53

Nima Arkani-Hamed had a series of talks at JHU roughly 6 months ago which I attended related to this topic. He discussed it at Stony Brook a little bit over a week ago (pointed out by Emilio Pisanty in the comments above in which he used the term "amplituhedron", but my understanding of this comes mostly from his earlier talks.
Update: I managed to track ...

37

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"...

35

My opinion is that physicists transferred from study of "individual objects" to that of "large systems" where the order arises from limit probability laws rather than from simple deterministic formulae and from the study of something "readily observable" to something that is, essentially, "a purely mathematical object" invisible to a direct experiment. This ...

35

I haven't read the paper carefully, but this appears to be a standard undecidability result, of the sort of which there are dozens if not hundreds in the literature, of the same ilk as the undecidability of Wang tilings, the undecidability of the existence of solutions to Diophantine equations, the word problem for groups, and many others.
It's a formal ...

33

One of my favorite non-physical applications of topological QFTs is a proof of Mednykh's formula
$$\frac{\left|\mathrm{Hom}(\pi_1(\Sigma),G)\right|}{\left|G\right|} = \frac{1}{\left|G\right|^{\chi(\Sigma)}}\sum_{V}\left(\dim V\right)^{\chi(\Sigma)},$$
where $\Sigma$ is a closed, connected, orientable surface with Euler characteristic $\chi(\Sigma)$, $G$ is ...

30

The answer of Jørgen Ellegaard Andersen only concerns the case of when the gauge group is $SU(n)$.
I will argue that all the ingredients for the equivalence between the two approaches (namely "geometric quantization of character varieties" and "quantum groups plus skein theory") are out there, for arbitrary simply connected gauge group.
First of all, let ...

answered Oct 25 '14 at 23:22

André Henriques

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30

You may be interested in reading this, by Michael Atiyah:
http://www.mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0031.0036.ocr.pdf (Internet Archive)
Edward Witten certainly is the master of finding impressive applications of QFT to Mathematics (for example TQFT has been instrumental for finding invariants in three-dimensional manifolds). His work has ...

30

As Robert Israel indicates, if you really wanted to define a QFT, it would certainly be very analytic, say, to define the path integral. So analytic, in fact, that no one can do it for even semicomplicated theories.
However, one of the big math/physics developments in the last few decades is a class of QFTs where the observables are topological in nature. ...

25

There is a recent construction of a fully extended 4d TQFT from a modular tensor category,
due to Dan Freed and Constantin Teleman (using Lurie's proof of the cobordism hypothesis).
It is described in Freed's lecture notes from the Segal 70th birthday conference here:
https://people.maths.ox.ac.uk/tillmann/ASPECTS.html
The idea is that braided tensor ...

25

The framing of your question is a bit ambiguous and perhaps there are two different questions
here depending on the context and interpretation. One could approach your question from the point of view of intrinsic scientific content and ask: why QFT seems to be intrinsically more related to topology than analysis? (Question A). But one can also
approach the ...

answered Apr 30 '17 at 17:47

Abdelmalek Abdesselam

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24

@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 ...

24

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 ...

21

When I was first learning QFT, I found it very helpful to start in the parts that rest on the most solid mathematical foundations, like topological quantum field theories and 2d conformal field theories. This might be an especially good place to start if you're aiming for the geometric applications of QFT, because I think many of those applications currently ...

20

I interpret your question to be asking about the transition from
computable undecidability to Gödelian or logical
undecidability, and furthermore about the extent to which this
logical undecidability might depend on which axioms of mathematics
we have adopted.
The answer is that one may quite generally deduce that there are
concrete instance of logical ...

answered Dec 12 '15 at 13:00

Joel David Hamkins

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19

Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum.
Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) manifold, whose metric can be found -- in principle -- by solving a system of PDEs. To be more specific, the metric $g$ is such that the classical action $S[g]$...

16

This question is very nicely discussed in the paper of Ruelle,
Ruelle, David(F-IHES)
Is our mathematics natural? The case of equilibrium statistical mechanics.
Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 259–268.
But he does not really have an answer, and perhaps nobody has an exact answer.
My own opinion is that a large part of mathematics (some ...

16

The relationship between the Ising model (spins on a lattice) and conformal field theory holds only in the immediate vicinity of the critical point, when correlation lengths go to infinity and all details on the scale of the lattice constant become irrelevant.
The relationship is explained, for example, in these lecture notes. Let me walk you through them.
...

answered May 26 '14 at 18:33

Carlo Beenakker

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16

Perhaps the main "analytic" area in Quantum Field Theory is known as Constructive Quantum Field Theory. This essentially emerged in the 1960's
with the Wightman axioms. There is still work going on there, but I think there is a feeling that the "easy" questions have been answered, and much of
what remains may be impossibly hard: e.g. the Millenium Prize ...

16

First of all, there are important differences between the notions of strict $n$-category, weak $n$-category, and $(\infty,n)$-category. The easiest notion is that of a strict $n$-category, and there's no doubt about the definition there: a strict $0$-category is a set, and by induction a strict $n$-category is a category enriched in the category of $(n-1)$-...

15

It seems to me the answer to:
"Do physicists have some tools/ideas/techniques which allow them to make insights, which are not seen for mathematicians?" is indeed yes. Not only such a tool exists but, in my opinion, it is also unique: functional integrals. Predictions based on that tool are what mathematicians have a hard time reproducing and justifying ...

15

The standard reference for constructive QFT is the classic book by J. Glimm and A. Jaffe, Quantum Physics: a Functional Integral Point of View (2nd. ed., Springer-Verlag, 1988). It is certainly more than satisfactory from the viewpoint of mathematical rigor, it has a lot of background material (specially the second edition linked above) and parts of it can ...

answered Jan 20 '14 at 1:00

Pedro Lauridsen Ribeiro

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15

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver:
"A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman integrals are not mixed Tate.
2) In 2008, Schnetz has compiled a list of Feynman integrals in Quantum periods: A census of $\phi^4$-transcendentals. These were ...

15

The renormalization group (RG) as a geometric flow (like the Ricci flow) is a very
special case of the RG, namely, the one corresponding to the nonlinear sigma-model (NLSM) in two dimensions with values in a Riemannian manifold.
Now the RG is much more general and applies to all sorts of models, not just the NLSM.
In order to find satisfactory answers to ...

answered Aug 5 '19 at 17:39

Abdelmalek Abdesselam

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14

There are two different things which are called "Riemann surface" in the literature.
The modern notion (introduced by Hermann Weyl): complex 1-dimensional manifold.
In older literature this is sometimes called "Abstract Riemann surface".
"Riemann surface spread over the plane" (or over the sphere, or over some other surface). Surface de Riemann etalee in ...

answered Jul 5 '13 at 7:51

Alexandre Eremenko

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14

Summary: the equivalence relies on the mathematical formalism of TQFTs, and sends a 2d TQFT $Z$ to the state space $Z(S^1)$, which is naturally a Frobenius algebra.
One potential point of confusion is that people think of TQFTs in different ways. They were originally defined as QFTs that only depend on topological information, so the emphasis was on the ...

rt.representation-theory mp.mathematical-physics quantum-field-theory topological-quantum-field-theory frobenius-algebras

13

The Weyl algebra construction can be done abstractly for any real vector space (even infinite-dimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C*-algebra construction using generators and relations ("Shape theory for C∗-algebras", Math. Scand. 56 (1985) 249–275). However, since you asked for a concrete ...

answered Aug 16 '13 at 1:11

Pedro Lauridsen Ribeiro

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13

If your goal is to get some understanding of the Clay Problem, you can't really go wrong with first reading the official problem statement and then reading the papers referred to in the document.
On the other hand, if your goal is not the quantum problem but more the classical problem, for the geometers and algebraists a good starting point is of course ...

13

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.

13

I keep seeing this question percolate up. I think it deserves at least one more or less correct answer.
First, the 1d integrals $I(a,b) = \int_{\mathbb{R}} \exp(-\frac{1}{2}x^2 - ax - b x^4) dx$ certainly exist for $b \geq 0$ and are analytic in $a$ and $b$. But they're not do-able in elementary terms. Noting that $\partial_bI = \partial_a^4I$ and ...

13

The partition function should assign to each possible field configuration $\Phi$ (or field history) in your quantum field theory a number $Z(\Phi)$. That is, it should be a function on the collection of fields configurations, and from that function you can derive lots of quantities in the field theory.
It can happen, however, that in order to come up with, ...

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