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I have read (mainly in the articles of Atiyah) that the partition function is the simplest topological invariant of a quantum field theory.

I have an arithmetic geometry background and know statistical physics quite well, but am a beginner in QFT. A reference for details on the above statement in mathematical language à la "Quantum Fields and Strings: A Course for Mathematicians" would be highly appreciated.

A sketch of the ideas involved would be even nicer.

If this question is too simple or vague, please let me know - I will transfer it to SE.

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    $\begingroup$ But if you have read Atiyah's papers, then you should know that he himself defined what's a called now a functorial field theory in math.ru.nl/~mueger/TQFT/At.pdf . In any case, I like the references ma.utexas.edu/users/dafr/OldTQFTLectures.pdf and arxiv.org/pdf/q-alg/9503002.pdf . A similar definition was also given by Segal for the conformal flavor math.upenn.edu/~blockj/scfts/segal.pdf. $\endgroup$
    – user40276
    Commented Jan 16, 2018 at 22:28
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    $\begingroup$ It is unclear to me what the question is about. Is it about the definition of the partition function of a QFT? Is it about topological invariants of manifolds which can be realized as partition functions of TQFT? (More precisely, I am not sure what "topological invariant of a QFT" means, is it something invariant under deformation of the QFT, or something else?) $\endgroup$
    – user25309
    Commented Jan 16, 2018 at 22:46
  • $\begingroup$ I agree with the comment of user25309. Perhaps you could give a citation to a particular claim in the literature that you are curious about. $\endgroup$
    – j.c.
    Commented Jan 16, 2018 at 22:50
  • $\begingroup$ @user40276 Thank you very much for references. This is exactly what I have been looking for. $\endgroup$
    – Dimitri
    Commented Jan 17, 2018 at 1:04

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This is one case where you might just want to go back to the original article that proved that a partition function is metric-independent and hence a topological invariant: Edward Witten, Quantum field theory and the Jones polynomial (1989) --- see page 361.

For a sketch of the idea, this introduction by Marcos Marino seems useful (page 25+26).

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  • $\begingroup$ Having browsed through that article myself, I believe it is not what the OP has asked for: he is looking for a rigorous text, "in mathematical language à la 'Quantum Fields and Strings: A Course for Mathematicians'" - which Witten's article is very far from (really, why was he awarded the Fields Medal for that... thing?). $\endgroup$
    – Alex M.
    Commented Jan 16, 2018 at 18:48
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    $\begingroup$ this is off-topic here, but if you want to know why Witten was awarded the Fields Medal, perhaps Atiyah's appraisal is helpful. $\endgroup$
    – Carlo Beenakker
    Commented Jan 16, 2018 at 18:58
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    $\begingroup$ Thank you very much for the reference. It gives an nice illustration of the idea. I will be a good exercise for me to translate it into the "mathematical language". Together with the references from user40276 this is exactly what I have been looking for. $\endgroup$
    – Dimitri
    Commented Jan 17, 2018 at 1:07

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