TITLE: Reference request: Partition function as a topological invariant of a QFT
QUESTION [8 upvotes]: 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.

REPLY [2 votes]: 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).