# nLab fiber integration in ordinary differential cohomology

### Context

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### Integration theory

integration

analytic integrationcohomological integration
measureorientation in generalized cohomology
Riemann/Lebesgue integration, of differential formspush-forward in generalized cohomology/in 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 $\left[\nabla \right]\in {H}_{\mathrm{diff}}^{n}\left(Y\right)$ a class in ordinary differential cohomology of degree $n$ on $Y$, its fiber integration

$\left[\mathrm{exp}\left(i{\int }_{Y/X}\nabla \right)\right]\in {H}_{\mathrm{diff}}^{n-k}\left(X\right)$\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 $\mathrm{dim}Y=k=n-1$ the element

$\mathrm{exp}\left(i{\int }_{Y}\nabla \right)\in {H}_{\mathrm{diff}}^{1}\left(*\right)\simeq U\left(1\right)$\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 $\stackrel{^}{\omega }$ in the ordinary differential cohomology of degree $k$ of $V$;

• such that for each $x\in X$ we have

${\int }_{{V}_{x}}\omega =±1\phantom{\rule{thinmathspace}{0ex}}.$\int_{V_x} \omega = \pm 1 \,.
###### Remark

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

1. A factorization through an embedding of smooth manifolds

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

for some $N\in ℕ$;

2. a tubular neighbourhood $W↪Y×{ℝ}^{N}$ 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 ${H}_{\mathrm{diff}}^{n}\left(-\right)$ 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 $Y×{ℝ}^{N}\to Y$ over a compact $Y$:

${\int }_{{ℝ}^{N}}:{H}_{\mathrm{diff},\mathrm{cpt}}^{n+N}\left(Y×{ℝ}^{n}\right)\to {H}_{\mathrm{diff}}^{n}\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $Hℤ$-orientation $U$, def. 2. Then the corresponding fiber integration of ordinary differential cohomology is the composite

${\int }_{X/Y}:{H}_{\mathrm{diff}}^{n+k}\left(X\right)\stackrel{\left(-\right)\cup U}{\to }{H}_{\mathrm{diff},\mathrm{cpt}}^{n+N}\left(X×{ℝ}^{N}\right)\stackrel{{\int }_{{ℝ}^{N}}}{\to }{H}_{\mathrm{diff}}^{n}\left(Y\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $H\left(X,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)$ for the Deligne complex in degree $\left(n+1\right)$ 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

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

${B}^{n}U\left(1{\right)}_{\mathrm{conn}}\in H\phantom{\rule{thinmathspace}{0ex}},$\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 $k\in ℕ$ without boundary. There is the internal hom in an (infinity,1)-topos

$\left[{\Sigma }_{k},{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\in H\phantom{\rule{thinmathspace}{0ex}},$[\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

$\mathrm{exp}\left(2\pi i{\int }_{\Sigma }\left(-\right)\right)\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left[{\Sigma }_{k},{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\to {B}^{n-k}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in H\phantom{\rule{thinmathspace}{0ex}}.$\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 $U×{\Sigma }_{k}$ to the $\left(n-k\right)$-connection on $U$ which is their fiber integration over ${\Sigma }_{k}$.

###### Proof

To see this, observe that

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

2. if $\left\{{U}_{i}\to {\Sigma }_{k}\right\}$ is a fixed good open cover of ${\Sigma }_{k}$, then $\left\{U×{U}_{i}\to U×{\Sigma }_{k}\right\}$ is also a good open cover, for every $U\in$ CartSp;

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

4. the (image under the Dold-Kan correspondence) of the Deligne complex $ℤ\left(n+1{\right)}_{D}^{\infty }$ 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 $\left[{\Sigma }_{k},{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]$ by an object of $\left[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj},\mathrm{loc}}$ is given by the simplicial presheaf

$U↦\mathrm{DK}ℤ\left(n+1{\right)}_{D}^{\infty }\left(C\left(\left\{U×{U}_{i}\right\}\right)\right)$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 $\left\{U×{U}_{i}\to U×{\Sigma }_{k}\right\}$.

On this def. 4 provides a morphism of simplicial sets

$\mathrm{DK}ℤ\left(n+1{\right)}_{D}^{\infty }\left(C\left(\left\{U×{U}_{i}\right\}\right)\right)\to \mathrm{DK}ℤ\left(n+1{\right)}_{D}^{\infty }\left(U\right)$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

(…)

### 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:

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

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

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

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

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

extended action functional of $\stackrel{^}{c}$-Chern-Simons theory in codimension $\left(n-k\right)$

$\mathrm{exp}\left(2\pi i{\int }_{{\Sigma }_{k}}\left[{\Sigma }_{k},\stackrel{^}{c}\right]\right):\left[{\Sigma }_{k},B{G}_{\mathrm{conn}}\right]\stackrel{\left[{\Sigma }_{k},\stackrel{^}{c}\right]}{\to }\left[{\Sigma }_{k},{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right]\stackrel{\mathrm{exp}\left(2\pi i{\int }_{{\Sigma }_{k}}\left(-\right)\right)}{\to }{B}^{n-k}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}}.$\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

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)