Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge 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 a bundle of compact smooth manifolds of dimension and a class in ordinary differential cohomology of degree on , its fiber integration
is a differential cohomology class on of degree less.
In the particular case that is the point and the element
is the higher holonomy of over .
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 .
Let be a smooth function of smooth manifolds.
An -orientation on is
A factorization through an embedding of smooth manifolds
for some ;
a tubular neighbourhood of ;
a differential Thom cocycle, def. 1, on .
This appears as (HopkinsSinger, def. 2.9).
Via differential Thom cocycles
Write 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 over a compact :
Let be a smooth function equipped with differential -orientation , def. 2. Then the corresponding fiber integration of ordinary differential cohomology is the composite
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 a smooth manifold, write for the Deligne complex in degree over .
(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 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 be a compact smooth manifold of dimension without boundary. There is the internal hom in an (infinity,1)-topos
which is the smooth moduli -stack of circle -connections on .
For all there is a natural morphism
which for SmthMfd a smooth test manifold sends -connections on on to the -connection on which is their fiber integration over .
To see this, observe that
by definition ;
if is a fixed good open cover of , then is also a good open cover, for every CartSp;
hence the Cech nerve is a natural (functorial in ) cofibrant object resolution of in the projective local model structure on simplicial presheaves which presents Smooth∞Grpd (as discussed there);
the (image under the Dold-Kan correspondence) of the Deligne complex is a is fibrant in this model structure (since every circle -bundle is trivializable over a contractible space CartSp).
This means that a presentation of by an object of is given by the simplicial presheaf
that sends to the Cech-Deligne hypercohomology chain complex with respect to the cover .
On this def. 4 provides a morphism of simplicial sets
which one directly sees is natural in , hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in .
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 .
-Chern-Simons functionals in higher codimension
Differential universal characteristic class / extended -Chern-Simons Lagrangian:
moduli -stack of higher gauge fields on a given :
Lagrangian of -Chern-Simons theory:
extended action functional of -Chern-Simons theory in codimension
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)
- 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