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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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer 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, Lustig'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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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...

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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer 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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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...

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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer index theorem, and Mathai-Quillen's 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, Lustig's use of ADHM to construct crystal 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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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...

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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer 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, Lustig'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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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...

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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer index theorem, and Mathai-Quillen's 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, Lustig's use of ADHM to construct crystal 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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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 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 pre-existing math conjectures? (And we're not allowed to mention 4-manifolds.) 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 heat-kernel proof of the Atiyah-Singer index theorem, and Mathai-Quillen's 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, Lustig's use of ADHM to construct crystal 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 Kac-Moody algebras at critical level.

• Kontsevich's construction of a universal finite-type 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...