Relation of knots and gauge fields?

Question

Hi! I would like to ask some heuristic (and not only) explanation of how are gauge fields related to knots. I understand what knots, links, skein relations etc are but reading through Marino's book, he suddenly says "... Given an oriented knot K in S^3, we can consider the trace of the holonomy of the gauge connection around K in a given irrep R of U(N)..."

Can someone please expand and explain this idea?

Thanks a lot.

Comments

Related question on MathOverflow: Applications of knot theory

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Possibly related: Gauge theory and knot homologies (lecture notes by Sergei Gukov)

Answer 1

In any gauge theory in any spacetime dimension, it is possible to construct a class of operators called the Wilson operators. For every loop $C$ in spacetime (i.e. for every embedding of a circle in spacetime), it is possible to define a Wilson operator as follows. Let us fix some classical configuration $A$ of the gauge field. Consider the holonomy $Hol(A, C)$ around $C$: if we fix a point $c$ of $C$, it is the element of the gauge group $G$ which defines the action of the parallel transport starting from $c$, following $C$ and returning to $c$. Formally, we have

$Hol(A,C) = Texp(\int_C A)$

where Texp denotes the path-ordered exponential.

The element $Hol(A,C)$ of $G$ depends of the choice of the base point $c$ but not its conjugation class. So, in order to obtain a well-defined (gauge invariant) quantity, a way to do is to take the trace $Tr_R Hol(A,C)$ in some representation R of the gauge group. For a fixed loop $C$ and a fixed representation $R$, $A \mapsto Tr_R Hol(A,C)$ is a gauge invariant function on the space of gauge fields. In particular, it is possible to take its vacuum expectation value in the quantum theory (say in Euclidean signature);

$W(C,R) = <Tr_R Hol(A,C)> = \int DA e^{-S(A)} Tr_R Hol(A,C)$

where $S(A)$ is the action of the gauge theory, $\int DA$ is the functional integral on the space of gauge fields. 

Physically, the Wilson operators $Tr_R Hol(A,C)$ are of central importance to understand the dynamics of gauge theories and are kind of "test objects" to detect various phenomena such as confinement.

If one is interested by knots, one can try to apply what preceeds by taking a 3-dimensional spacetime $M$ (such as the 3-sphere) and then $C$ is simply a knot in $M$ (the possibility for a loop to be knotted is special to the dimension 3). For a general gauge theory, a quantity such as $W(C,R)$ will depend on the precise embedding of $C$ on $M$ and not only of the "topology" (more precisely the isotopy class)  of the embedding. A knot theorist is interested by knot invariants, i.e. quantity associated to C depending only of the topology of the embedding. To construct such quantities using the preceding lines, we have to take a topological gauge theory. The simplest topological field theory in 3 dimensions is Chern-Simons theory, of action

$S(A,k) = k \int_M Tr(AdA + \frac{2}{3}A^3)$

where the coupling constant $k$ must be an integer (up to some universal normalization) in order to obtain a gauge invariant quantum field theory.

So, given a knot in the 3-sphere, a choice of gauge group $G$, a choice of representation $R$, and a choice of integer $k$, it is possible to define some knot invariant $W(C,R)$. For $G=SU(2)$, $R$ the fundamental representation, and $k$ varying, we obtain the famous Jones polynomial. The starting point of this story is the seminal paper of Witten http://projecteuclid.org/euclid.cmp/1104178138

From a purely mathematical point of view, the situation is not very satisfying because we don't know how to rigorously define the path integral defining the quantum Chern-Simons theory. Still, the Witten's paper had some enormous influence in knot theory and many things have been done: the Chern-Simons invariants can be rigorously defined combinatorially, it is possible to study the perturbative expansion of Chern-Simons theory...

Furthermore, the relation between knot theory and physics is still currently very active: by Witten, Chern-Simons theory is the same thing as the A-model of open topological string on the cotangent bundle of the 3-sphere, then by applying the large N transition of Gopakumar-Vafa, we obtain a relation with closed string amplitudes on the conifold... Some random names on these questions: Dijkgraaf, Aganagic, Marino, Vafa, Gukov... For another direction, more related to gauge theory, but this time in 4 or 5 dimensions, see the work of Witten on Khovanov homology http://arxiv.org/abs/1108.3103

Comments

This is amazingly nice, and it is so well explained that even I can get something out of it  :-)!. Even if I had the appropriate knowledge, it would probably take me forever to write such a great detailed answer ...
 

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To say "we don't know how to formally define the path integral for Witten Chern-Simons theory in 3d" is not completely accurate, because you can add an infinitesimal ordinary kinetic term, and then it is a topological mass perturbation of a standard gauge theory, which is superrenormalizable in 3d, and can be constructed by Glimm-Jaffe methods. Then the pure Chern-Simons theory can be defined as the limit of infinite topological mass of the ordinary gauge theory. The construction has not been carried out, but the methods presumably are fine, so this is not the same type of problem as defining 4d gauge theory, where you need a somewhat new method or limiting procedure.

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@ Ron Maimon Do you know if the construction of 3d Yang-Mills theory by Glimm-Jaffe methods has been carried out? I don't think it is the case but of course I agree that superrenormalizability makes this question qualitatively much easier than the definition of 4d gauge theory.

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Thank you very much. You gave a nice summary :)

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Some maybe useful additional reference on 3d gauge theories: a video by Witten 

http://media.scgp.stonybrook.edu/video/video.php?f=20120117_3_qtp.mp4

Slides: 

http://media.scgp.stonybrook.edu/presentations/20120117_3_Witten.pdf

See slides 70-95 for something about the approach suggested by Ron Maimon