nLab
fiber integration in ordinary differential cohomology

Idea

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 $Y\to X$ a bundle of compact smooth manifolds $S$ of dimension $k$ and $[\nabla ]\in H_{\mathrm{diff}}^n(Y)$ a class in ordinary differential cohomology of degree $n$ on $Y$, its fiber integration

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

is a differential cohomology class on $X$ of degree $k$ less.

In the particular case that $X=*$ is the point and $\dim Y=k=n-1$ the element

$$\exp (i{\int }_Y\nabla )\in H_{\mathrm{diff}}^1(*)\simeq U(1)$$\exp(i \int_{Y} \nabla) \in H^{1}_{diff}(*) \simeq U(1)

is the higher holonomy of $\nabla$ over $Y$.

Definition

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 .

Definition

For $X$ a compact smooth manifold and $V\to X$ a smooth real vector bundle of rank $k$ a differential Thom cocycle on $V$ is

• a compactly supported cocycle $\hat{\omega }$ in the ordinary differential cohomology of degree $k$ of $V$;

• such that for each $x\in X$ we have

  $${\int }_{V_x}\omega =\pm 1.$$\int_{V_x} \omega = \pm 1 \,.

Remark

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

Definition

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

An $H{\mathbb{Z}}_{\mathrm{diff}}$-orientation on $p$ is

1. A factorization through an embedding of smooth manifolds

   $$p:X\hookrightarrow Y\times {ℝ}^N\stackrel{}{\to }Y$$p : X \hookrightarrow Y \times \mathbb{R}^N \stackrel{}{\to} Y

   for some $N\in \mathbb{N}$;

2. a tubular neighbourhood $W\hookrightarrow Y\times {ℝ}^N$ of $X$;

3. a differential Thom cocycle, def. 1, $U$ on $W\to X$.

This appears as (HopkinsSinger, def. 2.9).

Via differential Thom cocycles

Write $H_{\mathrm{diff}}^n(-)$ 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 $Y\times {ℝ}^N\to Y$ over a compact $Y$:

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

Definition

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

$${\int }_{X/Y}:H_{\mathrm{diff}}^{n+k}(X)\stackrel{(-)\cup U}{\to }{H}_{\mathrm{diff},\mathrm{cpt}}^{n+N}(X\times {ℝ}^N)\stackrel{{\int }_{{ℝ}^N}}{\to }{H}_{\mathrm{diff}}^n(Y).$$\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 $X$ a smooth manifold, write $H(X,B^{n}U(1{)}_{\mathrm{conn}})$ for the Deligne complex in degree $(n+1)$ over $X$.

Definition

Let $X$ be a paracompact smooth manifold and let $F$ be a compact smooth manifold of dimension $k$ without boundary. Then there is a morphism

$${\int }_F:H(X,B^{n}U(1{)}_{\mathrm{conn}})\to H(X,B^{n-k}U(1{)}_{\mathrm{conn}})$$\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 $H≔$ 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

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

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

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

$$[{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}]\in H,$$[\Sigma_k, \mathbf{B}^n U(1)_{conn}] \in \mathbf{H} \,,

which is the smooth moduli $n$-stack of circle $n$-connections on ${\Sigma }_k$.

Proposition

For all $k\le n$ there is a natural morphism

$$\exp (2\pi i{\int }_{\Sigma }(-)):[{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}]\to B^{n-k}U(1{)}_{\mathrm{conn}}\in H.$$\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 $U\in$ SmthMfd a smooth test manifold sends $n$-connections on ${\Sigma }_k$ on $U\times {\Sigma }_k$ to the $(n-k)$-connection on $U$ which is their fiber integration over ${\Sigma }_k$.

Proof

To see this, observe that

1. by definition $H(U,[{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}])\simeq H(U\times {\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}})$;

2. if $\{U_i\to {\Sigma }_k\}$ is a fixed good open cover of ${\Sigma }_k$, then $\{U\times U_i\to U\times {\Sigma }_k\}$ is also a good open cover, for every $U\in$ CartSp;

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

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

This means that a presentation of $[{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}]$ by an object of $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ is given by the simplicial presheaf

$$U\mapsto \mathrm{DK}\mathbb{Z}(n+1{)}_D^{\mathrm{\infty }}(C(\{U\times U_i\}))$$U \mapsto DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))

that sends $U$ to the Cech-Deligne hypercohomology chain complex with respect to the cover $\{U\times U_i\to U\times {\Sigma }_k\}$.

On this def. 4 provides a morphism of simplicial sets

$$\mathrm{DK}\mathbb{Z}(n+1{)}_D^{\mathrm{\infty }}(C(\{U\times U_i\}))\to \mathrm{DK}\mathbb{Z}(n+1{)}_D^{\mathrm{\infty }}(U)$$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 $U$, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in $H$.

Properties

General

(…)

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 .

Examples

$\mathrm{\infty }$-Chern-Simons functionals in higher codimension

(…)

Differential universal characteristic class / extended $\mathrm{\infty }$-Chern-Simons Lagrangian:

$$\hat{c}:B{G}_{\mathrm{conn}}\to B^{n}U(1{)}_{\mathrm{conn}}$$\hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^{n}U(1)_{conn}

moduli $\mathrm{\infty }$-stack of higher gauge fields on a given ${\Sigma }_k$:

$$[{\Sigma }_k,B{G}_{\mathrm{conn}}]\in H$$[\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}

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

$$[{\Sigma }_k,\hat{c}]:[{\Sigma }_k,B{G}_{\mathrm{conn}}]\to [{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}]$$[\Sigma_k, \hat \mathbf{c}] : [\Sigma_k, \mathbf{B}G_{conn}] \to [\Sigma_k, \mathbf{B}^n U(1)_{conn}]

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

$$\exp (2\pi i{\int }_{{\Sigma }_k}[{\Sigma }_k,\hat{c}]):[{\Sigma }_k,B{G}_{\mathrm{conn}}]\stackrel{[{\Sigma }_k,\hat{c}]}{\to }[{\Sigma }_k,B^{n}U(1{)}_{\mathrm{conn}}]\stackrel{\exp (2\pi i{\int }_{{\Sigma }_k}(-))}{\to }{B}^{n-k}U(1{)}_{\mathrm{conn}}.$$\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} \,.

(…)

Related concepts

• higher parallel transport, higher holonomy

• fiber integration in differential K-theory

References

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

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology,and M-Theory

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)

and

• 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)