}U(1)$ from $P_1(X)$ to to the smooth groupoid $\mathbf{B} U(1)$ with one object and $U(1)$ as its smooth space of morphisms by sending each path $\gamma : [0,1] \to X$ to $\exp (2 \pi i\int_0^1 \gamma^* A)$. 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 $\eta_f : tra_A \to tra_{A'}$ between two such functors is in components precisely a smooth function $f : X \to U(1)$ such that $A' = A + d log f$.

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

$Funct^\infty(P_1(-), \mathbf{B}U(1)) : Op(X)^{op} \to LieGrpd \,.$

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

$Funct^\infty(\Pi_1(-), \mathbf{B}U(1)) \simeq \mathbb{Z}(2)^\infty_D \,.$

This way Deligne cohomology is realized as computing the stackification of the pre-stack $Funct^\infty(P_1(-), \mathbf{B}(1))$ of smooth $U(1)$-valued parallel transport functors.

The identification generalizes: for all $n$ there is a path n-groupoid $P_n(X)$ whose $k$-morphisms are $k$-dimensional smooth paths in $X$. Smooth $n$-functors $tra_C : _n(X) \to \mathbf{B}^n U(1)$ are canonically identified with smooth $n$-forms $C \in \Omega^n(X)$ 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

$n Funct^\infty(P_n(-), \mathbf{B}^n) \simeq \mathbb{Z}(n+1)^\infty_D \,.$

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 $U(1)$-principal bundles with connection. The Deligne complex $\bar \mathbf{B}U(1)$ in this case coincides with the groupoid of Lie-algebra valued forms for the Lie algebra of $U(1)$.

• 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 $SO(n)$-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)