fiber integration in ordinary differential cohomology


Differential cohomology

Integration theory



The special case of fiber integration in differential cohomology for ordinary differential cohomology is the partial higher holonomy operation for circle n-bundles with connection:

for YXY \to X a bundle of compact smooth manifolds SS of dimension kk and []H diff n(Y)[\nabla] \in H_{diff}^n(Y) a class in ordinary differential cohomology of degree nn on YY, its fiber integration

[exp(i Y/X)]H diff nk(X) \left[\exp(i \int_{Y/X} \nabla)\right] \in H^{n-k}_{diff}(X)

is a differential cohomology class on XX of degree kk less.

In the particular case that X=*X = * is the point and dimY=k=n1dim Y = k = n-1 the element

exp(i Y)H diff 1(*)U(1) \exp(i \int_{Y} \nabla) \in H^{1}_{diff}(*) \simeq U(1)

is the higher holonomy of \nabla over YY.


Differential orientation

The operation of fiber integration in generalized (Eilenberg-Steenrod) cohomology requires a choice of orientation in generalized cohomology. For fiber integration in differential cohomology this is to be refined to a differential orientation .

Accordingly, instead of a Thom class there is a differential Thom class .


For XX a compact smooth manifold and VXV \to X a smooth real vector bundle of rank kk a differential Thom cocycle on VV is


The underlying class [ω^]H compact k(V,)[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z}) in compactly supported integral cohomology is an ordinary Thom class for VV.


Let p:XYp : X \to Y be a smooth function of smooth manifolds.

An H diffH \mathbb{Z}_{diff}-orientation on pp is

  1. A factorization through an embedding of smooth manifolds

    p:XY× NY p : X \hookrightarrow Y \times \mathbb{R}^N \stackrel{}{\to} Y

    for some NN \in \mathbb{N};

  2. a tubular neighbourhood WY× NW \hookrightarrow Y \times \mathbb{R}^N of XX;

  3. a differential Thom cocycle, def. 1, UU on WXW \to X.

This appears as (HopkinsSinger, def. 2.9).

Via differential Thom cocycles

Write H diff n()H^n_{diff}(-) for ordinary differential cohomology. For any choice of presentation, there is a fairly evident fiber integration of compactly supported cocycles along trivial Cartesian space bundles Y× NYY \times \mathbb{R}^N \to Y over a compact YY:

N:H diff,cpt n+N(Y× n)H diff n(Y). \int_{\mathbb{R}^N} : H^{n+N}_{diff,cpt}(Y \times \mathbb{R}^n) \to H^n_{diff}(Y) \,.

Let XYX \to Y be a smooth function equipped with differential HH\mathbb{Z}-orientation UU, def. 2. Then the corresponding fiber integration of ordinary differential cohomology is the composite

X/Y:H diff n+k(X)()UH diff,cpt n+N(X× N) NH diff n(Y). \int_{X/Y} : H_{diff}^{n+k}(X) \stackrel{(-)\cup U}{\to} H_{diff, cpt}^{n+N}(X \times \mathbb{R}^N) \stackrel{\int_{\mathbb{R}^N}}{\to} H_{diff}^n(Y) \,.

This appears as (HopkinsSinger, def. 3.11).

In terms of Deligne cocycles

We discuss an explicit formula for fiber integration along product-bundles with compact fibers in terms of Deligne complex, following (Gomi-Terashima).

For XX a smooth manifold, write H(X,B nU(1) conn)\mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) for the Deligne complex in degree (n+1)(n+1) over XX.


Let XX be a paracompact smooth manifold and let FF be a compact smooth manifold of dimension kk without boundary. Then there is a morphism

F:H(X,B nU(1) conn)H(X,B nkU(1) conn) \int_F : \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \to \mathbf{H}(X, \mathbf{B}^{n-k} U(1)_{conn})

given by (…)

(Gomi-Terashima, section 2, corollary 3.2)

In terms of smooth homotopy types

The above formulation of fiber integration in ordinary differential cohomology serves as a presentation for a more abstract construction in smooth homotopy theory.

Let H\mathbf{H} \coloneqq Smooth∞Grpd be the ambient cohesive (∞,1)-topos of smooth ∞-groupoids/smooth ∞-stacks. As discussed there, the Deligne complex, being a sheaf of chain complexes of abelian groups, presents under the Dold-Kan correspondence a simplicial presheaf on the site CartSp, which in turn presents an object

B nU(1) connH, \mathbf{B}^n U(1)_{conn} \in \mathbf{H} \,,

discussed here: the smooth moduli ∞-stack of circle n-bundles with connection.

Let now Σ k\Sigma_k be a compact smooth manifold of dimension kk \in \mathbb{N} without boundary. There is the internal hom in an (infinity,1)-topos

[Σ k,B nU(1) conn]H, [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \in \mathbf{H} \,,

which is the smooth moduli nn-stack of circle nn-connections on Σ k\Sigma_k.


For all knk \leq n there is a natural morphism

exp(2πi Σ()):[Σ k,B nU(1) conn]B nkU(1) connH. \exp(2\pi i\int_\Sigma(-)) \; \colon \; [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k} U(1)_{conn} \;\;\; \in \mathbf{H} \,.

which for UU \in SmthMfd a smooth test manifold sends nn-connections on Σ k\Sigma_k on U×Σ kU \times \Sigma_k to the (nk)(n-k)-connection on UU which is their fiber integration over Σ k\Sigma_k.


To see this, observe that

  1. by definition H(U,[Σ k,B nU(1) conn])H(U×Σ k,B nU(1) conn)\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn});

  2. if {U iΣ k}\{U_i \to \Sigma_k\} is a fixed good open cover of Σ k\Sigma_k, then {U×U iU×Σ k}\{U \times U_i \to U \times \Sigma_k\} is also a good open cover, for every UU \in CartSp;

  3. hence the Cech nerve C({U×U i})C(\{U \times U_i\}) is a natural (functorial in UCartSpU \in CartSp) cofibrant object resolution of U×Σ kU \times \Sigma_k in the projective local model structure on simplicial presheaves [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} which presents H=\mathbf{H} = Smooth∞Grpd (as discussed there);

  4. the (image under the Dold-Kan correspondence) of the Deligne complex (n+1) D \mathbb{Z}(n+1)^\infty_D is a is fibrant in this model structure (since every circle nn-bundle is trivializable over a contractible space UU \in CartSp).

This means that a presentation of [Σ k,B nU(1) conn][\Sigma_k, \mathbf{B}^n U(1)_{conn}] by an object of [CartSp op,sSet] proj,loc[CartSp^{op}, sSet]_{proj,loc} is given by the simplicial presheaf

UDK(n+1) D (C({U×U i})) U \mapsto DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))

that sends UU to the Cech-Deligne hypercohomology chain complex with respect to the cover {U×U iU×Σ k}\{U \times U_i \to U \times \Sigma_k\}.

On this def. 4 provides a morphism of simplicial sets

DK(n+1) D (C({U×U i}))DK(n+1) D (U) DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\})) \to DK \mathbb{Z}(n+1)^\infty_D(U)

which one directly sees is natural in UU, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in H\mathbf{H}.




Abstract formulation

At least the fiber integration all the way to the point exists on general grounds for the intrinsic differential cohomology in any cohesive (∞,1)-topos: the general abstract formulation is in the section Higher holonomy and Chern-Simons functional and the implementation in smooth ∞-groupoids is in the section Smooth higher holonomy and Chern-Simons functional .


\infty-Chern-Simons functionals in higher codimension


Differential universal characteristic class / extended \infty-Chern-Simons Lagrangian:

c^:BG connB nU(1) conn \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^{n}U(1)_{conn}

moduli \infty-stack of higher gauge fields on a given Σ k\Sigma_k:

[Σ k,BG conn]H [\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}

Lagrangian of c^\hat \mathbf{c}-Chern-Simons theory:

[Σ k,c^]:[Σ k,BG conn][Σ k,B nU(1) conn] [\Sigma_k, \hat \mathbf{c}] : [\Sigma_k, \mathbf{B}G_{conn}] \to [\Sigma_k, \mathbf{B}^n U(1)_{conn}]

extended action functional of c^\hat \mathbf{c}-Chern-Simons theory in codimension (nk)(n-k)

exp(2πi Σ k[Σ k,c^]):[Σ k,BG conn][Σ k,c^][Σ k,B nU(1) conn]exp(2πi Σ k())B nkU(1) conn. \exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \hat \mathbf{c}] ) : [\Sigma_k, \mathbf{B}G_{conn}] \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to} [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i\int_{\Sigma_k} (-))}{\to} \mathbf{B}^{n-k} U(1)_{conn} \,.



A discussion in the general sense of fiber integration in generalized (Eilenberg-Steenrod) cohomology is in section 3.4 of

and around prop. 2.1 (in the context of Chern-Simons theory) in

  • Daniel Freed, Classical Chern-Simons theory II. Special issue for S. S. Chern. Houston J. Math. 28 (2002), no. 2, 293–310.

Explicit formulas for fiber integration of cocycles in Cech-Deligne cohomology are given in

  • Kiyonori Gomi and Yuji Terashima, A Fiber Integration Formula for the Smooth Deligne Cohomology International Mathematics Research Notices 2000, No. 13 (pdf)

and their generalization from higher holonomy to higher parallel transport in

  • Kiyonori Gomi and Yuji Terashima, Higher dimensional parallel transport Mathematical Research Letters 8, 25–33 (2001) (pdf)


  • David Lipsky, Cocycle constructions for topological field theories (2010) (pdf)

See also

  • Johan Dupont, Rune Ljungmann, Integration of simplicial forms and Deligne cohomology Math. Scand. 97 (2005), 11–39 (pdf)

The observation that the construction in Gomi-Terashima induces refines to smooth higher moduli stacks is discussed in

for the case without boundary and for the general case in

Revised on October 2, 2013 15:58:20 by Urs Schreiber (