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I have a question concerning the so called Local Index Formula by Connes in noncommutative geometry. First issue:
why it is called Index Formula?
I spoke to one person about this and he gave me the following explanation: one is interested in computing the pairing between K-theory and K-homology. It is known that this pairing is computed as an index of some operator. But this pairing could be computed in different way: there is a map from K-homology to cyclic cohomology (Connes Chern character) and there is also a map from K-theory to cyclic homology. Then pairing this two elements from cyclic cohomology and cyclic homology gives the same value as if we start from K-theory and K-homology. Then the local index formula gives another expression for the Connes-Chern character (the expression involves residues of some meromorphic function). But it is really true that one can derive the classical Atiyah Singer index theorem from Local Index Formula?
1. Local Index Theorem is for p-summable spectral triples: if you start with smooth (say compact) manifold $M$ is it always possible to construct spectral triple $(A,D,H)$ for $M$ in such a way that the underlying algebra $A$ is $C^{\infty}(M)$? The classical construction where $D$ is Dirac operator works only for spin manifolds. So this is first problem.
2. Second problem is that Atiyah Singer index theorem is for arbitrary elliptic operator: so the question arises whether for any such operator it is possible to construct a spectral triple?
3. What about the condition of p-summability? Would it be automatically fulfilled?
Why it is called local?
The person which I spoke to said that the term local reffers to the fact that in classical situation the index formula would involve integration and could be computed from the local data. On the other hand, Connes' explanation sounds somehow mysterious, he says that "we are working in momentum space and local means therefore asymptotic".
I would be interested in any explanation.

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  • $\begingroup$ I can't give a complete answer from my mobile device, but here's a start. You can get a spectral triple from pretty much any first order elliptic differential operator, e.g. de rham, signature, dolbeault. It comes down to analysis more than algebra. $\endgroup$ Jun 7, 2014 at 0:28
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    $\begingroup$ It is called the "local" index theorem because it gives you the index as an "integral" (really some sort of trace) of noncommutative differential forms. On a manifold, this expresses the index in terms of genuinely local data. This contrasts with the global index theorem, which asserts that two different maps from K-homology to the integers coincide. $\endgroup$ Jun 7, 2014 at 0:35

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