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
ci(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) = −\(1\over16\pi^2\)Fa ∧ Fa = \(1\over8\pi\)tr(F ∧ F) .
* For a tangent bundle TM:
The ci(TM)
are invariants of the complex structure of M.
* For a principal fiber bundle P with
curvature F: Given by ci(P)
= Pi(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), cn(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, Cj(P):=
∫c cj(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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