Chern Classes and Numbers

Chern Classes
* Idea: They are characterisic classes, used for G = U(k).
* Notation: They belong to f* Hⁱ(Gr(n, k, \(\mathbb C\)); \(\mathbb R\)) = Hⁱ(B; \(\mathbb R\)), and they are non-zero only if i is even, so they are often written c_i(P) ∈ H²ⁱ(B; \(\mathbb R\)).

And Bundle Operations
* For the Whitney sum: c(E ⊕ F) = c(E) c(F) (in terms of forms, this means exterior product).

Examples
* For SU(2):

c₁(P) = 0 ,    c₂(P) = −\(1\over16\pi^2\)F^a ∧ F^a = \(1\over8\pi\)tr(F ∧ F) .

* For a tangent bundle TM: The c_i(TM) are invariants of the complex structure of M.
* For a principal fiber bundle P with curvature F: Given by c_i(P) = P_i(F); In particular, if n is the dimension of the group representation,

c₀(P) = 1,   c₁(P) = \({\rm i}\over2\pi\)tr(F),   c₂(P) = \(\big({{\rm i}\over2\pi}\big)^2\)\(1\over2\)(tr F ∧ tr F − tr F ∧ F),   c_n(P) = \(\big({{\rm i}\over2\pi}\big)^n\)det F .

References > s.a. non-commutative geometry.
@ In terms of curvature, etc: Briggs gq/99.
@ And physics: Yang PT(12)jan [and quantum numbers].
> Online resources: see Wikipedia page.

Chern Numbers
* Idea: Roughly speaking, the number of times a closed surface is wrapped around another closed surface.
$ Def: Integrals of Chern classes, C_j(P):= ∫_c c_j(P), where c is a j-chain.
* Chern character: The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection; Its cohomology class does not depend on the particular connection chosen.
@ References: Leonforte et al a1806 [Uhlmann number extension, and fermion systems].

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