Chern–Weil Theory: Fundamentals & Extensions

• Chern–Weil theory is a framework in differential geometry that produces characteristic classes from principal bundles using connections and invariant polynomials.
• The theory extends to Lie groupoids, stacks, noncommutative settings, and higher categorical structures, broadening its mathematical and physical applications.
• Its functorial nature, independence from specific connections, and explicit computational tools link it to topology, index theory, and representation theory.

Chern–Weil theory is a central construction in differential geometry and mathematical physics, associating canonical de Rham cohomology classes ("characteristic classes") to principal bundles with connection via invariant polynomials on the Lie algebra of the structure group. It subsumes classical results about Chern, Pontryagin, and Euler classes, and has far-reaching generalizations encompassing Lie groupoids, stacks, noncommutative geometry, infinite-dimensional Lie groups, and higher or derived contexts.

1. Foundations: Principal Bundles, Invariant Polynomials, and the Chern–Weil Homomorphism

Let $G$ be a (typically compact) Lie group with Lie algebra $\mathfrak{g}$, and $\pi : P \to M$ a smooth principal $G$-bundle over a manifold $M$. A connection is expressed globally as a $\mathfrak{g}$-valued 1-form $\omega \in \Omega^1(P; \mathfrak{g})$ satisfying the standard equivariance and normalization identities. Its curvature is

$$\Omega = d\omega + \tfrac{1}{2}[\omega,\omega] \in \Omega^2(P; \mathfrak{g}),$$

which descends (via horizontality and $G$-equivariance) to a well-defined 2-form $F \in \Omega^2(M; \mathrm{Ad} P)$ on the base manifold.

Given an $\mathrm{Ad}$-invariant symmetric polynomial $P \in \mathrm{Sym}^k(\mathfrak{g}^*)^G$, the Chern–Weil form is defined by

$$P(\Omega) = P(\underbrace{F, \dots, F}_{k\;\mathrm{copies}}) \in \Omega^{2k}(M),$$

which is closed by virtue of the Bianchi identity and the invariance of $P$, and its de Rham class is independent of the choice of connection. This yields the Chern–Weil homomorphism:

$$\mathrm{CW}: \mathrm{Sym}^k(\mathfrak{g}^*)^{G} \longrightarrow H^{2k}_{\mathrm{dR}}(M),\qquad P \mapsto [P(\Omega)].$$

The construction extends to holomorphic and Hermitian vector bundles by using the canonical ("Chern") connection, and to classes in Dolbeault cohomology in the complex-analytic setting (Pingali et al., 2011).

2. Functoriality, Independence, and Uniqueness

The Chern–Weil assignment is functorial: for $f : N \to M$, $f^*P(\Omega_M) = P(\Omega_N)$ under the natural pull-back of bundles and connections. The cohomology class $[P(\Omega)]$ is independent of the connection $\omega$: for two connections $\omega_0, \omega_1$ one constructs a (secondary) transgression form (Chern–Simons form) $T$ such that

$P(\Omega_1) - P(\Omega_0) = dT,$

so the de Rham class is unchanged under deformations of the connection (Hong, 2015, Pingali et al., 2011).

Uniqueness holds in a strong sense: any natural rule assigning closed differential forms to principal $G$-bundles with connection must arise via the Chern–Weil construction using invariant polynomials (Freed et al., 2013). This is formalized using abstract functoriality and homotopy-theoretic structures (e.g., simplicial sheaves), where the only closed, natural forms on the universal $G$-bundle arise from invariant polynomials via the Weil algebra model (Freed et al., 2013).

3. Generalizations: Lie Groupoids, Stacks, and Singular Geometric Contexts

Chern–Weil theory extends to geometric structures beyond manifolds:

• Lie groupoids and differentiable stacks: For a Lie groupoid $[X_1 \rightrightarrows X_0]$ with integrable connection $\mathcal{H} \subset TX_1$, one defines a "horizontal" de Rham complex $\Omega^\bullet(X,\mathcal{H})$, and Chern–Weil classes become characteristic elements in $H^\bullet_{\mathrm{dR}}(X,\mathcal{H})$ (Biswas et al., 2020). For étale groupoids presenting Deligne–Mumford stacks, this recovers stack-cohomological characteristic classes.
• Coherent analytic sheaves: Through admissible (barycentric) simplicial connections on Green resolutions on the Čech nerve, one constructs explicit representatives for characteristic classes ("simplicial Chern–Weil forms"). This procedure uniquely extends Chern class theory to coherent analytic sheaves, matching the Grothendieck–Hirzebruch axioms (Hosgood, 2020, Hosgood, 2020).
• Bundles with singular metrics: For Hermitian vector bundles with singular metrics (e.g., $\mathcal{I}$-good or full mass metrics), Chern–Weil type formulae are established in terms of b-divisors and intersection theory on the Riemann–Zariski space, unifying classical and singular Arakelov-theoretic contexts (Xia, 2022, Jespers et al., 2017).

4. Homotopy-theoretic, Noncommutative, and Higher Settings

Recent advances place Chern–Weil theory in derived and categorical frameworks:

• Homotopy-theoretic characterizations: The Weil algebra is interpreted as the de Rham complex of universal $G$-connection simplicial sheaves, and the Chern–Weil homomorphism is realized as pullback from the $G$-basic subalgebra. Unique naturality arises from the structure of the universal bundle in the derived category framework (Freed et al., 2013).
• Noncommutative geometry: There is a cyclic-homology analogue of Chern–Weil, where invariant polynomials are replaced by coalgebra cotraces and the de Rham/homology side is replaced by Hochschild/cyclic homology associated to H-unital "row extensions" (e.g., Ehresmann–Schauenburg quantum groupoids). This yields "cyclic-homology Chern–Weil" classes generalizing the Chern–Galois character and mirroring the classical situation (Hajac et al., 2017).
• Higher and derived structures: Chern–Weil theory is categorified by interpreting it as a DG- or $A_\infty$-functor between suitable categories of ∞-local systems. The "classical" Chern–Weil homomorphism is then recovered as the endomorphism algebra of the trivial object, with independence under connection yielding $A_\infty$-natural isomorphisms (Abad et al., 2021). Theory further generalizes to dg-manifolds and higher stacks, realizing the AKSZ sigma-model action functionals as higher Chern–Weil cocycles (Fiorenza et al., 2011).

5. Applications: Topological Group Cohomology, Index Theory, and Infinite-dimensional Contexts

The Chern–Weil construction underpins numerous applications:

• Topology of Lie groups and symmetric spaces: For a real semisimple Lie group $G$ with maximal compact $K$ and compact dual $G_u$, the Chern–Weil map for the canonical invariant connection on $G_u/K$ gives a ring isomorphism between symmetric invariants and de Rham cohomology: $$S^\ast(\mathfrak{g}_u^*)^{G_u} \xrightarrow{\sim} H^\ast(G_u/K;\mathbb{R})$$ (Wockel, 2014). These classes participate in the long exact sequence for topological group cohomology, and the Chern–Weil images can be identified with image of characteristic morphisms.
• Index theory and representation theory: Transversally elliptic Dirac operators lifted to principal $G$-manifolds yield distributional indices whose testings against suitably chosen functions recovers Chern–Weil integrals. This provides a direct geometrization of the Duflo isomorphism (passing from invariant polynomials to the center of the universal enveloping algebra) via index-theoretic and heat kernel techniques (Hong, 2015).
• Infinite-dimensional settings: For gauge bundles, mapping spaces, and pseudodifferential operator groups, the leading-order and Wodzicki-residue traces provide invariant functionals allowing extension of Chern–Weil theory to infinite-rank situations. Characteristic classes detect nontrivial cohomology in mapping spaces and, in special cases, for diffeomorphism groups (Rosenberg, 2013).

6. Secondary and Relative Chern–Weil Theory: Transgression, Bott–Chern, and Non-uniqueness

In situations where the primary characteristic form vanishes, or pairs of connections/metrics are considered, secondary classes such as Chern–Simons and Bott–Chern forms are defined. These capture differences of characteristic classes and provide explicit representatives for exact forms:

• Chern–Simons forms provide transgression connecting characteristic forms for different connections. Every exact odd form on a smooth manifold arises as a Chern–Simons form for a suitable trivial bundle with two connections (Pingali et al., 2011).
• Bott–Chern forms in the holomorphic/Hermitian context measure differences between Chern character forms for different Hermitian metrics, yielding secondary invariants that exhaust the space of $\partial\bar\partial$-exact forms modulo the sum $$\operatorname{Im}\,\partial + \operatorname{Im}\,\bar\partial$$ (Pingali et al., 2011).
• Non-uniqueness in secondary theory: While primary Chern–Weil classes are uniquely determined by naturality, secondary classes depend on the additional data of metric or connection interpolations, leading to a richer but less rigid structure (Pingali et al., 2011).

7. Explicit Computations and Practical Formulae

Chern–Weil theory provides local and global computational tools:

• Local trivializations: For a principal $G$-bundle over $U\subset M$, in a local section with gauge $g$, the connection 1-form is written as $\omega = \sigma^*A + g^{-1} dg$, curvature as $K = dA + \tfrac{1}{2}[A,A]$.
• Chern forms: The $i$-th Chern form can be locally written as

$$c_i = \frac{1}{(2\pi i)^i i!} \operatorname{Tr}(K^i) \in \Omega^{2i}(U),$$

which globalizes to a de Rham cohomology class (Biswas et al., 2020).

• Explicit Čech representatives: In the context of vector bundles over Stein covers, the exponential Atiyah classes evaluated on transition cocycles generate Čech cocycles for the Chern classes (Hosgood, 2020).

These formulae serve both theoretical and computational applications, facilitating explicit study of characteristic classes across a broad spectrum of geometric environments—manifolds, orbifolds, stacks, and beyond.

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The Chern–Weil paradigm thus articulates a unifying algebraic–differential mechanism for characteristic classes, extending across categorical, analytic, geometric, and topological generalizations, and deepening links with representation theory, index theory, and noncommutative geometry (Pingali et al., 2011, Biswas et al., 2020, Hajac et al., 2017, Freed et al., 2013, Abad et al., 2021, Rosenberg, 2013, Xia, 2022, Hosgood, 2020, Hosgood, 2020, Jespers et al., 2017, Wockel, 2014, Fiorenza et al., 2011, Hong, 2015).