Chern number

Chern number is a topological invariant arising in the study of complex vector bundles and connections on smooth manifolds. It assigns an integer to certain geometric data and plays a central role in linking global topology to local curvature information through index theorems and characteristic classes. The invariant has deep consequences across Differential geometry, Algebraic topology, Mathematical physics, and applied fields such as Condensed matter physics and Optics.

Definition and mathematical formulation

Let E → M be a complex vector bundle of rank r over a compact oriented 2n-dimensional manifold M equipped with a connection with curvature F. The Chern classes c_k(E) are elements of the cohomology groups H^{2k}(M; Z) constructed by Shiing-Shen Chern using curvature forms via the Chern–Weil homomorphism; the top Chern class c_n(E) is of degree 2n. The Chern number is obtained by evaluating the top-degree cohomology class on the fundamental homology class [M], yielding an integer: ∫_M c_n(E) ∈ Z. In the special case of a complex line bundle L over a compact oriented surface Σ, the first Chern class c_1(L) ∈ H^2(Σ; Z) integrates to the degree (or winding number) of L: deg(L) = ∫_Σ c_1(L). Shiing-Shen Chern's approach parallels constructions by Élie Cartan and connects with curvature forms used in the proof of the Gauss–Bonnet theorem. The Chern–Weil theory also relates Chern numbers to symmetric polynomials in the eigenvalues of the curvature 2-form, connecting to the Atiyah–Singer index theorem and work by Michael Atiyah and Isadore Singer.

Topological interpretation and properties

Chern numbers are homotopy invariants determined by the isomorphism class of the vector bundle and the oriented diffeomorphism type of the base manifold. For holomorphic vector bundles on compact complex manifolds, Chern numbers can be computed using sheaf cohomology and Riemann–Roch type formulas developed by G. H. Hardy’s contemporaries and formalized in the Hirzebruch–Riemann–Roch theorem by Friedrich Hirzebruch. Multiplicative and functorial properties include behavior under Whitney sum and tensor product, and integrality results follow from the integer lattice structure of singular cohomology classes. The Chern number generalizes classical invariants such as the Euler characteristic (viewed as a Pfaffian or top Chern class in certain contexts) and is constrained by signature theorems and congruences studied by Hirzebruch and John Milnor. For complex projective varieties, Chern numbers are deformation invariants under families preserved by projective morphisms studied by Alexander Grothendieck and Jean-Pierre Serre.

Computation and examples

Concrete computations often use explicit connections and curvature forms on standard bundles. For the tangent bundle of the complex projective space CP^n, Chern classes and hence Chern numbers can be computed from the hyperplane bundle O(1) using splitting principles and total Chern class manipulations; classic references include work by Élie Cartan and Hermann Weyl. For complex line bundles L over the 2-sphere S^2 (equivalently Riemann sphere), the integral of c_1 gives the integer degree which equals the winding number of a transition function around the equator; this calculation appears in examples by Bernhard Riemann and later expositions by Jean Leray. For higher-rank bundles over algebraic surfaces such as K3 surfaces or complex tori, Chern numbers are computable via intersection theory and Chern roots, with formulae appearing in the work of Friedrich Hirzebruch and Kunihiko Kodaira. Computational techniques employ spectral sequences introduced by Jean Leray and localization formulas due to Berline–Vergne and Atiyah–Bott in equivariant cohomology contexts.

Physical applications in condensed matter and optics

Chern numbers appear as quantized invariants classifying topological phases in Condensed matter physics, notably in the integer Quantum Hall effect where the Hall conductance equals an integer Chern number of a Bloch bundle over the Brillouin zone, as elucidated by Thouless, Kohmoto, Nightingale, and den Nijs and later developments by Brian David Josephson’s contemporaries. In topological insulators and superconductors classified by Kitaev and Schnyder et al., Chern numbers and related invariants predict robust edge states tied to bulk–boundary correspondence studied by Kenneth Wilson and Chetan Nayak. In photonics and cold-atom systems, engineered bandstructures realize nonzero Chern numbers leading to unidirectional edge modes; these implementations have been demonstrated in experiments influenced by proposals from John Pendry’s collaborators and groups led by Nader Engheta and Mikael Rechtsman. The Chern number also appears in semiclassical dynamics through the Berry phase and Berry curvature formalism introduced by Sir Michael Berry and in transport coefficients in solid-state physics research by Neil Ashcroft and N. David Mermin.

Generalizations and higher Chern classes

Chern numbers are instances of characteristic numbers obtained by evaluating products of Chern classes and other characteristic classes (such as Pontryagin classs and the Todd class) on fundamental cycles. Higher Chern classes c_k(E) for 1 ≤ k ≤ n yield a ring of characteristic classes with relations controlled by the splitting principle and formal Chern roots; these structures play roles in K-theory developed by Atiyah and Friedrich Hirzebruch. Secondary invariants and anomalies in quantum field theory connect to Chern–Simons forms linked to transgression of Chern classes studied by Shiing-Shen Chern and James Simons. In algebraic geometry, generalizations include Chern–Mather classes and Chern–Schwartz–MacPherson classes associated to singular varieties, with foundations laid by Robert MacPherson and subsequent work by Jean-Pierre Serre and Alexander Grothendieck.

Category:Differential topology