Differential Cohomology

• Differential cohomology is a framework that unifies classical cohomology with differential form data to bridge topology and geometry.
• It employs models like Cheeger–Simons, Deligne, and Hopkins–Singer to establish exact sequences linking integral classes, closed forms, and refined invariants.
• Applications span gauge theory, index theory, and TQFTs, providing tools for anomaly cancellation and the study of smooth geometric structures.

Differential cohomology provides a framework that refines classical (integral) cohomology by incorporating both topological data and geometric, differential form information. For a smooth manifold $M$, the differential cohomology group in degree $k$, frequently denoted $\widehat{H}^k(M)$, or in various contexts as $\mathcal{C}\widehat{H}^k(M)$, fits into exact sequences that encode both the integral cohomology group $H^k(M;\mathbb{Z})$ and the space of closed differential $k$-forms $\Omega^k_{\mathrm{cl}}(M)$, together with “secondary” or “refined” invariants such as flat classes or holonomy data. This refinement is essential for understanding the interaction of topology and geometry in fields such as gauge theory, characteristic class theory, index theory, topological quantum field theory, and string theory, where global quantization, connections, and curvature play equally fundamental roles.

1. Structural Foundations: Definitions and Models

Differential cohomology groups may be defined through several, ultimately equivalent, models. The Cheeger–Simons model introduces degree-$k$ differential characters as group homomorphisms

$$\chi : Z_{k-1}(M) \rightarrow \mathbb{R}/\mathbb{Z}$$

from smooth $(k-1)$-cycles, characterized by the property that there exists a unique closed $k$-form $\omega$ such that for any $k$-chain $c$

$\chi(\partial c) = \int_c \omega \pmod{\mathbb{Z}}$

Here, the “characteristic class” map $cc$ sends $\chi$ to its underlying class in $H^k(M; \mathbb{Z})$, while the “curvature” map $curv$ assigns the differential form $\omega$ to $\chi$ (Debray, 2023). Alternative, but equivalent, definitions involve:

• Deligne cohomology: Hypercohomology $H^n(M; \mathcal{D}(n))$ of the Deligne complex

$$0 \rightarrow \mathbb{Z} \rightarrow \Omega^0 \rightarrow \Omega^1 \rightarrow \cdots \rightarrow \Omega^{n-1} \rightarrow 0$$

• Hopkins–Singer model: Triples $(c,h,\omega)$, where $c$ is an integral cocycle, $h$ is a cochain with real coefficients, and $\omega$ is a closed differential form, related by $\delta h = \omega - c$, so the group consists of such triples modulo suitable coboundaries.

These models are related through canonical isomorphisms and each supports a universal hexagon diagram expressing compatibility between the integral, differential, and secondary (flat/holonomy) data (Debray, 2023, Ho, 2013).

2. Axiomatic and Homotopical Formalism

Modern treatments express differential cohomology as sheaves of spectra (or chain complexes) on the site of smooth manifolds (Bunke et al., 2013, Amabel et al., 2021), refining functors $E^*(-)$ representing generalized cohomology theories. Differential cohomology is then a homotopy pullback of the diagram: $\begin{tikzcd} & \mathcal{A}(E) \ \widehat{E}(M) \ar[r,"I"] \ar[u,"curv"'] & E^*(M) \end{tikzcd}$ where $\mathcal{A}(E)$ is the “cycle part” (e.g., a truncated de Rham complex), and the horizontal and vertical exact sequences encode the relationships between differential, topological, and form data (Bunke et al., 2013, Amabel et al., 2021). The canonical decomposition of any differential extension produces the classical differential cohomology diagram and grounds the subsequent refinement of characteristic classes, operations, and module structures.

The bicategory approach (Upmeier, 2012) recognizes differential cohomology refinements as symmetric monoidal groupoids and establishes uniqueness up to monoidal equivalence, supporting multiplicative (cup-product) structures and functoriality under maps of manifolds.

3. Exact Sequences, Hexagons, and Mayer–Vietoris

Differential cohomology groups fit into exact sequences relating them to integral cohomology, flat classes, and closed differential forms. A central diagram is the differential cohomology hexagon

$\begin{tikzcd}[column sep=small] & \Omega^{k-1}(M)/\Omega^{k-1}_\mathbb{Z}(M) \ar[dr] & \ H^{k-1}(M;\mathbb{R}/\mathbb{Z}) \ar[ur] \ar[dr] & & \Omega^k_\mathbb{Z}(M) \ & \widehat{H}^k(M) \ar[ur] \ar[ul] & \end{tikzcd}$

whose upper and lower rows are exact and that express the precise relationships among all the constituent structures (Ho, 2013, Debray, 2023).

The Mayer–Vietoris property, central for computational and locality considerations, is satisfied for all differential cohomology functors constructed from $J(-; \mathbb{Z})$ with finitely generated coefficients (and, in particular, for differential K-theory) on compact manifolds. Classes agreeing on overlaps of an open cover can be glued to global classes (Simons et al., 2010, Simons et al., 2010). Diagram chases in the relevant commutative diagrams involving integral, real, and $R/\mathbb{Z}$-valued theories realize this assertion.

4. Multiplicative and Higher Structures

Differential cohomology admits and extends multiplicative (cup or internal product) structures. In the Cheeger–Simons case, cup products and module structures are constructed via explicit lifts, barycentric subdivision, and chain homotopies, preserving naturality and compatibility with curvature and characteristic class maps (Becker, 2013). For generalized theories, the bicategory formalism encodes products as monoidal structures at the level of groupoids (Upmeier, 2012).

Higher operations, such as Massey products and Steenrod operations, admit differential refinements using stack-theoretic and homotopical machinery. Refined Massey products are constructed via simplicial or Dold–Kan models for stacks of bundles with connection, yielding secondary and tertiary invariants relevant for anomaly cancellation and higher trivializations (Grady et al., 2015). Explicit characterization results show that only certain primary operations (e.g., odd Steenrod squares) admit differential refinements, and those are computed as composites involving Bockstein homomorphisms, reductions, and inclusion of flat data (Grady et al., 2016).

5. Relative and Equivariant Forms

Relative differential cohomology systematically tracks relative geometric/topological data, leading to precise long exact sequences generalizing those of ordinary relative cohomology and Deligne cohomology, with several inequivalent definitions (absolute vanishing, trivialization, etc.) (Ruffino, 2014, Becker, 2013). These sequences are crucial for describing D-brane anomaly cancellation, boundary field theories, and fiber integration/transgression in field-theoretic contexts.

Equivariant differential cohomology integrates group symmetries (including gauge actions) via the Cartan or Borel models and Getzler resolutions, refining both topological (integral Borel) and equivariant differential form data (Kübel et al., 2015, Davighi et al., 2020). The equivariant theory fits into a refined hexagon diagram paralleling the nonequivariant case but encoding moment map data and nontrivial polynomial Lie algebra contributions. The lack of surjectivity in maps from equivariant to invariant differential cohomology quantifies the presence of ’t Hooft anomalies, providing a topological classification of gauge and global symmetry obstruction phenomena (Davighi et al., 2020).

6. Differential Characteristic Classes and Index Theory

Differential characteristic classes arise from refining the classical Chern–Weil construction for $G$-bundles with connection to populate differential cohomology classes whose curvature is a characteristic form and whose characteristic class matches the topological invariant (Debray, 2023, Amabel et al., 2021). This construction extends to the equivariant setting by incorporating both curvature and moment map terms in the Cartan complex (Kübel et al., 2015). Uniqueness of such differential refinements is established, and their behavior under smooth deformation of connections is governed by explicit transgression formulas.

Applications to index theory, invertible field theories, and the construction of centrally extended, anomaly-sensitive invariants (such as the differential Pontryagin and Chern–Simons classes) are immediate. Multiplicativity and the module structure over absolute differential cohomology are crucial for formulating fiber integration and push-forward operations (Becker, 2013).

7. Applications, Duality, and Geometric Structures

Differential cohomology is fundamental for quantization conditions in gauge/field theories, including the discrete quantization of electromagnetic or RR fields, topological terms (Chern–Simons, Wess–Zumino–Novikov–Witten), and the detection of global and perturbative anomalies (Davighi et al., 2020, Debray, 2023). Pontryagin duality is established between ordinary and compactly supported differential cohomology, generalizing classical dualities for abelian groups and extending Harvey–Lawson–Zweck’s results (Becker et al., 2015). Excision, functoriality, fiber integration, and smooth Fréchet–Lie group structures are rigorously constructed (Becker et al., 2015, Becker et al., 2014).

Recently, differential cohomology has been realized in the framework of skeletal diffeology and diffeological homotopy theory, offering a geometric model where the underlying “space” is refined (via skeletal diffeologies) rather than the coefficient object. This approach recovers Cheeger–Simons differential characters in terms of homotopy-theoretic invariants of simplicial presheaves and yields models suggestive of future generalizations to nonabelian differential cohomology (Scalbi, 5 Aug 2024).

8. Extensions, Generalizations, and Outlook

Differential cohomology unifies discrete topological invariants (integral cohomology, characteristic classes) and smooth geometric data (closed forms, connections, holonomy) within a single homotopy-theoretic and sheaf-theoretic structure (Amabel et al., 2021, Debray, 2023). It extends naturally to “generalized” cohomology theories, such as K-theory and cobordism, via stacky, bicategory, or spectral sheaf models (Bunke et al., 2013, Upmeier, 2012). These theories support all expected operations—pullback, cup-product, module structures, relative and fiber integration, duality, and smooth structures—making them central in the modern interface of geometry, topology, physics, and higher category theory.

Further directions include the paper of higher nonabelian analogues, refined infinity-topos approaches, stacky realization of smooth invariants, applications to TQFTs, index theory, and quantization of gauge and boundary field theories (Schreiber, 2013, Scalbi, 5 Aug 2024, Davighi et al., 2020).