Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
A differential function complex (HopkinsSinger) is a Kan complex of cocycle s for generalized differential cohomology, hence for differential refinements of generalized (Eilenberg-Steenrod) cohomology theories:
roughly, given a spectrum representing a given cohomology theory, its differential function complex over any given smooth manifold is the simplicial set whose -simplices are triples consisting of
a continuous function ;
a smooth differential form on whose corresponding real cohomology class (under the de Rham theorem) is that of the pullback of the real cohomology classes of along ;
an explicit coboundary in real cohomology exhibiting this fact.
(More precisely, in order for this construction to yield not just a single simplicial set (which will be a Kan complex) but a suitable spectrum object, there are conditons on the dependency of on the tangent vectors to the simplex.)
When applied to the Eilenberg-MacLane spectrum this construction reproduces, on cohomology classes, ordinary differential cohomology. Applied to the classifying space of topological K-theory it gives differential K-theory.
See also at differential cohomology diagram –Hopkins-Singer coefficients.
Cocycles with values in graded vector spaces
For the present purposes it will be convenient to collect cocycles of various degrees together to a single cocycle. For that purpose we make the following simple definition.
a differential function on a smooth manifold with values in is a triple with
a continuous map;
a smooth differential form on ;
a cochain in real cohomology on (the topological space underlying) ;
such that in the abelian group of singular cochains the equation
This is (HopkinsSinger, def.4.1).
In words this is: a continuous map to the topological space together with a smooth refinement of the pullback of the chosen singular cochain.
Differential function complexes
the differential function complex
of all differential functions is the simplicial set whose -simplices are differential functions, def, 2
For applications one needs certain sub-complex of this, filtered by the number of legs that has along the simplices.
for the sub-simplicial set of differential forms that vanish when evaluated on more than vector fields tangent to the simplex;
for the sub-simplicial set of those differential functions whose differential form component is in .
This is (HopkinsSinger, def. 4.5).
The complex is (up to equivalence, of course) the homotopy pullback
in sSet (regarded as equipped with its standard model structure on simplicial sets).
Here is the internal hom in Top and denotes the singular simplicial complex.
The following proposition gives the simplicial homotopy groups of these differential function complexes in dependence of the parameter .
We have generally
(for instance by the Dold-Kan correspondence).
The simplicial homotopy groups of are
This implies isomorphisms
This appears as HopkinsSinger, p. 36 and corollary D15.
Let be an Omega-spectrum. Let be the canonical Chern character class (…).
For a smooth manifold, and , the sequence of differential function complexes, def. 3,
forms an Omega-spectrum.
This is the differential function spectrum for , , .
This is ([HopkinsSinger, section 4.6]).
The differential -cohomology group of the smooth manifold in degree is
This is (HopkinsSinger, def. 4.34).
For reference, we repeat from above the central statements about the homotopy types of the differential function complexes, def. 3.
For an Omega-spectrum, a smooth manifold, we have for all , a weak homotopy equivalence
identifying the loop space object (at the canonical base point) of the differential function complex of at filtration level with that differential function complex of at filtration level .
Relation to differential cohomology in cohesive -toposes
The following is a simple corollary or slight rephrasing of some of the above constructions, which may serve to show how differential function complexes present differential cohomology in the cohesive (∞,1)-topos of smooth ∞-groupoids.
For a spectrum as above,
we have an (∞,1)-pullback square
By prop. 2 we have that
The statement then follows with the pasting law for homotopy pullbacks
For the Eilenberg-MacLane spectrum, is ordinary differential cohomology.
For the K-theory spectrum, is differential K-theory.
For the Thom spectrum, is differential cobordism cohomology;
For the tmf spectrum, is differential cohomology;
Line bundles with connection
Let be the Eilenberg-MacLane space that is the classifying space for -principal bundles. It carries the canonical cocycle representing in the class of the universal complex line bundle on .
Accordingly, for a continuous map, we have the corresponding line bundle on .
One checks (…details…Example 2.7 in HopSin) that a refinement of to a differential function corresponds to equipping with a smooth connection.
Now consider a morphism between two such -differential functions. By definition this is now a -principal bundle with connection on , whose curvature form is of the form , where is a 2-form on and is a smooth function on , both pulled back to and multiplied there.
But since is necessarily closed it follows with that is actually constant.
This means that that the parallel transoport of the connection on induces a insomorphism between the two line bundles on over the endpoints of that respects the connections.
Higher filtration degree
Differential function complexes were introduced and studied in
For further references see differential cohomology.