Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-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 $Y \to X$ 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
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
is the higher holonomy of $\nabla$ over $Y$.
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 $X$ 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
The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ in compactly supported integral cohomology is an ordinary Thom class for $V$.
Let $p : X \to Y$ be a smooth function of smooth manifolds.
An $H \mathbb{Z}_{diff}$-orientation on $p$ is
A factorization through an embedding of smooth manifolds
for some $N \in \mathbb{N}$;
a tubular neighbourhood $W \hookrightarrow Y \times \mathbb{R}^N$ of $X$;
a differential Thom cocycle, def. 1, $U$ on $W \to X$.
This appears as (HopkinsSinger, def. 2.9).
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 $Y$:
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
This appears as (HopkinsSinger, def. 3.11).
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 $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ for the Deligne complex in degree $(n+1)$ over $X$.
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
given by (…)
(Gomi-Terashima, section 2, corollary 3.2)
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
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
which is the smooth moduli $n$-stack of circle $n$-connections on $\Sigma_k$.
For all $k \leq n$ there is a natural morphism
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 $\Sigma_k$.
To see this, observe that
by definition $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;
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;
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);
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
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
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
(…)
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 .
(…)
Differential universal characteristic class / extended $\infty$-Chern-Simons Lagrangian:
moduli $\infty$-stack of higher gauge fields on a given $\Sigma_k$:
Lagrangian of $\hat \mathbf{c}$-Chern-Simons theory:
extended action functional of $\hat \mathbf{c}$-Chern-Simons theory in codimension $(n-k)$
(…)
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
Explicit formulas for fiber integration of cocycles in Cech-Deligne cohomology are given in
and their generalization from higher holonomy to higher parallel transport in
and
See also
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