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I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (APS) and the index theorems for families (Including Bismut's approach). I am less interested in the different proofs but rather on the setup for each result.

Thanks

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    $\begingroup$ The books by Berline–Getzler–Vergne, Booss–Bleecker, and Shanahan all offer introductions to the index theory from different viewpoints. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 28, 2020 at 15:29

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I am currently reading Dan Freed's "The Atiyah-Singer Index Theorem", which also has a corresponding Youtube video, and I am finding it very helpful. Here:

Paper: https://arxiv.org/pdf/2107.03557.pdf

Video: https://www.youtube.com/watch?v=AJHKp9kYm90

The video, of course, lacks the level of detail of the paper, but I found it quite enlightening. In the paper, I find that even the proof sketches are very nicely summarised. You might need (as me) to complement the read with some additional searches for definitions.

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Atiyah's expository paper, "Algebraic Topology and Elliptic Operators", Communications in Pure and Applied Mathematics 20 (1967) 237-249 is quite accessible.

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