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.
[Math] Mathematical applications of quantum field theory
mp.mathematical-physicsquantum-field-theory
Related Solutions
The path integral has many applications:
Mathematical Finance:
In mathematical finance one is faced with the problem of finding the price for an "option."
An option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or sell a specified asset, the underlying, on or before a specified future date, the option's expiration date, at a given price, the strike price. For example, an option may give the buyer the right but not the obligation to buy a stock at some future date at a price set when the contract is settled.
One method of finding the price of such an option involves path integrals. The price of the underlying asset varies with time between when the contract is settled and the expiration date. The set of all possible paths of the underlying in this time interval is the space over which the path integral is evaluated. The integral over all such paths is taken to determine the average pay off the seller will make to the buyer for the settled strike price. This average price is then discounted, adjusted for for interest, to arrive at the current value of the option.
Statistical Mechanics:
In statistical mechanics the path integral is used in more-or-less the same manner as it is used in quantum field theory. The main difference being a factor of $i$.
One has a given physical system at a given temperature $T$ with an internal energy $U(\phi)$ dependent upon the configuration $\phi$ of the system. The probability that the system is in a given configuration $\phi$ is proportional to
$e^{-U(\phi)/k_B T}$,
where $k_B$ is a constant called the Boltzmann constant. The path integral is then used to determine the average value of any quantity $A(\phi)$ of physical interest
$\left< A \right> := Z^{-1} \int D \phi A(\phi) e^{-U(\phi)/k_B T}$,
where the integral is taken over all configurations and $Z$, the partition function, is used to properly normalize the answer.
Physically Correct Rendering:
Rendering is a process of generating an image from a model through execution of a computer program.
The model contains various lights and surfaces. The properties of a given surface are described by a material. A material describes how light interacts with the surface. The surface may be mirrored, matte, diffuse or any other number of things. To determine the color of a given pixel in the produced image one must trace all possible paths form the lights of the model to the surface point in question. The path integral is used to implement this process through various techniques such as path tracing, photon mapping, and Metropolis light transport.
Topological Quantum Field Theory:
In topological quantum field theory the path integral is used in the exact same manner as it is used in quantum field theory.
Basically, anywhere one uses Monte Carlo methods one is using the path integral.
Although not mathematical per se, I personally like Kogut's works for Hamiltonian lattice gauge theory: there is an old RMP article and a rather good book which I purchased specifically for its presentation of the Hamiltonian theory. Of course Kogut introduced the subject along with Wilson in a PRD article, so his presentation of the material is not particularly exotic.
I will say that the advantage of the lattice is that rigor is not really much of an issue unless one wishes to take a continuous limit. Even so the incorporation of material on the Ising theory is helpful in this regard.
Best Answer
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.