Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
Deligne cohomology – or Deligne-Beilinson cohomology is an abelian sheaf cohomology that models ordinary differential cohomology.
The standard Deligne complex (of abelian sheaves) is under the Dold-Kan correspondence the sheaf of n-groupoids of smooth n-functors from the path n-groupoid to the -fold delooping :
Smooth Deligne cohomology in degree , of a smooth space is cohomology with coefficients in .
Here the notation on the right is as at the end of motivation for sheaves, cohomology and higher stacks.
This is a realization of the differential refinement (or smooth extension) of the integral cohomology of in terms of abelian sheaf cohomology.
Recall that analogous to how classifies line -bundles and equivalently line -gerbes on , classifies line -gerbes with connection.
Accordingly, the Deligne complex of sheaves is a complex of sheaves of differential forms.
For write for the [[sheaf] of smooth differential -forms on and for the sheaf of smooth -valued functions on .
The degree Deligne complex is the complex of sheaves
Often it is useful to consider the quasi-isomorphic complex
Here is the morphism of sheaves induced by regarding a -valued function locally as a -valued function and applying the deRham differential to that.
The obvious morphism of complexes
clearly induces isomorphism on homology groups: the homology in degree is locally constant -valued functions modulo locally constant -valued functions in the first case and constant -valued functions in the second case, which is the same.
Deligne cohomology in degree of is the cohomology (which is abelian sheaf cohomology in this case) with coefficients in .
Here the notation on the right is motivated from the discussion at the end of motivation for sheaves, cohomology and higher stacks.
Characteristic classes of Deligne cocycles
There are two natural morphisms of abelian cohomology groups out of Deligne cohomology:
- the map to the curvature characteristic class
These are induced from the canonical morphisms of coefficient objects
These two morphisms exhibit Deligne cohomology as a refinement in differential cohomology of ordinary (i.e. integral Eilenberg-MacLane) cohomology, in that the diagram
is the cohomology of a homotopy pullback diagram, i.e. satisfies the axioms described at differential cohomology.
Interpretation in terms of higher parallel transport
There is a natural way to understand the Deligne complex of sheaves as a sheaf which assigns to each patch the Lie -groupoid of smooth higher parallel transport n-functors. This perspective is helpful for understanding how Deligne cohomology relates to the bigger picture of differential cohomology.
We start by discussing this in low degree.
There is path groupoid whose smooth space of objects is and whose smooth space of morphisms is a space of classes of smooth paths in . Every smooth 1-form induces a smooth functor from to to the smooth groupoid with one object and as its smooth space of morphisms by sending each path to . This map from 1-forms to smooth functors turns out to be bijective: every smooth functor of this form uniquely arises this way. Similarly, one finds that smooth natural transformation between two such functors is in components precisely a smooth function such that .
Since the analogous statements are true for every open subset this defines a sheaf of Lie groupoids
By the Dold-Kan correspondence this sheaf of groupoids corresponds to a sheaf of complexes of groups. This complex of sheaves is nothing but the degree 2 Deligne complex
This way Deligne cohomology is realized as computing the stackification of the pre-stack of smooth -valued parallel transport functors.
The identification generalizes: for all there is a path n-groupoid whose -morphisms are -dimensional smooth paths in . Smooth -functors are canonically identified with smooth -forms and under the Dold-Kan correspondence the Deligne-complex in degree is identified with the sheaf of -groupoids of such smooth -functors
- John Baez, Urs Schreiber, Higher Gauge Theory (arXiv)
The full proof for this is in
- Urs Schreiber, Konrad Waldorf, Parallel transport and functors (arXiv);
- Urs Schreiber, Konrad Waldorf, Smooth functors versus differential forms (arXiv)
For more on this see infinity-Chern-Weil theory introduction.
For higher there is as yet no detailed proof in the literature, but the low dimensional proofs have obvious generalizations.
See Beilinson-Deligne cup-product.
Moduli and deformation theory
moduli spaces of line n-bundles with connection on -dimensional
The Deligne complex is naturally defined in smooth differential geometry as well as in complex analytic geometry as well as in algebraic geometry over the complex numbers. In the spirit of GAGA it is of interest to know how Deligne cohomology in these different settings relates.
One useful statement is: given an smooth algebraic variety over the complex numbers, then a sufficient condition for a complex-analytic Deligne cocycle over its analytification to lift to an algebraic Deligne cocycle is that its curvature form is an algebraic form (Esnault 89, corollary 1.3).
As described in some detail at electromagnetic field in abelian higher gauge theories the background field naturally arises as a Čech–Deligne cocycle, i.e. a Čech cocycle representative with values in the Deligne complex.
Degree 2 Deligne cohomology classifies -principal bundles with connection. The Deligne complex in this case coincides with the groupoid of Lie-algebra valued forms for the Lie algebra of .
- In physics the electromagnetic field is modeled by a degree 2 Deligne cocycle. See there for a derivation of Čech–Deligne cohomology from physical input.
Degree 3 Deligne cohomology classifies bundle gerbes with connection.
Degree 4 Deligne cohomology classifies bundle 2-gerbes with connection. In particular Chern-Simons bundle 2-gerbes whose degree 4 curvature characteristic class is a multiple of the Pontryagin 4-form on some -principal bundle.
Deligne cohomology was introduced in complex analytic geometry (by a chain complex of holomorphic differential forms) in
with applications to Hodge theory and intermediate Jacobians. The same definition appears in
Barry Mazur, William Messing, Universal extensions and one-dimensional crystalline cohomology, Springer lecture notes 370, 1974
Michael Artin, Barry Mazur, section III.1 of Formal Groups Arising from Algebraic Varieties, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131 numdam, MR56:15663
under the name “multiplicative de Rham complex” (and in the context of studying its deformation theory by Artin-Mazur formal groups). The theory was further developed in
with the application to Beilinson regulators. Later the evident version of the Deligne complex in differential geometry over smooth manifolds gained more attention and is still referred to as “Deligne cohomology”.
Surveys and introductions in the context of differential geometry include
Review with more emphasis on complex analytic geometry and the theory of (Beilinson 85) with more details spelled out is in
Hélène Esnault, Eckart Viehweg, Deligne-Beilinson cohomology in Rapoport, Schappacher, Schneider (eds.) Beilinson’s Conjectures on Special Values of L-Functions . Perspectives in Math. 4, Academic Press (1988) 43 - 91 (pdf)
Hélène Esnault, On the Loday-symbol in the Deligne-Beilinson cohomology, K-theory 3, 1-28, 1989 (pdf)
See also the references given at differential cohomology hexagon – Deligne coefficients.