nLab
Deligne 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 nn-fold delooping B nU(1)\mathbf{B}^n U(1):

(n+1) D B¯ nU(1) N[P n(),B nU(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 nn, of a smooth space XX is cohomology with coefficients in B¯ nU(1)\bar \mathbf{B}^n U(1).

Delignecohomology=H(X,B¯ nU(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) H¯ n(X,)\bar H^n(X,\mathbb{Z}) of the integral cohomology H n(X,)H^n(X, \mathbb{Z}) of XX in terms of abelian sheaf cohomology.

Recall that analgous to how H n(X,)H^n(X,\mathbb{Z}) classifies line (n1)(n-1)-bundles and equivalently line (n2)(n-2)-gerbes on XX, H¯ n(X,)\bar H^n(X, \mathbb{Z}) classifies line (n2)(n-2)-gerbes with connection.

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

Definition

Definition

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

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

(n+1) D :=(0C (,)C (,)dΩ 1()ddΩ n()). \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

B¯ nU(1):=(0C (,U(1))dlogΩ 1()ddΩ n()) \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 (,U(1))dlogΩ 1()C^\infty(-,U(1)) \stackrel{d log}{\to} \Omega^1(-) is the morphism of sheaves induced by regarding a U(1)/U(1) \simeq \mathbb{R}/\mathbb{Z}-valued function locally as a \mathbb{R}-valued function and applying the deRham differential dd to that.

The obvious morphism of complexes

C (,) C (,) dlog Ω 1() d d Ω n() ()/ Id Id 0 C (,U(1)) dlog Ω 1() d d Ω n() \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 nn is locally constant \mathbb{R}-valued functions modulo locally constant \mathbb{Z}-valued functions in the first case and constant U(1)U(1)-valued functions in the second case, which is the same.

Definition

Deligne cohomology in degree n+1n+1 of XX is the cohomology (which is abelian sheaf cohomology in this case) with coefficients in B¯ nU(1)\bar \mathbf{B}^n U(1).

H(X,(n+1) D )H(X,B¯ nU(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,B¯ nU(1))H(X,B nU(1))H(X,B n+1)H n+1(X,) 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,B¯ nU(1))H dR n+1(X). [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

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

given by

C (,) C (,) d Ω 1() d d Ω n() Id 0 0 0 C (,) 0 0 0 \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

B¯ nU(1)(n+1) D (Ω 0()dΩ 1()ddΩ n+1()) \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

C (,) C (,) d Ω 1() d d Ω n() 0 d d d C (,) d Ω 1() d Ω 2() d d Ω n+1() \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

H(X,B¯ U(1)) [F] H dR +1(X) cl H +1(X,) H +1(X,) \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 nn-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)P_1(X) whose smooth space of objects is XX and whose smooth space of morphisms is a space of classes of smooth paths in XX. Every smooth 1-form AΩ 1(X)A \in \Omega^1(X) induces a smooth functor tra A:P 1(X)BU(1)tra_A : P_1(X) \to \mathbf{B}U(1) from P 1(X)P_1(X) to to the smooth groupoid BU(1)\mathbf{B} U(1) with one object and U(1)U(1) as its smooth space of morphisms by sending each path γ:[0,1]X\gamma : [0,1] \to X to exp(2πi 0 1γ *A)\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 η f:tra Atra A\eta_f : tra_A \to tra_{A'} between two such functors is in components precisely a smooth function f:XU(1)f : X \to U(1) such that A=A+dlogfA' = A + d log f.

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

Funct (P 1(),BU(1)):Op(X) opLieGrpd. 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 (Π 1(),BU(1))(2) D . 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 (P 1(),B(1))Funct^\infty(P_1(-), \mathbf{B}(1)) of smooth U(1)U(1)-valued parallel transport functors.

The identification generalizes: for all nn there is a path n-groupoid P n(X)P_n(X) whose kk-morphisms are kk-dimensional smooth paths in XX. Smooth nn-functors tra C: n(X)B nU(1)tra_C : _n(X) \to \mathbf{B}^n U(1) are canonically identified with smooth nn-forms CΩ n(X)C \in \Omega^n(X) and under the Dold-Kan correspondence the Deligne-complex in degree n+1n+1 is identified with the sheaf of nn-groupoids of such smooth nn-functors

nFunct (P n(),B n)(n+1) D . 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=1n=1 this is in

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

for n=2n=2 in

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

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

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

Cup product

See Beilinson-Deligne cup-product.

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.

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)