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EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)). This well known article applies some tools developed by physicists (e.g. path integrals) to topology of manifolds.

I am a mathematician, but I am familiar with some necessary physical ideas, although probably on more elementary level than necessary to understand the paper. To be more specific, I cannot understand the computation of matrix elements of the (twisted) de Rham differential $d_t$ on p. 672-673. The exposition there is too concise for me.

Is there a more detailed exposition of Witten’s paper today? However mathematical rigor is not so important for me. I would like to understand the physicists tools and ideas.

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You can find much more on the specific family $d_t$ if you search for the key phrase "Witten deformation"; I would try to give some specific references here but I am a little puzzled by the statement that you are "NOT looking for a mathematically more rigorous exposition". Could you clarify; does this rule out e.g. anything with definitions, theorems and proofs? I guess the Helffer-Sjöstrand theory is not what you're after?

Perhaps Section 5 of this paper of Rogers, which uses (stochastic calculus) path integral techniques to compute those matrix elements could be helpful?

More generally this part of the computation has been generalized / abstracted as an instance of localization in equivariant cohomology. About this, there has been much written in both theoretical physics and mathematics; one physics source that I found inspiring (although also with few details) is Part II of this review of Cordes, Moore and Ramgoolam. Chapter 12 is all about SUSY QM, though let me warn that it's likely not readable on its own. It essentially specializes the "symmetries, fields, equations" viewpoint developed in chapters 9, 10, 11 to this example. Appendix 4 of Chapter 12 gives a brief resummation of Witten's paper on Morse theory that you cite.

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    $\begingroup$ Thanks for the answer, some references seem to be helpful. I mentioned that I am not looking for a mathematically more rigorous exposition not to rule out such papers, but to to say that non-rigorous exposition is ok with me. On the mathematical site this should be mentioned I think. Sorry if I was not clear. $\endgroup$ – MKO Apr 15 '18 at 16:08

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