# nLab fiber integration in ordinary differential cohomology

### Context

#### Differential cohomology

differential cohomology

# Contents

## 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 $[\nabla] \in H_{diff}^n(Y)$ a class in ordinary differential cohomology of degree $n$ on $Y$, its fiber integration

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

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

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

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

$\int_{V_x} \omega = \pm 1 \,.$
###### Remark

The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ 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 $H \mathbb{Z}_{diff}$-orientation on $p$ is

1. A factorization through an embedding of smooth manifolds

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

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

2. a tubular neighbourhood $W \hookrightarrow Y \times \mathbb{R}^N$ 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 $H^n_{diff}(-)$ 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 $Y \times \mathbb{R}^N \to Y$ over a compact $Y$:

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

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

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 00).

For $X$ a smooth manifold, write $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ 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

$\int_F \;\colon\; \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) \to \mathbf{H}(X, \mathbf{B}^{n-k} U(1)_{conn})$

given by (…)

### 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 $\mathbf{H} \coloneqq$ 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

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

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

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

which is the smooth moduli $n$-stack of circle $n$-connections on $\Sigma_k$.

###### Proposition

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

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

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 $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;

2. if $\{U_i \to \Sigma_k\}$ 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 $U \in CartSp$) cofibrant object resolution of $U \times \Sigma_k$ in the projective local model structure on simplicial presheaves $[CartSp^{op}, sSet]_{proj,loc}$ which presents $\mathbf{H} =$Smooth∞Grpd (as discussed there);

4. the (image under the Dold-Kan correspondence) of the Deligne complex $\mathbb{Z}(n+1)^\infty_D$ 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 $[\Sigma_k, \mathbf{B}^n U(1)_{conn}]$ by an object of $[CartSp^{op}, sSet]_{proj,loc}$ is given by the simplicial presheaf

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

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

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

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

Applications are to

## Properties

(…)

### 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

### $\infty$-Chern-Simons functionals in higher codimension

(…)

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

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

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

$[\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}$

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

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

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

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

(…)

## References

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, 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)