I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space. 


Coming from a background of studying Quantum Field Theory from the books like that of Weinberg, I have myself used Feynman Path Integrals to compute scattering of particles. 


Earlier I had done courses in Riemannian Geometry and these days I am also doing courses in Algebraic Topology and hence I think it would be very educative if I can see how exactly the calculation of topological invariants that one does here are related to Feynman's ideas. 


It would be helpful if someone can give me references which explain (hopefully starting with simple examples!) how one can use path integrals in geometry. 



Doing geometry using Feynman Path Integral?
Try:


Witten, Quantum field theory and the Jones polynomial


Witten, The index of the Dirac operator in loop space


I have found both of these papers quite difficult to understand. I don't know any easier references, and would greatly appreciate it if anybody could suggest some.


Anyway, I guess the basic idea is very simple: Take a manifold, consider some space of "fields" on the manifold (for example a space of sections of a vector bundle), do "integrals" over this space of fields. The results should be invariants of your manifold --- this is not always true, but this is the idea or the hope, anyway.


Edit: I want to also add that (T)QFT has applications not just to geometry/topology but also representation theory. For example check out these [nice](http://www.math.utexas.edu/users/benzvi/GRASP/lectures/NWTFT.html) [notes](http://ncatlab.org/nlab/show/Northwestern+TFT+Conference+2009) of David Ben-Zvi.


