Applications of TQFTs beyond physics
Solution 1:
Tqft has connections to moduli space:http://prd.aps.org/abstract/PRD/v42/i6/p2080_1 and Morse theory.
Also, this post:https://mathoverflow.net/questions/60550/usefulness-of-using-tqfts suggests that tqft invariants, together with the fundamental group, completely characterize three-manifolds, as well as solving many problems in enumerative geometry.