TITLE: Doing geometry using Feynman Path Integral?
QUESTION [37 upvotes]: 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.

REPLY [3 votes]: 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: