Alain Connes (pdf)

This fact is not difficult to prove, the point is that a deformation of idempotents is
always isospectral,
Ė = [X, E] for some X ∈ M n (A) . (38)
When we take A = C ∞ (M) for a manifold M and let
ϕ(f 0 , f 1 , f 2 ) = 〈C, f 0 df 1 ∧ df 2 〉 ∀f j ∈ A (39)
where C is a 2-dimensional closed de Rham current, the invariant given by the lemma
is equal to (up to normalisation)
〈C, c 1 (E)〉 (40)
where c 1 is the first chern class of the vector bundle E on M whose fiber at x ∈ M
is the range of E(x) ∈ M n (C). In this example we see that for any permutation of
{0, 1, 2} one has:
ϕ(f σ(0) , f σ(1) , f σ(2) ) = ε(σ)ϕ(f 0 , f 1 , f 2 ) (41)
where ε(σ) is the signature of the permutation. However when we extend ϕ to M n (A)
as ϕ n = ϕ ⊗ Tr,
ϕ n (f 0 ⊗ µ 0 , f 1 ⊗ µ 1 , f 2 ⊗ µ 2 ) = ϕ(f 0 , f 1 , f 2 )Tr(µ 0 µ 1 µ 2 ) (42)
the property 41 only survives for cyclic permutations. This is at the origin of the name,
cyclic cohomology, given to the corresponding cohomology theory.
In the example of the noncommutative torus, the cyclic cocycle that was giving an
integral invariant is
where τ is the unique trace,
ϕ(b 0 , b 1 , b 2 ) = τ(b 0 (δ 1 (b 1 )δ 2 (b 2 ) − δ 2 (b 1 )δ 1 (b 2 )) (43)
τ(b) = b 00 for b = ∑ b nm U n W m . (44)
The pairing given by the lemma then gives the Hall conductivity when applied to
a spectral projection of the Hamiltonian (see [12] for an account of the work of J.
Bellissard).
We then obtain in general the beginning of a dictionary relating usual geometrical
notions to their algebraic counterpart in such a way that the latter is meaningfull in
the general noncommutative situation.
Space
Vector bundle
Differential form
DeRham current
DeRham homology
Chern Weil theory
Algebra
Finite projective module
(Class of) Hochschild cycle
(Class of) Hochschild cocycle
Cyclic cohomology
Pairing 〈K(A), HC(A)〉
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