The best known version of differential cohomology is a differential refinement of generalized (Eilenberg-Steenrod) cohomology, hence of cohomology in stable homotopy theory (as opposed to more general nonabelian cohomology). For ordinary cohomology the refinement to ordinary differential cohomology is modeled for instance by complex line bundles with connection on a bundle, bundle gerbes with connection, etc, and generally by Deligne cohomology. Similarly, differential refinements of topological K-theory to differential K-theory may be presented by vector bundles with connection (e.g Simons-Sullivan 08).
A systematic characterization and construction of differential generalized (Eilenberg-Steenrod) cohomology in terms of suitable homotopy fiber products of the mapping spectra representing the underlying cohomology theory with differential form data was then given in (Hopkins-Singer 02) (motivated by discussion of the quantization of the M5-brane via holographically dual 7d Chern-Simons theory by Edward Witten).
In this stable case one characteristic property of differential cohomology that was first observed empirically in ordinary differential cohomology (Simons-Sullivan 07) and differential K-theory (Simons-Sullivan 08) is that it is a kind of cohomology theory which fits into a differential cohomology diagram which is an interlocking system of two fiber sequences that expresses how the data of a geometric bundle with connection may be decomposed into the underlying bundle and the curvature of the connection and how connections on trivial bundles as well as bundles with flat connections are examples of the general concept. (The homotopy fiber product used in (Hopkins-Singer 02) provides one of the two commuting squares in this diagram.)
Schematically the characteristic differential cohomology hexagon (see there for details) looks as follows
It turns out that such differential cohomology diagrams are exactly what all stable homotopy types in cohesive (∞,1)-toposes naturally sit in (Bunke-Nikolaus-Völkl 13). All the traditional differential cohomology theories and their joint generalization due to (Hopkins-Singer 02) are special cases of this for the case that is the tangent cohesive (∞,1)-topos of that of smooth ∞-groupoids (∞-stacks over the site of smooth manifolds).
It is therefore natural to define differential cohomology to be the intrinsic cohomology of cohesive (∞,1)-toposes. (Notice that while the differential cohomology diagram itself only involves the shape modality and the flat modality of cohesion, the sharp modality is needed to produce, via differential concretification, the correct moduli stacks of differential cocycles over a given base object from the mapping stack of that into the representing stable homotopy type).
Viewed in this generality, differential cohomology makes sense for instance also in supergeometry where it subsumes structures such as super line 2-bundles and more generally higher super gerbes with connection, which naturally appear as differential refinements of the geometric twists of (differential) K-theory and tmf.
The ordinary differential cohomology is modeled by
apart from that people studied mainly differential K-theory.
Cocycles in differential K-theory model the
For a differential cocycle representing a gauge field, one says that
its image in differential forms is the corresponding field strength;
the following are ancient notes that need to be brushed up and polished.
A standard definition of differential cohomology is in terms of a homotopy pullback of a generalized (Eilenberg-Steenrod) cohomology theory with the complex of differential forms over real cohomology:
Let be a generalized cohomology theory in the sense of the generalized Eilenberg–Steenrod axioms, and let be a morphism to real singular cohomology with coefficients in the ring of -cohomology of the point. Then the differential refinement of , the degree differential -cohomology is the homotopy pullback in
is the complex underlying real singular cohomology
is the complex underlying deRham cohomology
is deRham theorem morphism
is the Chern character map.
For more details on this see at differential function complex.
The following are old notes once taken in a talk by Thomas Schick at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology. Needs to be brushed-up, polished, improved, rewritten…
The following theory of differential cohomology (also called smooth cohomology) is developed and used in the work of Ulrich Bunke and Thomas Schick. Contrary to the above, it does not take the notion of homotopy limit as fundamental, but instead characterizes the universality of the above commuting diagram by other means. On the other hand, this means that their axiomatization at the moment only capture cohomology classes, not the representing cocycles. It is sort of known that for various applications specific cocycle representatives do play an important role, and one may imagining refining to discussion below eventually to accommodate for that.
so differential cohomology combines the ordinary cohomology with a differential form representative of its image in real cohomology.
projects a differential cohomology to its underlying ordinary cohomology class;
send the differential cohomology class to its curvature differential form data
we want an exact sequence
Given cohomology theory , a smooth refinement is a functor with transformations such that
is the graded non-torsion cohomology of on the point (so now all the gradings above denote total grading)
and such that there is a transformation
that gives the above kind of exact sequence.
If is multiplicative, we say is multiplicative with product if takes values in graded rings and the transformations are compatible with multiplicative structure, where
has -integration if there is a natural (in ) transformation
compatible with of forms and for it is given by the suspension isomorphism
Ordinary cohomology theories are supposed to be homotopy invariant, but differential forms are not, so in general the differential cohomology is not.
Given a smooth cohomology theory. The homotopy formula:
given a smooth homotopy we have
(i.e. flat cohomology) is a homotopy invariant functor.
It suffices to show
for all .
Observe if the left hand side vanishes, .
For general ; .
Stokes’ theorem gives .
On the other hand
A calculation: .
For each generalized cohomology theory a differential version as in the above definition does exist.
It’s not evident how to obtain more structure like multiplication.
Using geometric models, multiplicative smooth extensions with -integration are constructed for
MU-bordisms (unitary bordisms)
(Bunke–Schröder–Schick–Wiethaupts; and from there Landweber exact cohomology theories)
(Simons–Sullivan proved this for ordinary integral cohomology.)
Assume is rationally even, meaning that
plus one further technical assumption.
Then any two smooth extensions , are naturally isomorphic.
If required to be compatible with integration the ismorphism is unique.
If are multiplicative, then this isomorphism is, as well.
If we don’t require compatibility with -integration, then there are “exotic” abelian group structures on .
gauge field: models and components
|physics||differential geometry||differential cohomology|
|gauge field||connection on a bundle||cocycle in differential cohomology|
|instanton/charge sector||principal bundle||cocycle in underlying cohomology|
|gauge potential||local connection differential form||local connection differential form|
|field strength||curvature||underlying cocycle in de Rham cohomology|
|minimal coupling||covariant derivative||twisted cohomology|
|BRST complex||Lie algebroid of moduli stack||Lie algebroid of moduli stack|
|extended Lagrangian||universal Chern-Simons n-bundle||universal characteristic map|
Differential cohomology was first developed for the special cases of ordinary differential cohomology and of differential K-theory (interest into which came from discussion of D-brane charge in type II superstring theory). See the references there. A survey is in
The suggestion that differential cohomology is or should be characterized by the differential cohomology diagram is due to
(discussed there for ordinary differential cohomology)
(discussed there for differential K-theory).
A first systematic account characterizing and constructing stable differential cohomology in terms of homotopy fiber products of bare spectra with differential form data (which is really the homotopy-theoretic refinement of one of right square in the differential cohomology diagram) was given in
with further reviews including
Comprehensive lecture notes on the developments up to this stage are in