I understand that quantum field theories are interesting as physics; however, there is also a large community of mathematicians who are interested in them. For someone who is not at all interested in physics, what are some compelling mathematical applications of this work? I've search for such things on the internet, but all I find are speculation and philosophy, neither of which interest me very much. I prefer concrete theorems about concrete mathematical objects (eg in topology, algebraic geometry, number theory, etc). The only counterexample to "not finding stuff" I have seen concerns gauge theory and its applications to geometry and topology (especially in dimension 4). Since this is so well-documented, I'd prefer to exclude it from this discussion.