Knots, Energy and…Wilson Loops?

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Apologies for the delayed post. I’ve been on the east coast suffering through an intense heatwave. But I’m back, baby!

This week I want to talk about something I don’t know a lot about–knots. Knot theory is a deep and beautiful topic. And it’s nice because it is ~relatively~ intuitive, as far as I can tell–it’s all about embeddings of S^1 into \mathbb{R}^3. For the non-math minded people reading this, a knot is…well, exactly what you think it is.

Why knots? Why not? But seriously I saw this video in one of my many math facebook pages (s/o >implying we can discuss mathematics) about the equivalence of two genus 2 2-surfaces (a donut with 2 holes for the non-experts). Someone in the comments posted a truly bizzare animation demonstrating the equivalence (aside: seriously everyone watch that video because it’s very cool and very trippy). This video was apparently made for a math video festival at the 1998 International Math Conference in Berlin. Lucky for me there was a whole playlist of videos from this fest and so of course I spend the next hour watching them all.

One of those videos was about…you guessed it: knots! This particular video (and the accompanying paper which was surprisingly hard to find online–email me and I’ll send you a PDF if you want) was demonstrating the time evolution of knots with respect to their Mobius energy. Under this evolution, the knots would untangle into a configuration in the same isotopy class which minimizes the energy–sometimes landing in metastable configurations before evolving to the true ground state. For those that don’t know–an isotopy is a way to say that two knots are “the same”. That is, you can do a series of moves called Reidemeister moves which can bring one into the other (although isotopy only refers to the 2nd and 3rd moves if I’m not mistaken). I’ve described this in very physics-y language, which shouldn’t be surprising to anyone reading this. So when I say “time evolve with respect to Mobius energy” I am really talking about solving the gradient flow problem \dot{x}(t)=-\nabla E(x(t)) where E(x(t)) is the Mobius energy of the system. My language seems to imply that this energy is a Hamiltonian–which it might be. Mathematically speaking this is the problem they are solving.

The energy function in question is the following: given a knot K \subset \mathbb{R}^3, the Mobius energy is defined by E(K)=\iint_{K\times K} \frac{f(x,y)}{|x-y|^{2}} dvol_K(x) dvol_K(y) . The regulator f(x,y) is chosen to vanish as x\rightarrow y faster than the denominator so as to render the energy finite. For certain choices of f the energy is invariant under the Mobius group of \mathbb{R}^3. I am not sure why they require this other than things which have symmetry are very nice, but perhaps one of my readers can enlighten me on the technicalities involved here. For this particular video, the regulator they choose is f=1-\cos(\alpha) where I BELIEVE \alpha is supposed to be the angle between x and y, but I am not 100% sure–they never specify in the paper. They do claim that this is related to a slightly more intuitive regularization which gives \tilde{E}(K)=\iint_{K\times K} \left( \frac{1}{|x-y|^{2}}-\frac{1}{d_K(x,y)^2} \right) dvol_K(x) dvol_K(y) where d_K(x,y) is the shortest distance between the points on the knot itself, by E=\tilde{E}-4. So if you want you can just think of this regulator instead. This energy is non-negative, zero for the unknot and infinite if the knot contains self-crossings–that sounds like a good energy to me!

Another way to think of this energy is in the following: discretize the knot using your favorite triangulation, put an electric charge at each vertex and compute the mean field potential induced from all the other charges. Then sum up all of these contributions. This gives essentially the same thing and is indeed a slightly more intuitive way to think about this energy.

Physicists love minimizing energy. It’s one of our top 5 favorite things to do. This got me thinking–is there a way to formulate this problem in a more physics-y language? And will that do anything useful? One first has to wonder whether this energy can indeed be interpreted as a Hamiltonian. This does not seem very easy to do, as the gradient flow equations are not necessarily Hamiltonian–but perhaps a regulator can be chosen such that it becomes Hamiltonian. (I found a set of lecture notes that says a system x'=F(x) can be both Hamiltonian and Gradient flow iff F(x) is a harmonic function. With some work I’m sure an appropriate f(x,y) and F(x) can be found such that E(K) is the Hamiltonian of the system. Whether or not the solution to this problem will preserved the full Mobius invariance or just some subgroup is a whole different ball game).

Now, one of the most amazing connections between knot theory and physics comes from the mind of Ed Witten. In his (amazing) paper titled Quantum Field Theory and the Jones Polynomial, Witten shows that expectation values of Wilson lines in pure Chern-Simons theory on a compact three manifold give a generalization of the Jones polynomial for that line. Let’s break this statement down.

Ok, take a gauge theory with gauge group SU(N). On a three dimensional Euclidean manifold, one can define the Chern-Simons action as S=\frac{k}{4\pi} \int _{M}\text{Tr} (AdA+\frac{2}{3}A\wedge A\wedge A ) . For the mathematicians reading, A is Lie algebra valued the connection 1-form on a principle SU(N) bundle over M. It depends on the local trivialization. For the physicists reading A is the usual covariant vector potential. It is not gauge invariant. For the non-technical people reading, A tells you information about the “electric” and “magnetic” field defined on the space M–a certain combination of derivatives exactly gives you these fields (I put “electric” and “magnetic” in quotes because this is a generalized version of electricity and magnetism which is more complication (and more interesting) but nonetheless the same terms can be used). The theory I have just described is an example of a topological field theory. It does not depend on the metric on M, it’s excitations are non-local and it does weird things on manifolds with boundaries–three very important characteristics of a topological field theory.

A Wilson line is ~basically~ just a line of electric flux. But more specifically it is the holonomy of A around a curve. For simplicity let’s just consider things that are in the fundamental representation of SU(N). We then define the Wilson line as W(K)=\text{Tr}\mathcal{P}e^{i\int _K A} where K is the knot of interest, the trace is taken in the fundamental representation and \mathcal{P} is something technical called path ordering that I don’t want to get into. Consider a collection of n possibly intersecting knots K_i. The remarkable insight of Witten was that the quantity Z \langle W(K_1)W(K_2)...W(K_n)\rangle=\int \mathcal{D}Ae^{iS}W(K_1)W(K_2)...W(K_n) is equal to the Jones polynomial of the link defined by the K_i's. Z is the partition function of Chern-Simons theory and acts as the normalization. We care about the unnormalized expectation value, hence why it is on the left side and not the right. This is very, very, very cool for a LOT of reasons. Incidentally these theories are also the main object of my research. How about that!! The weird \mathcal{D}A is the measure for the path integral over the field A (look it up!) and is the crux of quantum field theory (canonical quantization gang can get bent!!). If you can compute this integral you can solve the full quantum theory. It is very hard to do, though, and doesn’t actually have a very mathematically rigorous formulation (outside of lattice gauge theory where the measure is given by the Haar measure on the gauge group).

Now is there a way to connect these ideas? More specifically, can we define a Mobius energy on the configuration space of the knots such that the expectation value of a Wilson loop defined by a specific knot will evolve to a minimum in time? I do not know. Probably knot, but maybe! There doesn’t seem to be a clear relation between the Jones polynomial and the Mobius energy. One is a topological invariant and the other is certainly not (since it depends on the configuration of the knot), and it is not clear how one should affect the other.

One thing we might be able to do here is find extremize the Wilson loop with respect to it’s path. That is, we take d \langle W(K) \rangle /dK=0 and solve for the minimum configuration K. This may possibly tell us the “minimum energy” configuration for a given Wilson loop in its isotopy class. We can do the same with d \langle W(K_1)W(K_2)...W(K_n) \rangle /d(K_1...K_n). It is far from clear if this will actually work. If we include the “unnormalization” Z it probably won’t work because the unnormalized expectation value is a topological invariant and shouldn’t care about the specific configuration–it only cares about the linking numbers and things like that. If we consider the normalized expectation value, it might work.

To make things even more difficult, if we’re going to want to talk about time evolving something with a Hamiltonian we will definitely have to move out of 3 dimensions. This is because time is its own dimension and if we want to start with a certain Wilson loop in 3 dimensions at time t=0 and evolve it forward in time using the Hamiltonian associated to the Mobius energy to some time t we need to be talking about 3+1 dimensional spacetime. The problem now is that Chern-Simons theory doesn’t exist in 3+1 d and so the connection to the knot invariants is lost. Still, we could do two things: (1) use the standard Yang-Mills Hamiltonian and time evolve a Wilson loop such that it minimizes the YM energy, or (2) define a new type of gauge theory in 3+1 such that the Hamiltonian is the Mobius energy for a given knot and time evolve it using this Hamiltonian. It is not clear if (1) will give the results we want or if (2) is even possible. If it was it would be really cool!

Thanks for listening to my incoherent rambles on knots and gauge theory. These ideas truly are half baked (I’m not admitting I was high when I watched that video and came up with these ideas, but I’m also not not admitting it). If any readers wants to talk about and possibly develop these ideas further, please reach out to me at andrew.baums@gmail.com. I would love to think about this more if it means that I could write a paper about it someday and put it on my CV.

This week’s music recommendation is Denver’s City Hunter. I saw this band a couple weeks ago at the Black Lodge in Seattle and HOLY SHIT. They rip. Black metal style vocals over hardcore/power violence-y music. Inspired by your favorite 80’s slasher flicks the lead singer wore a leather ski mask, gloves and a gigantic coat. Rumor has it sometimes he pulls a knife out too, although that wouldn’t fly at the tiny DIY venue we were at. This band is really really good and if they are coming through your town you should check them out because they put on one helluva show.