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Are there more recent developments on connections between QFT and knot theory than Witten's initial 1989 paper?

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  1. What has happened in the field of QFT and knot theory since Witten's paper "Quantum field theory and the Jones polynomial" from 1989?

  2. What resources should one read to get up-to-date on this story?

  3. Or is this the only connection between QFT and knot theory?

This post imported from StackExchange Physics at 2016-06-13 21:34 (UTC), posted by SE-user Loffen
asked May 29, 2016 in Resources and References by Loffen (0 points) [ no revision ]
retagged Jun 13, 2016
Enter Khovanov homology arxiv.org/abs/1108.3103

This post imported from StackExchange Physics at 2016-06-13 21:34 (UTC), posted by SE-user Thomas

1 Answer

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I'm glad you asked! In the last decade, there have been incredibly exciting developments which have showed that the connection between QFT and knot theory is even deeper than previously expected.

Let me first give a bit of a conceptual sketch, both of past and recent developments:

In the 1980's, Witten realized that calculating Wilson loops in TQFTs in fact defines knot invariants. In the simplest case, let $\gamma$ be a loop/knot in 3D and let us take the abelian Chern-Simons action $S = \int A dA$. Then we can consider the observable $$\langle e^{i \oint_\gamma A}\rangle := \frac{1}{Z} \int e^{i \oint_\gamma A} e^{i \int A d A} \; \mathcal D A \; .$$
This is the expectation value of a Wilson loop, which naturally extends to non-abelian TQFTs. Witten realized that this quantity (which is a function of $\gamma$) is in fact invariant under smooth deformations of $\gamma$ and as such defines a knot invariant. In particular in the special case of abelian Chern-Simons it reduces to the well-known writhe number (or self-linking). It gave rise to more non-trivial knot invariants for non-abelian Chern-Simons, like the Jones polynomial, a knot invariants that mathematicians had struggled to understand for decades. (Mathematicians couldn't make sense of its chiral property, i.e. that it distinguished left from right, but from the Chern-Simons perspective this became very natural. The history of this is nicely told by C. Nash in ``Topology and physics - a historical essay'', for example discussing how Atiyah prodded Witten to investigate the Jones polynomial, which then led to the aforementioned realization.)

Some prophetic words from Penrose in 1988 [Topological QFT and twistors]: noting the above use of TQFTs by Witten, Penrose noted that ``the salient feature of TQFT is that there are no field equations'' (e.g. $S = AdA$ gives as equation of motion $F = dA = 0$). He pointed this out because by that point, he had developed a reformulation of physics in terms of twistor space. One can think of going into twistor space as analogous to doing a Fourier transform: it is simply a different way of expressing the degree of freedoms. Reformulating things in twistor space has various nice properties, one of which is that all field equations in some sense become equivalent to a condition of holomorphicity, i.e. $\bar \partial f = 0$. As such field equations in twistor space become trivial, in the same sense as those of topological field theories in real space. He thus mused:

``Thus the physically appropriate home for TQFT might well be twistor space rather than space-time.'' [Penrose, 1988]

Let me now fast forward to the last decade and tell you that Penrose was right. It turns out that if one translates Witten's previous insights into twistor space, something beautiful unfolds. In fact: it turns out that calculating scattering amplitudes in real space is in some sense equivalent to calculating knot invariants in twistor space!

More precisely,

 1. If one takes a collection of momenta $p_1, p_2, \cdots$ that sum to zero (i.e. the external legs for a Feynman diagram) and map them to twistor space, then it turns out that this defines a holomorphic knot $\gamma$. (Think of this as a complexified version of a usual knot.)
 2. And if one takes $\mathcal N=4$ SYM in real space and maps this to twistor space, it becomes a holomorphic version of Chern-Simons $S_\textrm{hCS}$.
 3. It now turns out that calculating the scattering amplitude in real space is equivalent to calculating the expectation value of the Wilson loop along $\gamma$ (using $S_\textrm{hCS}$ as the action).
 4. In other words, using Witten's insight form the 80's, one can say that calculating the scattering amplitude in real space is equivalent to calculating a knot invariant of $\gamma$ in twistor space!

These beautiful insights have been uncovered by David Skinner, Lionel Mason, Matthew Bullimore and others. It is on-going research to see how this extends to non-SUSY and/or gravitational theories. I wrote a review about these developments as my master's essay (usually choosing a more conceptual approach, going for breadth rather than depth). I hope it can be of help! There are of course quite some references in the bibliography.

answered Jun 14, 2016 by Ruben Verresen (205 points) [ no revision ]

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