Chern–Weil theory

Chern–Weil theory
Name	Chern–Weil theory
Field	Differential geometry
Introduced	1940s–1950s
Founders	Shiing-Shen Chern; André Weil
Related	de Rham cohomology; characteristic classes; principal bundles

Chern–Weil theory is a fundamental method in differential geometry and topology that produces characteristic class invariants of principal bundles and vector bundles from connections and curvature via invariant polynomials. The theory, developed by Shiing-Shen Chern and André Weil with antecedents in work by Hermann Weyl and Elie Cartan, bridges de Rham cohomology with algebraic topology constructions such as Chern classes, Pontryagin classes, and the Euler class, interfacing with later developments by Michael Atiyah, Raoul Bott, and Isadore Singer.

Introduction

Chern–Weil theory assigns cohomology classes in de Rham cohomology of a smooth base manifold to principal bundles with structure group a compact Lie group such as U(n), SO(n), SU(n), or Sp(n), using connections inspired by work of Elie Cartan and Élie Joseph Cartan while relying on algebraic foundations from invariant theory and Lie group representation theory developed by Hermann Weyl and Élie Cartan. The construction yields topological invariants equivalent to the Chern classes defined by Steenrod and Hurewicz-era homotopy-theoretic methods, and it plays a central role in index theorems of Atiyah–Singer via contributions by Michael Atiyah, Isadore Singer, and Raoul Bott.

Differential Forms and Connections

Chern–Weil uses differential-geometric objects: a connection 1-form on a principal G-bundle (with G a compact Lie group like SO(3), SU(2), U(1), Sp(1)) and its curvature 2-form, concepts shaped by Elie Cartan and refined in contexts such as Riemannian geometry by Bernhard Riemann and Georg Friedrich Bernhard Riemann. The curvature, a Lie algebra-valued 2-form related to the Maurer–Cartan form and the structure constants of the Lie algebra of G, is paired with invariant polynomials coming from representation theory of groups studied by Weyl and Cartan. In practical computations one uses tools from differential forms and exterior algebra developed by Élie Cartan and topological input from de Rham and Hodge theory as advanced by Georges de Rham and W.V.D. Hodge.

Invariant Polynomials and Characteristic Classes

Invariant polynomials on the Lie algebra of G, originating in invariant theory and the classification of symmetric functions by Newton and later algebraists such as David Hilbert, produce closed differential forms when evaluated on curvature. These closed forms represent cohomology classes identified with classical characteristic classes like Chern classes (for U(n) and GL(n, C)), Pontryagin classes (for SO(n)), and the Euler class (for oriented real vector bundles), concepts linked historically to work by Shiing-Shen Chern, Jean-Pierre Serre, and W. R. Thompson. The relation between these differential representatives and topological characteristic classes was clarified by comparisons with cohomology rings of classifying spaces such as BG and computations by Bott and Samelson.

Chern–Weil Homomorphism

The Chern–Weil homomorphism maps the ring of Ad-invariant polynomials on the Lie algebra of G to the de Rham cohomology ring of the base manifold, respecting cup product structures analogous to the cohomology of classifying spaces studied by Henri Cartan and Armand Borel. This homomorphism is natural under pullback of bundles, compatible with operations like direct sum and tensor product encountered in the work of Alexander Grothendieck on K-theory and consistent with index-theoretic formulas in the Atiyah–Singer index theorem developed by Michael Atiyah and Isadore Singer. The independence of the resulting cohomology class from the choice of connection follows from homotopy arguments and transgression techniques used by Bott and Tu.

Examples and Computations

Classic examples include computing first and second Chern classes for complex vector bundles over manifolds such as CP^n (complex projective space) using Fubini–Study metric connections, Pontryagin classes for tangent bundles of S^n spheres and Lie groups like SO(n) and SU(n), and the Euler class of oriented bundles exemplified by the tangent bundle of S^2 related to the Gauss–Bonnet theorem originally proved by Carl Friedrich Gauss and extended by Shiing-Shen Chern. Explicit curvature computations often use connections from Levi-Civita connection in Riemannian geometry, the Chern connection on Hermitian bundles, and gauge-theoretic connections in Yang–Mills theory developed by Yang Chen-Ning and Robert Mills with later analyses by Donaldson and Taubes. Calculations on homogeneous spaces exploit structure theory by Élie Cartan, representation data from Weyl character formula by Hermann Weyl, and cohomology computations by Borel–Weil techniques.

Extensions and Generalizations

Extensions include Chern–Simons theory introduced by Shiing-Shen Chern and James Simons, equivariant Chern–Weil theory linked to actions of compact groups studied by Berline–Vergne and applications to localization formulas of Atiyah–Bott, and differential-character refinements in the sense of Cheeger–Simons connecting to Deligne cohomology and Arakelov theory by Gillet and Soulé. Noncommutative generalizations appear in noncommutative geometry pioneered by Alain Connes, and higher-categorical refinements relate to gerbes and bundle gerbe theory investigated by Murray and Stevenson as well as to elliptic cohomology and work linking TMF by Hopkins and Strickland. Applications permeate areas including the Atiyah–Singer index theorem, topological quantum field theory developed by Edward Witten, and modern studies in mirror symmetry and string theory connecting to contributions by Maxim Kontsevich, Cumrun Vafa, and Juan Maldacena.

Category:Differential geometry