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 welldocumented, I'd prefer to exclude it from this discussion.

8$\begingroup$ Would the downvoter like to explain why he/she does not like this question? $\endgroup$ – Sarah Nov 25 '16 at 19:49

8$\begingroup$ It's curious that you consider an algebraic variety more concrete than a quantum field. One is a complete abstraction, the other might arguably be considered the very fabric of reality. As a mathematician I understand your point of view, but to a physicist this must look completely backwards. (To avoid misunderstanding: I am not the downvoter.) $\endgroup$ – Federico Poloni Nov 25 '16 at 20:36

24$\begingroup$ @FedericoPoloni: I'm just not interested in the "very fabric of reality" (whatever that is). I'm a mathematician, not a physicist. $\endgroup$ – Sarah Nov 25 '16 at 20:37

10$\begingroup$ @FedericoPoloni I would say that this is one of the current mathematical models of the very fabric of reality :) $\endgroup$ – Alex Degtyarev Nov 25 '16 at 20:39

9$\begingroup$ Some conjectures in rational conformal field theory (and VOAs) imply (conjecturally) the existence of braided equivalences between representation categories of certain twisted doubles of groups, and a lot of work has been done to exhibit and classify such equivalences. This work itself involves almost no CFT/QFT in any direct fashion, though; it simply provides a motivation for bothering with the idea. I think this may be characteristic of a lot of things: the "physics" of CFT/QFT suggests intriguing research directions which can be investigated independently. $\endgroup$ – zibadawa timmy Nov 25 '16 at 21:33
Thomae's formula is a theorem about the properties of Riemann theta functions corresponding to hyperelliptic surfaces. In a paper, Fermionic fields on ${\mathbb Z}_N$ curves by Bershadsky and Radul, this formula is rederived and generalised from hyperelliptic surfaces to $N$fold covers of the sphere. Their argument works by computing the "partition function" for a quantum field theory describing fermions on the surface. The generalised result can also be derived without reference to QFT (that was part of my PhD thesis) but the result might not have been discovered without intuition coming from physics. There were a number of papers in a similar vein published at that time.

2$\begingroup$ Thanks, this is exactly the kind of answer that I was looking for! I hope to hear many more things along these lines. $\endgroup$ – Sarah Nov 25 '16 at 20:35
One of my favorite nonphysical applications of topological QFTs is a proof of Mednykh's formula
$$\frac{\left\mathrm{Hom}(\pi_1(\Sigma),G)\right}{\leftG\right} = \frac{1}{\leftG\right^{\chi(\Sigma)}}\sum_{V}\left(\dim V\right)^{\chi(\Sigma)},$$
where $\Sigma$ is a closed, connected, orientable surface with Euler characteristic $\chi(\Sigma)$, $G$ is a finite group, and the sum on the right is over irreducible complex representations of $G$.
This is a very concrete thing  if you take $\Sigma = S^2$, then the $\mathrm{Hom}$ vanishes and you get the firstcourseinrepresentationtheory fact that the sum of the squares of the dimensions of the complex irreps of a finite group $G$ is equal to the order of $G$. And if you take $\Sigma = T^2$ then you get (with some work) that the number of complex irreps is the number of conjugacy classes.
The connection is explained very well in these notes of Qiaochu Yuan (where I first learned about this), which are presumably based off of this paper of Dijkgraaf and Witten. I first stumbled across all this in this answer of Vectornaut.

$\begingroup$ Shouldn't "$\mathrm{Hom}$ vanishes" be "$\mathrm{Hom}$ trivialises" or something like that? You only get the identity so $\mathrm{Hom}=1$ rather than $\mathrm{Hom}=0$. $\endgroup$ – AccidentalFourierTransform Mar 30 '18 at 18:01

$\begingroup$ A Hom set between groups can never be empty, so there is no ambiguity. $\endgroup$ – dvitek Mar 30 '18 at 19:19

$\begingroup$ Of course there is no ambiguity, but saying that a group "vanishes" as a synonym for it being trivial seems unnatural to me. But if you like that terminology, it's all right. $\endgroup$ – AccidentalFourierTransform Mar 30 '18 at 19:28

$\begingroup$ Sure, it's probably not optimal given that it's only a set, not a group. $\endgroup$ – dvitek Mar 30 '18 at 19:35
You may be interested in reading this, by Michael Atiyah:
http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf
Edward Witten certainly is the master of finding impressive applications of QFT to Mathematics (for example TQFT has been instrumental for finding invariants in threedimensional manifolds). His work has inspired many mathematicians and has created even new areas in mathematics (SeibergWitten theory, for example, another impressive application of QFT to mathematics). You see, luckily there are mathematicians who are able to appreciate the invaluable insight that physics can give into very hard mathematical problems. These mathematicians may have, in my opinion, advantage over the rest of their colleagues. For example, Donaldson theory is inspired in physics and in Atiyah's vision of the importance of studying the moduli space of YangMills equations. Without the insight from physics it would have been very difficult to develop such theory. I recommend you to read the preface of the classical book of "Instantons and FourManifolds" by Freed and Uhlenbeck, where they explicitly say: "we mathematicians need physics!" and explain why. Of course, I am not implying that physics is useful for every mathematical problem, but it is indeed so for a good number of problems in geometry.
Note added: let me give a very explicit example of how QFT (in this case through String Theory) strongly influenced mathematics (in this case algebraic geometry) and indeed solved an outstanding open problem in AG. I am of course talking about mirror symmetry. In the following paper:
https://www.staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf
Philip Candelas, Xenia C. DE LA Ossa, Paul S. Green and Linda Parkes computed and predicted, using a mix of String Theory, Supergravity and SCFT, the number of rational curves on a generic Quintic threefold. This is the basis of Mirror Symmetry. Algebraic geometers were flabbergasted. The paper is indeed not easy to understand neither for mathematicians nor for string theorists (for general physicists is simply out of any reach), since it mixes the previous mentioned areas with relatively sophisticated mathematical arguments. Algebraic geometers immediately jumped into the topic, trying to understand what the heck was going on. Without String Theory, QFT and Supergravity, the outstanding conjecture encoded in Mirror Symmetry most probably would have not been proposed in "our era" (as I have heard from various algebraic geometers).

8$\begingroup$ I mentioned gauge theory in my question as something I am bracketing off. I certainly am impressed by the SeibergWitten invariants (but I'm not sure that they are really the kinds of things that mathematicians mean these days when they talk about QFT's). The TQFT invariants of knots and 3manifolds you mention are basically useless for mainstream 3manifold topology (notice that I am definitely not talking here about invariants of knots that come in some way out of gauge theory, eg Heegaard Floer invariants, but rather about descendents of the Jones polynomial). $\endgroup$ – Sarah Nov 25 '16 at 20:29

4$\begingroup$ The rest of your answer is just rhetoric about how I am closedminded. I won't bother to respond to it, but it certainly convinces me of nothing. $\endgroup$ – Sarah Nov 25 '16 at 20:29

4$\begingroup$ I read it many years ago. I stand by everything I wrote. Appeals to authority are meaningless. $\endgroup$ – Sarah Nov 25 '16 at 20:31

3$\begingroup$ @Sarah: What about mirror symmetry, is that important enough for you? Here you have the original article: staff.science.uu.nl/~beuke106/HypergeometricFunctions/COGP.pdf $\endgroup$ – Bilateral Nov 25 '16 at 23:24

7$\begingroup$ Sarah, I wouldn't read this answer as implying you are closeminded (maybe the "openminded" should be edited out to remove this impression). I think both your question and this answer are good ones. For my part, the answer is a kind of stinging reminder that I would be a richer mathematician if I had more physical insight  I am at least open to that :). $\endgroup$ – Todd Trimble♦ Nov 25 '16 at 23:36
I think a lot of the trouble people are having with this question comes from the phrase 'applications of quantum field theory'. Quantum field theory is a collection of ideas under active development; it's not so much a codified set of techniques like Class Field Theory or Sheaf Cohomology that you can just point at a problem.
Anyways, if I understand correctly, you're looking for examples where mathematicians have used ideas and techniques from QFT to prove preexisting math conjectures? (And we're not allowed to mention 4manifolds.) This is tricky to answer, since it's not always clear from the literature in exactly what sense Mathematician M understood QFT when proving conjecture C. Let's say rather that we're looking for cases where the ideas and techniques naturally belong to whatever Quantum Field Theory turns out to be, as it gets fleshed out.
There are tons of examples. Here are the first few that come to mind...
Borcherds' proof of Conway's Moonshine Conjecture, as a corollary of the construction of a certain QFT. (I think this is probably the ideal answer to your question.)
Witten's trivialization of Morse theory. (Also: Getzler's heatkernel proof of the AtiyahSinger index theorem, and Mathai & Quillen's novel construction of the Thom class.)
Drinfeld's introduction of quantum groups, which has lead to an enormous amount of pure math, with applications as far afield as combinatorics. (See also, the ADHM construction of instantons, Lusztig's use of ADHM to construct canonical bases, and Nakajima's introduction of quiver varieties.)
Beilinson & Drinfeld's partial proof of the geometric Langlands conjectures, building on Feigin & Frenkel's use of free field realizations to characterize the centers of enveloping algebras of KacMoody algebras at critical level.
Kontsevich's construction of a universal finitetype knot invariant, via monodromy of the KZ equation.
Deligne's completion of Grothendieck's proof of the Weil Conjectures. (Don't shoot! I'm joking!)
I could probably go on like this for some time, but I think that's a reasonable list of headline 'applications'. Note also that these mathematical developments feed back into QFT and inspire further internal development, as those concerned with physics try to work out just what the mathematicians have done...
@Sarah: I have some reservations about how the question is framed since it already answers itself and makes QFT seems like a subject which is completely separate from mathematics with occasional and anecdotic applications to mathematics. In one of your comments above you said "I'm looking for precisely that motivation (within mathematics as opposed to physics; I'm sure there is plenty of important physics here, but I'm not really interested in physics per se)." I think this is the right question to ask.
If the question is indeed about motivations to study QFT that are internal to mathematics, I think the answers is that much of QFT can be seen as the continuation of the development of calculus which started in the 17th century. I like to call it Calculus V (as in this paper or this older one). In Calc I one learns how to differentiate functions of one variable. In Calc II one learns how to integrate them. In Calc III one does both with finitely many variables. The natural continuation would be Calc IV: differentiation of functions on infinitedimensional spaces. Finally, Calc V would be integration on infinitedimensional spaces. Calc IV is taken care of by the theory of differential calculus in say Banach spaces, the calculus of variations etc. and this is now mathematics textbook material. Calc V is still work in progress. Stochastic calculus and modern probability theory make a good portion of what is known in mathematics under the heading of Calc V: essentially the theory of Gaussian Feynman path/functional integrals. However, there is much more to be explored and physicists have preceded mathematicians in figuring out a calculus for nonGaussian functional integrals. Some of it has been made rigorous but this is tiny in comparison with what has not. This makes QFT/Calc V a field of research for mathematicians.
One could try to enumerate applications of QFT to mathematics such as the geometric Langlands program etc. but this would, in my opinion, miss the main reason, internal to mathematics, why QFT is important to mathematics.
This being said, I feel I should at least give one such application which is not well known. By a remark of Burnol (in this article), the Riemann Hypothesis is equivalent to the existence of a Euclidean QFT (whose twopoint function is given by the Weil distribution).
Edit: another application worth mentioning is to PDEs. In some badly illposed cases one can nevertheless prove almost sure global wellposedness if one has an invariant measure for the PDE. For the nonlinear Schrödinger equation on the 2D torus, these measures come from rigorous QFT: they are the $P(\phi)_2$ models studied by Nelson, Glimm, Jaffe and many others. This application to PDEs was developed by Bourgain in this article (see also this more recent work in the area).
The study of the renormalization procedure in QFT led to the discovery of the Hopf algebra of rooted trees. Details are given in this paper.

1$\begingroup$ Isn't it peculiar how a seemingly arbitrary choice of regularization leads to interesting relations among common period numbers? I can never tell if this is deep or merely luck. $\endgroup$ – user1504 May 20 '17 at 0:20
I am aware of two books on quantum field theory written by and for mathematicians:
Quantum Field Theory: A tourist guide for mathematicians (Gerald Folland, 2008)
Quantum Field Theory for Mathematicians (R. Ticciati, 1999)
The most profitable mathematical developments have been in the context of topological quantum field theory and more specifically conformal field theories, because of their finitedimensional space. Some open problems in mathematical twodimensional conformal field theory are listed by YiZhi Huang (2000). One area of mathematics that apparently has nothing to do with QFT but which has benefited greatly from QFT techniques is knot theory (see String theory and Math: Why this marriage may last).

7$\begingroup$ I've seen these books, and they seem be attempts to explain what quantum field theories are, but they seem to assume that their readers are already motivated to learn this topic. I'm looking for precisely that motivation (within mathematics as opposed to physics; I'm sure there is plenty of important physics here, but I'm not really interested in physics per se). $\endgroup$ – Sarah Nov 25 '16 at 19:45

2$\begingroup$ In other words, I'm looking for something like: "<Mathematician X> used QFT in <Paper Y> to prove <Preexisting Conjecture that has nothing to do with QFT>." $\endgroup$ – Sarah Nov 25 '16 at 19:48

1$\begingroup$ Can you spell out these applications of TQFT's? $\endgroup$ – Sarah Nov 25 '16 at 19:50

3$\begingroup$ @ToddTrimble: I'm reacting to the final paragraph, where Carlo says that TQFT has been the most profitable from a mathematical standpoint. $\endgroup$ – Sarah Nov 25 '16 at 19:52

2$\begingroup$ @AbdelmalekAbdesselam  the energymomentum tensor of a TQFT vanishes, because the action does not depend on the metric so the variation of the action with respect to the metric is zero. So energy and momentum are both identically zero in a TQFT. $\endgroup$ – Carlo Beenakker Nov 26 '16 at 21:10
I'm not really sure that I totally got your question, so if I'm out of topic please forgive me. Since actually QFT is a rooted and mainstream theory that has been in the game for more than 50 years, a big part of the Mathematical Physics of the last 50 years is in some sort of sense originated from applications or needs coming from QFT. Moreover if I can add a personal opinion Theoretical Physics and Mathematical Physics worked quite well togheter during this time.
Atiyah and Witten works were of course cited. If you want some other example in addition to Atiyah's work I would say that most of the work of Alain Connes on noncommutative geometry, is in fact motivated from the need of combining QFT and Gravity as he concisely explains here. Moreover a lot of Quantum Algebra is originated from needs coming from QFT. I'm not sure how much of the work of Jimbo was motivated by QFT on his article anyway surely a lot of developments of Quantum Groups are. You might want to have a look at the works of Majid. Last but not least one of the most important Quantum Field Theories is Yang Mills theories and there's still a Clay prize on the foundations of it, so I guess someone is thinking is wortwhile studying these theories from a mathematical point of view.
I will stay conservative and I won't quote others examples because as I said since QFT has ruled physics for more than 50 years really a lot of Mathematical Physics is rooted and motivated by it. The fact that the term "Quantum Field Theory" is not always appearing in the abstracts or in the text should not mislead you.
Anyway as I said before I'm not sure this was really your question since the answer is really generic. If you want a more specific answer just write in the comments.
GromovWitten theory and DT theory have a lot of these sorts of results. For a sample, you may have a look at:
https://arxiv.org/abs/math/0312059
Abstract of "Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory" by Marcolli and Port:
Graph grammars extend the theory of formal languages in order to model distributed parallelism in theoretical computer science. We show here that to certain classes of contextfree and contextsensitive graph grammars one can associate a Lie algebra, whose structure is reminiscent of the insertion Lie algebras of quantum field theory. We also show that the Feynman graphs of quantum field theories are graph languages generated by a theory dependent graph grammar.