Doing geometry using Feynman Path Integral? 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. 
 A: "Feynman Path Integral can be used to compute geometric invariants of a space."
There several different approaches doing this. Let me try to explain one of them, but remember it is not the only.
The point is that first you should omit the world "Feynman" !
Just integrals are useful to compute geometric invariants - for example Gauss-Bonnet theorem expresses the Euler characteristics as integral over manifold.
Word "Feynman" appears when we consider infinite-dimensional manifolds - so we need to "integrate" over infinite-dimensional spaces.
However we are NOT really interested in geometry of infinite-dimensional manifolds - we are interested in finite-dimensional manifolds. 
It appears that in some situations infinite-dimensional manifolds are either contractable to finite-dim ones or there is some heuristics which relates invariants of infinite-dimensional manifolds and finite-dim. For example if you consider loop space of M, manifold itself is embeded into loops(M) as subset of constant loops. If you consider the rotations of loops - then constant loops  are fixed-point of this action - so in this case manifold is inf-dim but fixed point set is finite-dim - so we considering equivariant calculations we can get the result on finite-dim results.
So the red-line is the following - 
in finite-dim case you integrate closed forms on manifold - and get invariant
in Feynman setup certain integrals reminds closed forms on some inf-dim spaces (loop space or whatever) so integrating it you get invariant.
(In some situations "closed form"  menas with respect to BRST differential).

The classical examples are related to Mathai-Quillen formalism and interpretation 
in terms of QFT.
Let me suggest to look a
M. Blau The Mathai-Quillen Formalism and Topological Field Theory
http://arxiv.org/abs/hep-th/9203026
And cite the abstract:
"These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections."
A: You might find Witten's lectures on the The Dirac index on manifolds and loop spaces from the IAS course on quantum field theory useful.
A: If you read French, Henniart's survey Les inégalités de Morse. Séminaire Bourbaki, 26 (1983--1984), Exposé No. 617, 19 p. might be a good place to start.  He explains Witten's analytic proof of the Morse inequalities and calls it natural and elegant. 
A: Try this article: 
Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions by 
Pavel Putrov, Juven Wang, Shing-Tung Yau
Annals of Physics 384C, 254-287 (2017)
https://arxiv.org/abs/1612.09298
It explains many interesting link invariants and their relations to braiding statistics of anyons in 3d spacetime and anyonic strings in 4d spacetime in Quantum Matter (condensed matter and lattice models) and Quantum Field Theories:

A: Witten, Supersymmetry & Morse theory is probably the most accessible reference on "physical methods" in topology & geometry.
Witten, Two Dimensional Gauge Theory Revisited -- contains a path integral construction of the intersection numbers of the moduli space of flat connections
Witten, Topological Quantum Field Theory -- contains a path integral construction of the Donaldson invariants
Witten, Topological sigma models, and Witten, Mirror Manifolds and topological field theories -- use path integrals to compute the intersection numbers of moduli spaces of holomorphic maps.
Anyone seeing a pattern yet?
A: 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 notes of David Ben-Zvi.
