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Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?

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    Who voted to close this as belonging to physics.SE? Both knot theory and (T)QFT are well-known areas of mathematics... – Najib Idrissi Nov 07 '15 at 18:47
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    @NajibIdrissi I did as I thought OP would get a better response there (as they have some related questions there like [this](http://physics.stackexchange.com/questions/48514/understanding-cherns-simons-witten-theory)). Yes QFT/knot theory is studied by pure mathematicans, but how many people working on this is on this site? I have no idea, but we'll find out soon. – Winther Nov 07 '15 at 18:55
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    @Winther: A positive number. I don't know if I can give a so-called layman's explanation, though. This might be a good MO question. –  Nov 07 '15 at 19:34
  • @MikeMiller Hopefully someone is up to the task here. If not then OP can try his/her luck there instead. – Winther Nov 07 '15 at 19:37
  • Here are some short thoughts. First, there's the tidy notion of a TQFT. It was noticed in the 80s that Chern-Simons theory fits into this formalism; after the development of Floer homology it was noticed that Yang-Mills does too. In the direction of knots one has two orthogonal developments. First, one might think "Well, it was an effective idea in 4 dimensions to consider instantons that were singular along an embedded surface. What about the 3-dimensional analogue of instantons singular along knots?" This leads one to instanton knot homology. –  Nov 07 '15 at 19:50
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    Then one can think: well, I see no reason this tidy TQFT formalism shouldn't also apply to *pairs* $(Y,K)$ where $K$ is a knot (or link) in a 3-fold $Y$, and my things are functorial in terms of cobordisms $(W,\Sigma)$. Indeed, one can use instantons singular along a surface to get functoriality of the above theory, so one has developed a knot TQFT. Now one takes the graded Euler characteristic of this knot homology and sees what happens. I don't remember what happens in the instanton case, but for Heegaard Floer knot homology you get the Alexander polynomial, a famous knot invariant. –  Nov 07 '15 at 19:54
  • So then one thinks, well, what about the Jones polynomial? Where can I get that from? And we know that it can be obtained as the Euler characteristic of Khovanov homology. Witten proved that you can find the Jones polynomial in terms of gauge theory (via another type of 'singular along a knot' construction as above), leading one to wonder: is there a gauge-theoretic knot homology like above that gives it? Which leads to a lot of current research on $SL_2(\Bbb C)$-Floer homology, which is not a thing that exists yet, but to my understanding should be the tool that answers the above question. –  Nov 07 '15 at 19:56
  • I warn that much of the above is how I think of the story; I have no idea if that's actually how the story developed. I cannot say much about Witten's ideas for Khovanov homology, since I don't know much about them - I've only glanced at that paper. I think the key idea in terms of all of that was: Floer homology arises from the dimensional reduction of some 4-dim equation (Yang-Mills, Seiberg-Witten, e.g.). Then knot Floer homology arises as the dimensional reduction of studying solutions to that equation that are singular along some surface. –  Nov 07 '15 at 19:59

2 Answers2

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I am a topologist who uses gauge theory. This answer reflects my taste and background. I make no claims of historical accuracy. It would be nice to see answers from people with other perspectives.

In the 80s, Donaldson used Yang-Mills theory to great effect in studying the topology of smooth 4-manifolds. In it, one (in some sense) counts solutions to the PDE $F_A^+ = 0$ defined on the space of connections on a smooth manifold $M$. I will call solutions to this instantons and the equation the instanton equation. Later, Seiberg and Witten introduced the (related) Seiberg-Witten equations, which led to much less technical proofs of many of the same theorems. Solutions to these equations are called monopoles, and the invariants you get from counting them are called the Seiberg-Witten invariants.

In studying the Thom conjecture and more general adjunction inequalities, Kronheimer and Mrowka were led to the idea of studying monopoles that were singular along an embedded surface $\Sigma$. This was an extraordinarily successful avenue of research.

Orthogonally, after Floer's original invention of Floer homology groups, it was realized that the instanton and monopole invariants fit into the tidy formalism of a TQFT (well, mostly). By using the technique of dimensional reduction, and following the idaes of Floer's construction, every connected orientable 3-manifold $Y$ gets homology groups $HF(Y)$ (different depending on whether you're using instanton or monopole invariants; there are other related constructions), and for every cobordism $W: Y \to Y'$ we get a map $HF(Y) \to HF(Y')$ satisfying certain criteria. Equivalently from the perspective of TQFT, every 4-manifold $W$ with boundary $Y$ has an (instanton or monopole) invariant living in $HF(Y)$; one obtains the original invariants by deleting a 4-ball and looking at the invariant in $HF(S^3) \cong \Bbb Z$. There are numerous technical difficulties I have either avoided or lied about in this paragraph.

One begins to think: well, that idea of studying monopoles singular along a surface seemed pretty good. Why don't I take the dimensional reduction of that idea? Do so, and you obtain $HF(Y,K)$, the "knot Floer homology" of a knot $K \subset Y$. This is functorial with respect to cobordisms $(W,\Sigma): (Y,K) \to (Y',K')$ and gets you a sort of knot TQFT.

I'm a little uncomfortable saying this with complete certainty, but in the monopole case, the (graded) Euler characteristic of $HF(S^3,K)$ should be the Alexander polynomial of $K$. (I don't remember what it is for the instanton knot Floer homology; I would not guess it's the Alexander polynomial) This is a well-known knot invariant. One might ask: well, what about the Jones polynomial? Does that come as the Euler characteristic of some group? Well, it does, but not one that's obviously defined via gauge theory: it's the Euler characteristic of Khovanov homology. Now, separately, in a story I don't know very well, Witten gave a construction of the Jones polynomial in a way defined by Chern-Simons theory; my understanding is that it was defined by studying solutions to an equation that are singular along a knot (but I really have not looked at this much at all).

This suggests that one should expect there to be a gauge-theoretically defined version of Khovanov homology, which Witten described here. This is not yet known to work mathematically, but to my understanding it should be some sort of "$SL_2(\Bbb C)$-Floer homology" as opposed to the $SU(2)$ or $SO(3)$ or $S^1$-based ones above. (Note that $SL_2(\Bbb C)$ is not compact. This leads to an extraordinary amount of technical trouble - which is why this Floer homology doesn't really exist yet.) Then the Jones polynomial for knots in arbitrary manifolds should be the Euler characteristic of this thing. This is an active area of research, and when completed, should be the natural place where Witten's work lives.

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This will not be by any means a "layman's" explanation, but here's something short that might be helpful. There also won't be much physics here, if that's what you're looking for.

Let's take for granted, following Witten, that if $G$ is a simply connected compact Lie group (for simplicity) and $k$ is an integer, then there is a 3d topological field theory called "Chern-Simons theory with gauge group $G$ and level $k$." Chern-Simons theory is so named because its heuristic physical description involves path integrals where the action has something to do with the Chern-Simons 3-form. I won't get into all of the structure that this TFT has; it's enough for now that it assigns (up to some subtleties I don't want to get into)

  • a finite-dimensional complex vector space $Z(\Sigma)$ to every closed surface $\Sigma$, and
  • a linear map $Z(\Sigma_1) \to Z(\Sigma_2)$ to every 3d cobordism $M$ between $\Sigma_1$ and $\Sigma_2$

and these assignments satisfy various compatibilities. Also, you'll need to know one more thing: the vector space $Z(T^2)$ assigned to the torus has a natural basis that can be identified with either the irreducible representations of the corresponding loop group ("at level $k$") or with the irreducible representations of a version of the corresponding quantum group (for some root of unity $q$ depending on $k$). In physics language these are "Wilson loop operators."

How do you get knot invariants out of this? Given a knot $K$ sitting inside $S^3$, consider a tubular neighborhood around it. The complement of this tubular neighborhood describes a cobordism from the torus $T^2$ to the empty manifold, and hence hitting it with Chern-Simons gives a linear map

$$Z(T^2) \to \mathbb{C}.$$

Now evaluating this linear map at an irreducible representation of, according to taste, either the loop group or the quantum group produces a knot invariant. When $G = SU(2)$ and the representation is the standard one, you get the Jones polynomial evaluated at some root of unity $q$ depending on $k$.

Qiaochu Yuan
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  • Could you please provide a specific reference (or references) for this discussion please? – Malkoun Dec 28 '20 at 02:53
  • @Malkoun: the original reference for this sort of thing, as far as I know, is Witten's *Quantum field theory and the Jones polynomial* (https://projecteuclid.org/euclid.cmp/1104178138). You can look up search terms like "knot invariants" and "topological quantum field theory" to find more modern references. – Qiaochu Yuan Dec 28 '20 at 02:55
  • Thank you! I shall attempt to read more of that article. The topic is really interesting. – Malkoun Dec 28 '20 at 03:03