TITLE: Path integrals outside QFT
QUESTION [16 upvotes]: The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical theory of functional integration is not (yet) rigorous.
My question is: what are the applications of path integrals outside QFT? By "outside QFT" I mean non-QFT physics as well as various branches of mathematics.
(a similar question is Doing geometry using Feynman Path Integral?, but it concerns only one possible application)

REPLY [2 votes]: These articles explain many interesting link invariants and their relations to braiding statistics of anyons in 3d spacetime and anyonic strings in 4d spacetime in 

Entangled Quantum Matter (condensed matter and lattice models). 

Try these articles:
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) - doi.org/10.1016/j.aop.2017.06.019
and
Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
Juven Wang, Kantaro Ohmori, Pavel Putrov, Yunqin Zheng, Zheyan Wan, Meng Guo, Hai Lin, Peng Gao, Shing-Tung Yau
Prog. Theor. Exp. Phys. 053A01 (2018) - doi.org/10.1093/ptep/pty051
The careful uses of path integral are performed in order to identify the phases of matters beyond the Ginzburg-Landau theory. These new phases include: 

symmetry protected topological [SPT] orders, 
interacting topological superconductors/insulators, 
invertible topological orders 
intrinsic topological orders.